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Algebra 1
Concepts and Skills
^McDougal Littell
A HOUGHTON MIFFLIN COMPANY
Evanston, Illinois ♦ Boston ♦ Dallas
Copyright © 2004 by McDougal Littell, a division of Houghton Mifflin Company.
Warning. No part of this work may be reproduced or transmitted in any form or by any
means, electronic or mechanical, including photocopying and recording, or by any
information storage or retrieval system, without prior written permission of McDougal
Littell, a division of Houghton Mifflin Company, unless such copying is expressly
permitted by federal copyright law. With the exception of not-for-profit transcription in
Braille, McDougal Littell, a division of Houghton Mifflin Company, is not authorized
to grant permission for further uses of copyrighted selections reprinted in this text
without the permission of their owners. Permission must be obtained from the
individual copyright owners as identified herein. Address inquiries to Manager, Rights
and Permissions, McDougal Littell, a division of Houghton Mifflin Company,
P.O. Box 1667, Evanston, IL 60204.
ISBN: 0-618-37420-5 123456789-DWO-07 06 05 04 03
Internet Web Site: //www.classzone.com
► Ron Larson is a professor of mathematics at Penn State
University at Erie, where he has taught since receiving his
Ph.D. in mathematics from the University of Colorado in 1970.
He is the author of a broad range of instructional materials for
middle school, high school, and college. Dr. Larson has been an
innovative writer of multimedia approaches to mathematics,
and his Calculus and Precalculus texts are both available in
mL interactive form on the Internet.
► Laurie Boswell is a mathematics teacher at Profile Junior-
Senior High School in Bethlehem, New Hampshire. A recipient
of the 1986 Presidential Award for Excellence in Mathematics
Teaching, she is also the 1992 Tandy Technology Scholar and the
1991 recipient of the Richard Balomenos Mathematics Education
Service Award presented by the New Hampshire Association
of Teachers of Mathematics.
► Timothy D. Kanold is Director of Mathematics and
a mathematics teacher at Adlai E. Stevenson High School in
Lincolnshire, Illinois. In 1995 he received the Award of Excellence
from the Illinois State Board of Education for outstanding
contributions to education. A 1986 recipient of the Presidential
Award for Excellence in Mathematics Teaching, he served as
President of the Council of Presidential Awardees of Mathematics.
► Lee Stiff is a professor of mathematics education in the College
of Education and Psychology of North Carolina State University
at Raleigh and has taught mathematics at the high school and
middle school levels. He is the 1992 recipient of the W. W. Rankin
Award for Excellence in Mathematics Education presented by
the North Carolina Council of Teachers of Mathematics, and a
1995-96 Fulbright Scholar to the Department of Mathematics of
the University of Ghana.
All authors contributed to planning the content, organization, and instmctional design
of the program, and to reviewing and writing the manuscript. Ron Larson played
a major role in writing the textbook and in establishing the program philosophy.
Hi
>>y
(8,4)
rise =
(3, 2)
4-
-2
= 2
run =
8 "
-3
= 5
1
3
5
1
9
11 X
► Reviewers
Pauline Embree Diego Gutierrez
Mathematics Department Chair Mathematics Teacher
Rancho San Joaquin Middle School Crawford High School
Irvine, CA San Diego, CA
The reviewers read and commented on textbook chapters in pre-publication format,
particularly with regard to classroom needs.
► Teacher Panel
Courteney Dawe
Mathematics Teacher
Placerita Junior High School
Valencia, CA
Diego Gutierrez
Mathematics Teacher
Crawford High School
San Diego, CA
Dave Dempster
Mathematics Teacher
Temecula Valley High School
Temecula, CA
Roger Hitchcock
Mathematics Teacher
Buchanan High School
Clovis, CA
Pauline Embree
Mathematics Department Chair
Rancho San Joaquin Middle School
Irvine, CA
Louise McComas
Mathematics Teacher
Fremont High School
Sunnyvale, CA
Tom Griffith
Mathematics Teacher
Scripps Ranch High School
San Diego, CA
Viola Okoro
Mathematics Teacher
Laguna Creek High School
Elk Grove, CA
The Teacher Panel helped plan the content, organization, and instructional design
of the program.
California Consulting Mathematicians
Kurt Kreith Don Chakerian
Professor of Mathematics Professor of Mathematics
University of California, Davis University of California, Davis
The California Consulting Mathematicians prepared the Mathematical Background Notes
preceding each chapter in the Teacher’s Edition of this textbook.
iv
Av
STUDENTHELP
Study Tip 2 , 8 , 16 , 26 , 36 ,
37 , 42 , 44 , 49
Skills Review 7 , 16 , 32,43
Reading Algebra 9 , 11 , 30 ,
31,32
Writing Algebra 3
Vocabulary Tip 5 , 24 , 31 ,
43
Keystroke Help 13
Test Tip 14 , 41,60
APPLICATION
HI6HLI6HTg
Scuba Diving 1 , 53
Race Cars 4
Basketball 17
Veterinarians 26
Mach Number 28
Northwest Territory 34
Plant Growth 40
Braking Distance 46
Hot-air Ballooning 49
Chisholm Trail 52
^INTERNET
7, 7, 10 , 19 , 25 , 34 , 38 , 40 , 42 ,
44 , 46 , 50 , 52 , 61
► Chapter Study Guide 2
1.1 Variables in Algebra 3
1.2 Exponents and Powers 9
1.3 Order of Operations 15
QUIZ I, 21
1.4 Equations and Inequalities 24
► DEVELOPING CONCEPTS: Finding Patterns, 22
1.5 Translating Words into Mathematical Symbols 30
QUIZ Z, 35
1.6 A Problem Solving Plan Using Models 36
1.7 Tables and Graphs 42
1.8 An Introduction to Functions 48
QUIZ 3, 54
ASSESSMENT
Chapter Readiness Quiz, 2
Quizzes, 27, 35, 54
Standardized Test Practice, 8, 14,
20, 29, 35, 41, 47, 53
Chapter Summary and Review, 55
Chapter Test, 59
Chapter Standardized Test, 60
Maintaining Skills, 61
Ljaiibaj liamly
Pre-Course Test xviii
A diagnostic test on key skills
from earlier courses, referenced
to the Skills Review (pp. 759-782)
Pre-Course Practice xx
Additional practice on the skills in
the Pre-Course Test, also referenced
to the Skills Review
Contents
STUDENT HELP
Study Tip 66,86,93,94,
101, 102, 108, 109, 113,
114
Skills Review 66, 111
Reading Algebra 65,71,
107
Writing Algebra 94
Vocabulary Tip 100
Look Back 72, 79, 84, 87,
97,99, 119, 120
Keystroke Help 80
Test Tip 126
APPLICATION
mUUGUTS
Helicopters 63, 75
Nome, Alaska 67
Stars 69
Space Shuttle 73
Planets 75
Golf Scores 82
Stock Market 88
Water Cycle 90
Flying Squirrels 95
Rappelling 97
► Chapter Study Guide 64
2.1 The Real Number Line 65
2.2 Absolute Value 71
2.3 Adding Real Numbers 78
► DEVELOPING CONCEPTS: Addition of Integers, 77
QUIZ1, 83
2.4 Subtracting Real Numbers 86
► DEVELOPING CONCEPTS: Subtraction of Integers, 84
2.5 Multiplying Real Numbers 93
► DEVELOPING CONCEPTS: Multiplication of Integers, 92
2.6 The Distributive Property 100
► DEVELOPING CONCEPTS: The Distributive Property, 99
QUIZ Z, 106
2.7 Combining Like Terms 107
2.8 Dividing Real Numbers 113
QUIZ3, 118
Extension: Inductive and Deductive Reasoning 119
^INTERNET
63, 67, 69, 73, 75, 80, 90, 91,
95,97, 101, 104, 108, 115,
117, 127
ASSESSMENT
Chapter Readiness Quiz, 64
Quizzes, 83, 106, 118
Standardized Test Practice, 70, 76, 82,
91, 98, 105, 112, 117
Chapter Summary and Review, 121
Chapter Test, 125
Chapter Standardized Test, 126
Maintaining Skills, 127
Contents
’Solving Linear Equations
STUpENTHELP
Study Tip 132, 133, 138,
146, 151, 157, 158, 178,
184
Skills Review 165
Writing Algebra 177
Vocabulary Tip 144
Look Back 131, 152, 171
TestTip 143,194
APPLICATION
UIGUUGUTS
Bald Eagles 129, 181
City Parks 136
Newspaper Recycling 142
Thunderstorms 142
Earth's Temperature 145
Steamboats 155
Cheetahs 155
Moons of Jupiter 162
Cocoa Consumption 167
Bottle-nosed Whales 175
^INTERNET
129, 133, 136, 139, 142, 145,
148, 153, 155, 159, 161, 162,
164, 167, 170, 172, 173, 179,
181, 182, 185, 187, 195
► Chapter Study Guide 130
3.1 Solving Equations Using Addition and Subtraction 132
► DEVELOPING CONCEPTS: One-Step Equations, 131
3.2 Solving Equations Using Multiplication and Division 138
3.3 Solving Multi-Step Equations 144
QUIZ1, 149
3.4 Solving Equations with Variables on Both Sides 151
► DEVELOPING CONCEPTS: Variables on Both Sides, 150
3.5 More on Linear Equations 157
3.6 Solving Decimal Equations 163
QUIZ 2, 169
► GRAPHING CALCULATOR: Solving Multi-Step Equations, 170
3.7 Formulas 171
3.8 Ratios and Rates 1 77
3.9 Percents 183
QUIZ3, 188
Project: Planning a Car Wash 198
ASSESSMENT
Chapter Readiness Quiz, 130
Quizzes, 149, 169, 188
Standardized Test Practice, 137,143,
148,156, 162,168, 176, 182, 188
Chapter Summary and Review, 189
Chapter Test, 193
Chapter Standardized Test, 194
Maintaining Skills, 195
Cumulative Practice, Chapters 1-3 ,196
Contents
STUpENTHELP
Study Tip 203,211212,
216,223, 231,238, 250,
251,253, 254
Reading Algebra 230,236,
254
Look Back 218,252
Keystroke Help 250
Test Tip 264
APPLICATION
HIGHLIGHTS
Cable Cars 201, 247
Wing Length 207
Triathlon 214
Boiling Point 214
Mount St. Helens 218
Zoo Fundraising 226
U.S.S. Constitution 234
Gold Bullion 237
Violin Family 240
Monarch Butterflies 257
^INTERNET
201,204, 207,214,215,217,
224,226, 227,232, 237,245,
248,250, 256,265
► Chapter Study Guide 202
4.1 The Coordinate Plane 203
4.2 Graphing Linear Equations 210
► DEVELOPING CONCEPTS: Linear Equations, 209
4.3 Graphing Horizontal and Vertical Lines 216
QUIZ1, 227
4.4 Graphing Lines Using Intercepts 222
4.5 The Slope of a Line 229
► DEVELOPING CONCEPTS: Investigating Slope, 228
4.6 Direct Variation 236
QUIZ*, 241
4.1 Graphing Lines Using Slope-Intercept Form 243
► DEVELOPING CONCEPTS: Slope-Intercept Form, 242
► GRAPHING CALCULATOR: Graphing a Linear Equation, 250
4.8 Functions and Relations 252
QUIZ3, 258
ASSESSMENT
Chapter Readiness Quiz, 202
Quizzes, 221, 241, 258
Standardized Test Practice, 208, 215, 221, 227, 235, 240, 249, 257
Chapter Summary and Review, 259
Chapter Test, 263
Chapter Standardized Test, 264
Maintaining Skills, 265
VIII
Contents
► Chapter Study Guide
268
STUDENT HELP
5.1 Slope-Intercept Form
269
Study Tip 270,271,273,
5.2 Point-Slope Form
278,279, 285,292, 293,
295, 298, 299,307, 308
Vocabulary Tip 274,291
Look Back 300
Test Tip 318
278
► DEVELOPING CONCEPTS: Point-Slope Form, 276
QUIZ 1, 284
5.3 Writing Linear Equations Given Two Points
285
APPLICATION
5.4 Standard Form
291
HIGHLIGHTS
QUIZ2, 297
Archaeology 267,303
Hurdling 273
Old Faithful 274
5.5 Modeling with Linear Equations
298
Water Pressure 282
5.6 Perpendicular Lines
306
Airplane Descent 289
The Chunnel 289
► DEVELOPING CONCEPTS: Perpendicular Lines, 305
Speed of Sound 290
Movie Theaters 298,299
Car Costs 302
QUIZ 3. 312
City Street Plan 311
ASSESSMENT
INTERNET
Chapter Readiness Quiz, 268
Quizzes, 284, 297, 312
Standardized Test Practice, 275, 283, 290, 296, 304, 311
267,271, 273,274,280,282,
287,289, 295,298,303,306,
310,319
Chapter Summary and Review, 313
Chapter Test, 317
Chapter Standardized Test, 318
Maintaining Skills, 319
Contents
STUDENT HELP
Study Tip 324,330,337,
338, 342, 343,344, 348,
349, 350, 356,362,363,
368,369
Skills Review 340
Reading Algebra 323
Writing Algebra 325,331
Vocabulary Tip 367
Keystroke Help 374
Test Tip 380
APPLICATION
HI6HLI6HTS
Music 321
Astronomy 325,359
Mercury 327
Fly-fishing 338
Mountain Plants 343
Steel Arch Bridge 346
Water Temperature 352
Poodles 357
Fireworks 365
Nutrition 371
INTERNET
321,327, 332,337,340,346,
347,349, 359,365,369,371,
372,374, 381
► Chapter Study Guide 322
6.1 Solving Inequalities Using Addition or Subtraction 323
6.2 Solving Inequalities Using Multiplication or Division 330
► DEVELOPING CONCEPTS: Investigating Inequalities, 329
6.3 Solving Multi-Step Inequalities 336
QUIZ1, 341
6.4 Solving Compound Inequalities Involving “And" 342
6.5 Solving Compound Inequalities Involving “Or" 348
6.6 Solving Absolute-Value Equations 355
► DEVELOPING CONCEPTS: Absolute-Value Equations, 354
QUIZ2, 360
6.1 Solving Absolute-Value Inequalities 361
6.8 Graphing Linear Inequalities in Two Variables 367
QUIZ3, 373
► GRAPHING CALCULATOR: Graphing Inequalities, 374
Project: Investigating Springs 384
Assessment
Chapter Readiness Quiz, 322
Quizzes, 341, 360, 373
Standardized Test Practice, 328, 335,
341, 347, 353, 360, 366, 372
Chapter Summary and Review, 375
Chapter Test, 379
Chapter Standardized Test, 380
Maintaining Skills, 381
Cumulative Practice, Chapters 1-6 ,382
Contents
Systems of Line;
and Inequal ities
STUpENTHELP
Study Tip 397,409,424
Reading Algebra 391
Look Back 390,415,417,
426
Keystroke Help 395
Test Tip 436
► Chapter Study Guide 3ss
7.1 Graphing Linear Systems 389
► GRAPHING CALCULATOR: Graphing Linear Systems, 395
7.2 Solving Linear Systems by Substitution 396
7.3 Solving Linear Systems by Linear Combinations 402
QUIZ1, 408
APPLICATION
HIGHLIGHTS
Housing 387, 413
Web Sites 391
Softball 400
Volume and Mass 406
Beehive 407
Chemistry 410
Salary Plan 411
Gardening 413
Jewelry 421
Carpentry 421
7.4 Linear Systems and Problem Solving 409
7.5 Special Types of Linear Systems 417
► DEVELOPING CONCEPTS: Special Types of Systems, 415
7.6 Systems of Linear Inequalities 424
► DEVELOPING CONCEPTS: Systems of Inequalities, 423
QUIZ 2, 430
ASSESSMENT
^INTERNET
387,389, 391,395,398, 400,
404,411,413,421,428,429,
437
Chapter Readiness Quiz, 388
Quizzes, 408, 430
Standardized Test Practice, 394, 401, 407, 414, 422, 429
Chapter Summary and Review, 431
Chapter Test, 435
Chapter Standardized Test, 436
Maintaining Skills, 437
Contents
I
r
STUpENTHELP
Study Tip 444,445,456,
462, 463, 469, 476, 477,
478
Reading Algebra 441
Writing Algebra 449, 477
Look Back 443, 444,471
Keystroke Help 451,461,
471
Test Tip 494
APPLICATION
HI6HLI6HTg
Bicycle Racing 439,480
Irrigation Circles 445
Alternative Energy 447
U.S. History 453,471,473
Shipwrecks 457
World Wide Web 459
Baseball Salaries 467
Compound Interest 477
Car Depreciation 482
Pharmacists 486
INTERNET
439, 447, 450, 457, 461, 464,
466, 470, 473, 480, 486, 487,
495
► Chapter Study Guide 440
8.1 Multiplication Properties of Exponents 443
► DEVELOPING CONCEPTS: Investigating Powers, 441
8.2 Zero and Negative Exponents 449
8.3 Graphs of Exponential Functions 455
QUIZ1, 460
► GRAPHING CALCULATOR: Exponential Functions , 461
8.4 Division Properties of Exponents 462
8.5 Scientific Notation 469
QUIZ 2, 474
8.6 Exponential Growth Functions 476
► DEVELOPING CONCEPTS: Exponential Functions, 475
8.7 Exponential Decay Functions 482
QUIZ3, 488
ASSESSMENT
Chapter Readiness Quiz, 440
Quizzes, 460, 474, 488
Standardized Test Practice, 448, 454, 459, 468, 474, 481, 487
Chapter Summary and Review, 489
Chapter Test, 493
Chapter Standardized Test, 494
Maintaining Skills, 495
Contents
► Chapter Study Guide 498
STUDENT HELP
9.1
Square Roots
499
Study Tip 500,505,507,
511,512,521,522,528,
534, 535, 537, 540, 548
9.2
Solving Quadratic Equations by Finding Square Roots
505
Skills Review 516
Reading Algebra 499, 500,
533
9.3
Simplifying Radicals
QUIZ 1, 517
511
Look Back 542, 546
Keystroke Help 501
Test Tip 504,558
9.4
Graphing Quadratic Functions
► DEVELOPING CONCEPTS: Graphing Quadratic Functions, 518
520
APPLICATION
HIGHLIGHTS
9.5 Solving Quadratic Equations by Graphing 526
► GRAPHING CALCULATOR: Approximating Solutions, 532
Baseball 497,538
Chess 503
Minerals 509
Sailing 513
Tsunamis 515
Dolphins 524
Golden Gate Bridge 528
Microgravity 530
Red-tailed Hawk 537
Financial Analysis 544
9.6 Solving Quadratic Equations by the Quadratic Formula 533
QUIZ*, 539
9.7 Using the Discriminant 540
9.8 Graphing Quadratic Inequalities 547
► DEVELOPING CONCEPTS: Graphing Quadratic Inequalities, 546
QUIZ3, 552
INTERNET
497, 503, 506, 509, 512, 513,
524, 525, 527, 532, 534, 537,
538, 542, 544, 549, 559
Project: Designing a Stairway
562
Assessment
Chapter Readiness Quiz, 498
Quizzes, 517, 539, 552
Standardized Test Practice, 504, 510,
516, 525, 531, 538, 545, 551
Chapter Summary and Review, 553
Chapter Test, 557
Chapter Standardized Test, 558
Maintaining Skills, 559
Cumulative Practice, Chapters 1-9, 560
Contents
XIII
STUpENTHELP
Study Tip 570,582,588,
596, 597, 604, 610
Skills Review 590, 616
Reading Algebra 568
Look Back 567, 569, 575,
583, 605
Test Tip 628
► Chapter Study Guide see
10.1 Adding and Subtracting Polynomials 568
► DEVELOPING CONCEPTS: Addition of Polynomials, 567
10.2 Multiplying Polynomials 575
► DEVELOPING CONCEPTS: Multiplying Polynomials, 574
10.3 Special Products of Polynomials 581
QUIZ1, 587
APPLICATION
HI6HLI6HTS
Radio Telescope 565, 592
Picture Framing 579
Genetics 584,586
Meteorite Crater 592
Landscape Design 598
Taj Mahal 600
Cliff Diver 605
Block and Tackle 612
Pole-vaulting 614
Terrarium 619
^INTERNET
565,569,576, 579, 583,586,
589, 592,598,600, 605,607,
612,618,629
10.4 Solving Quadratic Equations in Factored Form 588
10.5 Factoring x 2 + bx + c 595
► DEVELOPING CONCEPTS: Factoring x 2 + bx + c, 594
10.6 Factoring ax 2 + bx + c 603
► DEVELOPING CONCEPTS: Factoring ax 2 + bx + c, 602
QUIZ2, 608
10.7 Factoring Special Products 609
10.8 Factoring Cubic Polynomials 616
QUIZ3, 622
ASSESSMENT
Chapter Readiness Quiz, 566
Quizzes, 587, 608, 622
Standardized Test Practice, 573, 580, 587, 593, 601, 608, 615, 622
Chapter Summary and Review, 623
Chapter Test, 627
Chapter Standardized Test, 628
Maintaining Skills, 629
Contents
► Chapter Study Guide
632
STUDENT HELP
ii.i
Study Tip 634,639,640,
11.2
647, 648, 652, 653, 659,
664, 666, 670, 671, 672,
673, 679
Skills Review 663
Reading Algebra 633, 678
Vocabulary Tip 633,640
Keystroke Help 645
Test Tip 686
11.3
11.4
APPLICATION
11.5
HIGHLIGHTS
11.6
Scale Models 631, 637
Clay Warriors 635
11.7
Fence Mural 637
Bicycling 641
Snowshoes 643
Ocean Temperatures 643
Air Pressure 650
Car Trip 666
Batting Average 675
► GRAPHING CALCULATOR: Modeling Inverse Variation, 645
Simplifying Rational Expressions
QUIZ1, 651
QUIZ Z, 677
Extension: Rational Functions
ASSESSMENT
INTERNET
631 535, 641, 643, 645, 647,
654, 656, 659, 661, 665, 672,
675, 687
Chapter Readiness Quiz, 632
Quizzes, 651 , 677
Standardized Test Practice, 638 , 644 , 650 , 657 , 662 , 669 , 676
Chapter Summary and Review, 681
Chapter Test, 685
Chapter Standardized Test, 686
Maintaining Skills, 687
633
639
646
652
658
663
670
678
Contents
► Chapter Study Guide 690
STUDENT HELP
12.1 Functions Involving Square Roots
692
Study Tip 692,693,699,
717, 724, 726, 736, 741
► DEVELOPING CONCEPTS: Functions with Radicals, 691
Reading Algebra 710
Vocabulary Tip 722,731
12.2 Operations with Radical Expressions
698
Look Back 698, 715, 726,
742
1 2.3 Solving Radical Equations
704
Test Tip 752
QUIZI, 709
APPLICATION
12.4 Rational Exponents
710
HIGHLIGHTS
12.5 Completing the Square
716
Centripetal Force 689, 706
Dinosaurs 696
► DEVELOPING CONCEPTS: Completing the Square, 715
Sailing 700
12.6 The Pythagorean Theorem and Its Converse
724
De-icing Planes 708
Penguins 718
► DEVELOPING CONCEPTS: The Pythagorean Theorem , 722
Diving 720
Staircase Design 728
QUIZ 2, 729
Soccer 732
Maps 734
12.7 The Distance Formula
730
Pony Express 739
12.8 The Midpoint Formula
736
^INTERNET
12.9 Logical Reasoning: Proof
740
689, 693, 696, 702, 705, 708,
712, 720, 725, 728, 732, 734,
QUIZ3, 746
737, 739, 742, 743, 744
Project: Investigating the Golden Ratio
756
ASSESSMENT
Chapter Readiness Quiz, 690 Chapter Test, 751
Quizzes, 709, 729, 746 Chapter Standardized Test, 752
Standardized Test Practice, 697, 703, 709, Cumulative Practice, Chapters 1-12, 754
714, 721, 729, 735, 739, 745
Chapter Summary and Review, 747
Contents
Skills Review Handbook
pages 759-782
II 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 1|
• Decimals
759-760
• Factors and Multiples
761-762
• Fractions
763-766
• Writing Fractions and Decimals
767-769
• Comparing and Ordering Numbers
770-771
• Perimeter, Area, and Volume
772-773
• Estimation
774-776
• Data Displays
777-779
• Measures of Central Tendency
780
• Problem Solving
781-782
Extra Practice for Chapters 1-12
pages 783-794
End-of-Course Test
pages 795-796
Tables
pages 797-802
• Symbols 797 • Squares and Square Roots 800-801
• Formulas 798 • Measures
802
• Properties 799
Appendices
pages 803-816
Glossary
pages 817-823
English-to-£panish Glossary
pages 824-831
Index
pages 832-844
..
Selected Answers
Contents
Decimals
Skills Review
pp. 759-760
Find the sum, difference, product, or quotient.
1.3.4 + 6.005 2.27.77 - 18.09
3. 23.7 X 13.67
4. 9.744 - 0.87
Factors and Multiples
Skills Review
pp. 761-762
Find the greatest common factor of the pair of numbers.
5.8,28 6.36,42 7.54,81
Find the least common multiple of the pair of numbers.
9.6,7 10.10,15 11.24,38
Find the least common denominator of the pair of fractions.
13.
I 1L
2 ’ 10
14.
5 6
8’ 7
15.
5 1_
9’ 12
8 . 50, 150
12 . 12, 36
16.
n is
20’ 32
Fractions
Find the reciprocal of the number.
17.12
18.
19.
16 — 5
Add, subtract, multiply, or divide. Write the answer in simplest form.
21 .
1
__ 1 . 1
22 ' 2 + 8
6 . 5
23 ‘7 + 9
Skills Review
pp. 763-766
1
20 . 2
04 11J— 9—
4 8
25 -i x 7i
7 . 3
26 ' 11 ' 5
27 — 4- —
15 3
28 - 4 i x f
Fractions, Decimals, and Percents
Write the percent as a decimal and as a fraction in simplest form.
29.7% 30.26% 31.48%
Skills Review
pp. 768-769
32. 84%
Write the decimal as a percent and as a fraction in simplest form.
33.0.08 34.0.15 35.0.47
36. 0.027
Write the fraction as a decimal and as a percent.
37.
10
38.
39.
40.
11
20
xviii Pre-Course Test
COMPARING AND ORDERING NUMBERS
Skills Review
pp. 770-771
Compare the two numbers. Write the answer using <, >, or =.
41. 138 and 198
42. 781 and 718
43. 8.4 and 8.2
44. —7.88 and —4.88
45.^- and |
3 4
46. | and £
O 8
47. | and l|
48. I 67 and I 67 -
J O
Write the numbers in order from least to greatest.
49. 47, 74, 44, 77
50. 80, 808, 88 , 8
51. 0.19, 0.9, 0.49, 0.4
52. -6.5, -5.4, 6.4, -6
53 1 4 3 I
8 ’ 7’ 5’ 2
54 9 6 5 A
7’ 4’ 4’ 13
5 3 13 7
55 i_ ii _ _
9’ 4’ IT 5
56. -16^, -15|, -16|,
Perimeter, Area, and volume
Skills Review
pp. 772-773
Find the perimeter.
57. a triangle with sides of length 18 feet, 27 feet, and 32 feet
58. a square with sides of length 4.7 centimeters
Find the area.
59. a square with sides of length 13 yards
60. a rectangle with length 7.7 kilometers and width 4.5 kilometers
Find the volume.
61 . a cube with sides of length 19 meters
62. a rectangular prism with length 5.9 inches, width 8.6 inches, and height 1.2 inches
Data Di splays
Skills Review
pp. 777-779
63. The list below shows the distribution of gold medals for the 1998 Winter Olympics.
Choose an appropriate graph to display the data. ►Source: International Olympic Committee
Germany 12
United States 6
South Korea 3
Switzerland 2
Norway 10
Japan 5
Finland 2
Bulgaria 1
Russia 9
Netherlands 5
France 2
Czech Republic 1
Canada 6
Austria 3
Italy 2
Measures of Central Tendency
Skills Review
p. 780
Find the mean, median, and mode(s) of the data set.
64.1,3, 3, 3, 4,5,6, 7, 7,9
66.5, 23, 12,5,9, 18, 12, 4, 10,21
65. 17, 22, 36, 47, 51, 58, 65, 80, 85, 89
67. 101, 423, 564, 198, 387, 291, 402, 572, 222, 357
Pre-Course Test
xix
Decimals
Skills Review
pp . 759-760
Find the sum or difference.
1 . 14 + 7.1
2 . 11 - 0.003
3. 19.76 + 48.19
4. 73.8 - 6.93
5. 10.2 + 3.805 + 1.1
6 . 7.2 - 3.56
Find the product or quotient.
7. 17 X 3.9
8 . 6.08 X 3.15
9. 15.2 X 5.02
10.0.019 X 0.27
11. 45.28 X 16.1
12 . 26.01 4- 5.1
13.7.03 - 1.9
14. 21.84 4- 0.84
15. 0.0196 4- 0.056
Factors and Multiples
Skills Review
pp . 761-762
List all the factors of the number.
1.12 2.41 3.54 4.126
Write the prime factorization of the number if it is not a prime number.
If a number is prime, write prime.
5. 54
6 - 60
7. 35
8 . 47
List all the common factors of the pair of numbers.
9. 16, 20
10. 24, 36
11 . 28,42
12 . 60, 72
Find the greatest common factor of the pair of numbers.
13. 8 , 12
14. 10, 25
15. 15,24
16. 24, 30
17. 36, 42
18. 54, 81
19. 68 , 82
20 . 102, 214
Find the least common multiple of the pair of numbers.
21.9, 12
22 . 8 , 5
23. 14, 21
24. 24, 8
25. 12, 16
26. 70, 14
27. 36, 50
28. 22, 30
Find the least common denominator of the pair of fractions.
5
29 ' 8’6
„ 5 7
30 ' 12’ 8
7 9
31 — —
12 ’ 20
„ 5 8
32 ‘ 6 ’ 15
__ 3 15
33 ‘ 4’ 28
34 ———
11’ 13
35 ^20
6 ’ 27
,.1721
40’ 52
XX
Pre-Course Practice
Fractions
Skills Review
pp. 763-766
Find the reciprocal of the number.
1 . 8
2 -J~6
3 -f
4. 3y
Add or subtract. Write the answer as a fraction or a mixed number in
simplest form.
5
12 12
6 1+3
6 ' 8 + 8
9 3
7 — + —
10 10
8 A
15 15
9 ± + ^
3 9
17 3
10 ' 20 5
ii — + —
6 8
12 1— — —
1 3 9
Multiply or divide. Write the answer as a fraction or a mixed number in
simplest form.
3 1
13 -5 X 2
2 3
14 — X —
3 8
3 1
15 -5 X1 2
16. 2f x jf
17 — h- —
5 5
18 — — —
lO- 3 • g
c 1.7
19 ‘ 5 4 ' 8
4 1
20 . 4^ - l|
Add, subtract, multiply.
or divide. Write the answer as a fraction or a
mixed number in simplest form.
21-1 +f
3 1
22 9— — 5—
8 4
2 3
23. 8f + 5~
J o
24 -j x I '
25. if if
26. 5§ X 3|
27 — h- —
‘ £/ - 4 5
if - f
Fractions, Decimals, and Percents
Skills Review
pp. 768-769
Write the percent as a decimal and as a fraction or a mixed number in
simplest form.
1 . 8 %
2 . 25%
3. 38%
4. 73%
5. 135%
6 . 350%
7. 6.4%
8 . 0.15%
Write the decimal as a percent and as a fraction or a mixed number in
simplest form.
9. 0.44
10. 0.09
11 . 0.13
12 . 0.008
13. 1.6
14. 3.04
15. 6.6
16. 4.75
Write the fraction or mixed number as a decimal and as a percent. Round
decimals to the nearest thousandth. Round percents to the nearest tenth
of a percent.
17 —
5
18 f
^ 17
19 ‘ 25
2°-7l
2i -4
22 . 2 f
23 - 3 -k
24. 8 y
Pre-Course Practice
>oci
continued from page xxi
COMPARING AND ORDERING NUMBERS
Skills Review
pp. 770-771
Compare the two numbers. Write the answer using <, >, or =.
1. 13,458 and 14,455
2 . 907 and 971
3. -8344 and -8434
4. —49.5 and —49.05
5. 0.58 and 0.578
6 . 0.0394 and 0.394
, 15 ,9
7 'T6 and To
_ 13 ,1
8 ' 20 and 4
9.|f and f
10 . 7 ' and
11 2
11 . -2^ and -3 jr
16 9
12 . 18|and 18|
Write the numbers in order from least to greatest.
13. 1507, 1705, 1775, 1075 14. 38,381, 30,831, 38,831, 30,138
15. -0.019, -0.013, -0.205, -0.035 16. 6.034, 6.30, 6.33, 6.34
17.
1 2 J_ 5
TV IV 8
„ 3 5 7 ,4
8’4’ 9’ 7
18.
4 3 3 4
5’ 4’ V 9
19.
4 _2 _4 _3
2’ 3’ 3’ 2
2i i— — — i —
^ '■ 1 5’ 5’ 3’ a 5
22 . 15|, 14|, 14y, 15§
Perimeter, Area, and volume
Skills Review
pp. 772-773
4. a rectangle with length of 12.5 centimeters and width of 11.6 centimeters
5. a regular pentagon with sides of length 19 feet
Find the area.
6 . a square with sides of length 1.67 yards
7. a rectangle with length 1.4 inches and width 2.8 inches
8 . a triangle with base 15 centimeters and height 10 centimeters
Find the volume.
9. a cube with sides of length 34 feet
10 . a rectangular prism with length 18 meters, width 6 meters, and height 3 meters
11 . a rectangular prism with length 6.5 millimeters, width 5.5 millimeters, and
height 2.2 millimeters
>ocii Pre-Course Practice
Data Displays
Skills Review
pp. 777-779
In Exercises 1 and 2, use the table shown below. Hurricane categories are
determined by wind speed, with Category 5 the most severe.
U.S. Mainland Hurricane Strikes by Category from 1900-1996
Category
One
Two
Three
Four
Five
Number
57
37
47
15
2
1 . The data range from 2 to 57. The scale must start at 0. Choose a reasonable
scale for a bar graph.
2 . Draw a bar graph to display the number of hurricane strikes by category.
In Exercises 3 and 4, use the table shown below.
U.S. Mainland Hurricane Strikes by Decade from 1900-1989
Decade
1900-
1909
mo-
1919
1920-
1929
1930-
1939
1940-
1949
1950-
1959
1960-
1969
1970-
1979
1980-
1989
Number
16
19
15
17
23
18
15
12
16
► Source: National Hurricane Center
3. The data range from 12 to 23. The scale must start at 0. Choose a reasonable
scale for a histogram.
4. Draw a histogram to display the number of hurricane strikes by decade.
Choose an appropriate graph to display the data. Draw the graph.
Reported House Plant Sales for One Week
Type
Violets
Begonias
Coleus
Orchids
Cacti
Number
90
46
39
70
60
Republicans in the Senate by Congress Number
Congress
100th
101st
102nd
103rd
104th
105th
106th
Republicans
45
45
44
43
52
55
55
► Source: Statistical Abstract of the United States: 1999
Measures of Central Tendency
Skills Review
p. 780
Find the mean, median, and mode(s) of the data set.
1- 1, 3, 7, 2, 6, 3, 7, 9, 4, 7 2. 16, 19, 15, 17, 23, 18, 15, 12, 16, 7
3. 10, 48, 86, 32, 58, 73, 89, 39, 59, 27 4. 53, 54, 53, 45, 45, 44, 43, 52, 55, 55
Pre-Course Practice xxiii
uftttHA RMm
r ~
A Guide to
y
4
J
Jaui UaJp
►Each chapter begins with a Study Guide
Chapter Preview
gives an overview of
what you will be
learning.
Key Words
lists important new
words in the chapter.
Readiness Quiz
checks your under¬
standing of words and
skills that you will use
in the chapter, and
tells you where to
go for review.
Study Tip
suggests ways to
make your studying
and learning easier.
,
3 j Study Guide
What’s the chapter a bout?
+ Solving a 11 nea r equation systematically
+ Using ratios, rates, and percent
Key Wows
■ eqidvaloi t equal ons H
p 112
■ Inverse operations,
p 111
■ lines' equation, p. ill
■ propties of equally;
p. MO
■ Idend ly, p 151
■ rouidng aTor, p 1 H
■ formda, p 171
■ ratio, p. 177
rat^ p. 177
unltrat^ p. 177
uni t andysls, p. 175
potent, p. 15 1
base number, p. 151
Chapter Readiness Quij
Take this quick quiz. If you are unsure of an answer, lookback rtthe
reference pages lor help.
Vocabulary Check (rtfsr topp.
1. What is the. opposite of ~"i V
® -i <E> | ® 3
3
Z Which number is the reciprocal of -V
<E> -f <E> <E) 1 CE> f
Skill Check (xtfkr to pp 24,102,1C8)
GL Which of the following is a solution of the equation -11 = -4y +1?
® -i ® 2 ® 3 CEE 1 4
4. Which expression is equivalent to - 3 - A)?
"ix - 12 ~2x +12 -ix - A (J5} ~"ix -12
3. Simplify the expression Ax^ — 5x — ?? + 2x.
- 2x £!£> 2x? + %x "i.? - 2\ (J5} .?
^EESEM Make Ffri-wula Cards
Writs a formula and a sample
problem on each card, hfake
sure you know what each
algebraic sym bol represents
in a formula.
Chapter 3 Sbfwg Icaeor fqodftnf
xxiv
►Student Help notes
throughout the book
Study TIPS help you under¬
stand and apply concepts and
avoid common errors.
More Examples indicates
that there are more worked-out
examples on the Internet.
Reading Algebra
Student MeCp
guides you in reading
and understanding your
textbook.
► Reading Algebra
Order is important for
subtraction. "4 less
than a number" means
y — 4, not 4 — y.
Skills Review
refers you to the pages
where you can go for
review and practice of
topics from earlier
courses.
Student HeCp
►Skills Review
For help with writing
fractions in lowest
terms, see p. 763.
\ _/
SIMPLIFIED EXPRESSIONS Tht distnbutiv& pro peity allows you to comb irae
iifce teynis by adding their coefficients. An expression is slm pllded if it has no
grouping symbols and if all the hie terms have been combined.
SthfbHl HaCp
►Srjoy Tp
In Example £ the
distributive property
has been extended to
three terms:
(t>+ t+ d)5 =
<?? + da
EXAMPLE
Combih* L9ci Ttrms
Simplify the expression
a. 3* + h* b.2f + lf-f+2
Solution
a. 3 i+3ie= (H3)e Ufidistjitwtife property.
= 1 l.v Add corffkfeits.
b. 2f + if — f + 2 = 2y* + Ty 1 — ly* + 2
= (2+7 - 1 "lf+2
= $f+2
Coeftcieat of -f is -1.
Us distriwtl'e property.
Add corifcfents.
StMbHUAtGp
► i EXAf.FLES
More examples
are ayjjishle at
vwwv.mcdougallittell.com
Simplify Express w ns with Grouping Symbols
Simplify the expression
a. 8 - 2(v + 4) lx 2(s + h) + h(5 - *)
Solution
a. 8 - 2(v + 4)
= 8 - 2(v) + (-2)(4)
= 8 - 2 * - 8
= -2v + 3 - 3
= -2v
lx 2(x + 2) + 3(5 - x)
= 2(x) + 2 (i) + 2$) + 2(-x)
= 2x + 6 + \5 - 2x
= 2i-ar+6+l 5
= -x+21
U£ distrlbvtt'e property.
MlMf-
GDipIktfllU
ComhiK IfctfTIE.
M<M*.
GDipIktfmt
■iii 11 , a
Simpiffy Expreifioat
Simplify the egression.
3 . 5x - 2x 4. 8th. - m - ira + .5
6. 3(y + 2) - 4y 7. - 4(2* - 1)
5L - x 2 +5x+x 2
a -(z + 2 } - 2(1 - z)
io&
Chapters PnpertiEf o/lHilMN^en
Homework Help tells you which
textbook examples may help you
with homework exercises, and lets
you know when there is extra help
on the Internet.
Student HeCp
1 " n
► Homework Help
Extra help with
problem solving in
Exs. 34-39 is available at
www.mcdougallittell.com
Other notes included are:
• Writing Algebra • Test Tip
• Keystroke Help • Look Back
Vocabulary Tips
explain the meaning and
origin of words.
Student McCp
►Vocabulary Tip
Equation comes from
a Latin word that
means "to be equal".
XXV
Connections to
How much does it cost to rent
scuba diving equipment?
APPLICATION: Scuba Diving
Number of days Rental charge
Scuba divers must be certified divers and must use
scuba equipment in order to breathe underwater.
Equipment such as wet suits, tanks, buoyancy
compensator devices, and regulators can be rented at
a cost per day at many sporting good stores.
Think & Discuss
The table shows the cost per day for renting a regulator.
1. You decide to rent a regulator for 4 days. What is
the rental charge?
2. Use the pattern in the table to predict the rental
cost if you rent a regulator for 10 days.
Learn More About It
You will learn more about the price of renting scuba
equipment in Exercise 24 on page 53.
APPLICATION LINK More about the prices of renting scuba
diving equipment is available at www.mcdougallittell.com
1
$ 12.00 X 1
2
$ 12.00 X 2
3
$ 12.00 X 3
4
$ 12.00 X 4
K ] Study Guide
Ril l I f 11
PREVIEW
PREPARE
What’s the chapter about ?
• Writing and evaluating variable expressions
• Checking solutions to equations and inequalities
• Using verbal and algebraic models
• Organizing data and representing functions
Key Words
-^
• variable, p. 3
• base, p. 9
• inequality, p. 26
• variable expression, p. 3
• order of
• modeling, p. 36
• numerical expression, p. 3
operations, p. 15
• function, p. 48
• power, p. 9
• equation, p. 24
• domain, p. 49
• exponent, p. 9
i_
• solution, p. 24
• range, p. 49
_ j
Chapter Readiness Quiz
Student Hedfp
" > i
► Study Tip
"Student Help" boxes
throughout the chapter
give you study tips and
tell you where to look
for extra help in this
book and on the
Internet.
I J
Take this quick quiz. If you are unsure of an answer, look at the
reference pages for help.
Vocabulary Check ( refer to pp. 3, 9)
1. Which of the following is not a variable expression?
(A) 9 — 4 y (ID 10 — 4(2) CD 2x + 3 CD 2m + 3 n
2. Which term describes the expression 7 3 ?
(A) power CD exponent CD base Co) variable
Skill Check (refer to pp. 772, 770)
3. Find the perimeter of the figure.
(A) 30 feet CD 60 feet
CD 120 feet CD 200 feet
4. Complete the statement 0.5 > ? .
20 ft
10 ft
10 ft
20 ft
(5)^ CD^ C©| d) |
STUDY TIP
Keep a Math
Notebook
Keeping a notebook will
help you remember new
concepts and skills.
a *-■
Keeping a Math Notebook
• keep a notebook of math notes about each
chapter separate from your homework exercises
• Review your notes each day before you start your
next homework assignment
Variables in Algebra
Goal
Evaluate variable
expressions.
Key Words
• variable
• value
• variable expression
• numerical expression
• evaluate
How many miles has a race car traveled?
A race car zooms around the
Indianapolis Motor Speedway
at 180 miles per hour. In
Example 3 you will find how
many miles the car will travel
in 2 hours.
ASSIGNING VARIABLES In algebra, you can use letters to represent one or
more numbers. When a letter is used to represent a range of numbers, it is called
a variable. The numbers are called values of the variable. For example, the
distance traveled by the race car in the picture above can be expressed as the
variable expression 180t, where t represents the number of hours the car has
traveled.
A variable expression consists of constants, variables, and operations. An
expression that represents a particular number is called a numerical expression.
For example, the distance traveled by the race car in two hours is given by the
numerical expression 180 X 2.
Student HeCp
^
►Writing Algebra
The multiplication
symbol x is usually
not used in algebra
because of its possible
confusion with the
variable x.
^ _ J
i Describe the Variable Expression
Here are some variable expressions, their meanings, and their operations.
VARIABLE EXPRESSION MEANING OPERATION
8 y, 8 • y, (8)(y) 8 times y Multiplication
16 + b 16 divided by b Division
4 + s 4 plus s Addition
9 — x 9 minus x Subtraction
State the meaning of the variable expression and name the operation.
1.10 + x 2. 13-x 3. tt" 4. 24x
1.1 Variables in Algebra
EVALUATING EXPRESSIONS To evaluate a variable expression, you write the
expression, substitute a number for each variable, and simplify. The resulting
number is the value of the expression.
Write expression.
Substitute numbers.
Simplify.
2 Evaluate the Variable Expression
Evaluate the variable expression when y — 2.
Solution
EXPRESSION
SUBSTITUTE
SIMPLIFY
a. 5y
= 5(2)
= 10
b.^
_ io
2
= 5
c. y + 6
= 2 + 6
= 8
d. 14 -y
= 14-2
= 12
Evaluate the Variable Expression
RACE CARS The fastest
average speed in the
Indianapolis 500 is 185.981
miles per hour set by Arie
Luyendyk, shown above.
Evaluate the variable expression when x = 3.
5. lx 6. 5 + x 7. — 8. x - 2
The variable expression r times t can be written as rt , r • t, or (r)(7).
3 Evaluate rt to Find Distance
RACE CARS Find the distance d traveled in 2 hours by a race car going an
average speed of 180 miles per hour. Use the formula: distance equals
rate r multiplied by time t.
Solution
d = rt Write formula.
= 180(2) Substitute 180 for rand 2 for t.
= 360 Simplify.
ANSWER ^ The distance traveled by the race car was 360 miles.
Evaluate rt to Find Distance
9. Using a variable expression, find the distance traveled by a car moving at an
average speed of 60 miles per hour for 3 hours.
Chapter 1 Connections to Algebra
Student HeCp
►Vocabulary Tip
The word perimeter
comes from peri-
meaning around and
-meter meaning
measure. Perimeter
is a measure of the
distance around a
geometric figure.
v j
4 Find the Perimeter
GEOMETRY LINK The perimeter P of
a triangle is equal to the sum of the
lengths of its sides:
P — a + b + c
Find the perimeter of the triangle in feet.
Solution
0 Write the formula. P = a + b + c
© Substitute the side lengths of 8, 15 and 17. = 8 + 15 + 17
© Add the side lengths. = 40
ANSWER ^ The triangle has a perimeter of 40 feet.
Find the Perimeter
10. Find the perimeter of a square with each side 12 inches long.
mS3MM 5
Estimate the Area
GEOGRAPHY LINK The area A of a triangle is equal to half the base b times
the height h: A = ^ bh . Use this formula to estimate the area (in square miles)
of Virginia.
h =
200 mi
Solution
0 Write the formula. A = \bh
© Substitute 410 for b and 200 for h. = |-(410)(200)
© Simplify the formula. = 41,000
ANSWER ► The area of Virginia is about 41,000 square miles.
I_
Find the Area
11. Find the area of the triangle in square meters.
b= 12
1.1 Variables in Algebra
_Exercises
Guided Practice
Vocabulary Check Identify the variable or variables.
1. y + 15 2.20 -5 3. 4. rt
5. Complete: You ? an expression by substituting numbers for variables
and simplifying. The resulting number is called the ? of the expression.
Skill Check State the meaning of the variable expression and name the operation.
6. ^ 7. p — 4 8-5 + n 9- ( 8 )(x)
Evaluate the variable expression when k = 3.
10.11+Jfc 11. £ - 2 12.7 k
k 18
13. £ 14. - 7 ^ 15. 18 • k
33 k
16. Geometry Link / Find the perimeter of each triangle.
b.
Practice and Applications
DESCRIBING EXPRESSIONS Match the variable expression with its
meaning.
17. y + 8
18. y — 8
<
20. 8 y
A. 8 times y
B. y divided by 8
C. y plus 8
D. v minus 8
EVALUATING EXPRESSIONS Evaluate the expression for the given value
Student He dp
of the variable.
21.9 + p when p — 11
22. y + t when t = 2
23. Tj when b — 14
>
► Homework Help
Example 1 : Exs. 17-20
24. when d = 36
25. (4)0z) when n — 5
26. 8 a when a — 6
Example 2: Exs. 21-32
Example 3: Exs. 33-39
27. 12 — x when x = 3
28. 9 — y when y — 8
29. 1 Or when r = 7
Example 4: Exs. 40-42
Example 5: Exs. 43-46
^ j
30. 13c when c = 3
31. — whenx = 3
32. when k — 9
k
Chapter 1 Connections to Algebra
p
Student Hedp
► Homework Help
Extra help with
problem solving in
Exs. 34-39 is available at
www.mcdougallittell.com
33. DRIVING DISTANCE You are driving across the country at an average speed
of 65 miles per hour. Using an appropriate formula, find the distance you
travel in 4 hours.
FINDING DISTANCE Find the distance traveled using d = rt.
34. A train travels at a rate of 75 miles per hour for 2 hours.
35. An athlete runs at a rate of 8 feet per second for 5 seconds.
36. A horse trots at 8 kilometers per hour for 30 minutes.
37. A racecar driver goes at a speed of 170 miles per hour for 2 hours.
38. A plane travels at a speed of 450 miles per hour for 3 hours.
39. A person walks at a rate of 4 feet per second for 1 minute.
Student Hedp
► Skills Review
The perimeter of any
geometric figure is
the sum of all its
side lengths. To
review perimeter
and area formulas,
see p. 772.
L J
Ge ometry Link y Find the area of each triangle.
43.
\ h = 4m
44. y
\. h = 6 mi 45.
K
/
h = 3yd
iA
/
=lA
b = 5m
b = 10 mi
46. Geography Link / To find the area A of a rectangle, you multiply the
length times the width:
A = I • w
Use the formula to estimate the area of Wyoming.
47. CHALLENGE A tsunami is a huge fast-moving series of water waves that
can be caused by disturbances such as underwater earthquakes or volcanic
explosions. If a tsunami is traveling at a speed of 500 miles per hour across
the Pacific Ocean, how far has it gone in 15 minutes? HINT: Convert 15
minutes to hours.
1.1 Variables in Algebra
Unit Analysis
Writing the units of measure helps you determine the units for the answer. This
is called unit analysis. When the same units occur in the numerator and the
denominator, you can cancel them.
Use unit analysis to evaluate the expression. The letter h is an abbreviation for
hours, while mi stands for miles.
a. (3 h)l
25 mi \
ih )
b. (90 mi) -h
/ 45 mi \
l lh )
P
Student HeCp
► Study Tip
When you divide by a
fraction, you multiply
by the reciprocal. See
Skills Review, p. 765. - -
Solution
a. ^ ) = 75 mi
IX
b. (90 mi) -
Cancel hours.
45 mi
1 h
(90ij*i)
lh
45 rX
= 2h
48, Evaluate the expression (4 h)( ^ ™
49- Evaluate the expression (80 mi) -r-
20 mi
1 h
Cancel miles.
Standardized Test
Practice
50. MULTIPLE CHOICE How many miles does Joyce travel if she drives for
6 hours at an average speed of 60 miles per hour?
(A) 66 miles Cb ) 180 miles Cep 360 miles CD) 420 miles
51. MULTIPLE CHOICE The lengths of the sides of a triangle are 4 centimeters,
8 centimeters, and 7 centimeters. What is the perimeter of the triangle?
CE) 7 cm (G) 16 cm (TT) 19 cm (T) 28 cm
OPERATIONS WITH DECIMALS Find the value of the expression.
(Skills Review p. 759)
52. 32.8 - 4 53. 3.98 + 5.50 54. 0.1(50)
SIMPLIFYING EXPRESSIONS Simplify the expression without using a
calculator. (Skills Review p. 765)
55.
56. (60)
57.
Maintaining Skills
ADDING DECIMALS
Add. (Skills Review p. 759)
58. 2.3 + 4.5
59. 16.8 + 7.1
60. 0.09 + 0.05
61. 1.0008 + 10.15
62. 123.8 + 0.03
63. 46 + 7.55
64. 0.32 + 0.094
65. 6.105 + 7.3
66. 2.008 + 1.10199
Chapter 1 Connections to Algebra
Exponents and Powers
Goal
Evaluate a power.
Key Words
• power
• exponent
• base
• grouping symbols
How much water does the tank hold?
How much water do you
need to fill a fish tank?
You will use a power to find
the answer in Example 5.
An expression like 2 3 is called a power. The exponent 3 represents the number
of times the base 2 is used as a factor.
base exponent
y ^
2 3 = 2 • 2 • 2
power 3 factors of 2
The expression 2 3 means “multiply 2 by itself 3 times.” The numbers you
multiply are factors. In general, a n — a • a • a • ... • a.
K V J
n times
Student MeCp
► Reading Algebra
Note that x 1 is
customarily written as
x with the exponent
omitted.
\ _ /
Read and Write Powers
Express the power in words. Then write the meaning.
EXPONENTIAL FORM WORDS
MEANING
a. 4 2
four to the second power
4 • 4
b. 5 3
or four squared
five to the third power
5-5-5
c. x 6
or five cubed
x to the sixth power
x-x-x-x-x-x
( Write the Power
Write the expression in exponential form.
1 _ 3 squared 2. x to the fourth power 3- s' cubed
1.2 Exponents and Powers
Student HeCp
p Mori Examples
More examples
are available at
www.mcdougallittell.com
2 Evaluate the Power
Evaluate x 4 when x = 2.
Solution
O Substitute 2 for x.
0 Write out the factors.
0 Multiply the factors.
x 4 = 2 4
= 2*2
= 16
ANSWER ► The value of the power is 16.
2 • 2
GROUPING SYMBOLS Parentheses () and brackets [ ] are grouping symbols.
They tell you the order in which to do the operations. You must do the operations
within the innermost set of grouping symbols first:
First multiply. Then add. First add. Then multiply.
(3 • 4) + 7 = 12 + 7 = 19 3 • (4 + 7) = 3 • 11 = 33
You will learn more about the order of operations in the next lesson.
3 Evaluate Exponential Expressions
Evaluate the variable expression when a = 1 and b = 2.
b. (a + b ) 2
a. (a 2 ) + (b 2 )
Solution
a. (a 2 ) + ( b 2 ) = (l 2 ) + (2 2 )
= (1 • 1 ) + (2
= 1+4
= 5
b. (a + b) 2 = (1 + 2) 2
= (3) 2
= 3*3
= 9
Substitute 1 for a and 2 for b.
• 2) Write factors.
Multiply.
Add.
Substitute 1 for a and 2 for b.
Add within parentheses.
Write factors.
Multiply.
Evaluate Exponential Expressions
Evaluate the variable expression when s = 2 and t = 4.
4. ( t — s ) 3 5. (s 2 ) + (t 2 ) 6. ( t + s ) 2
7. (t 2 ) - (.s 2 ) 8. (.s 2 ) + t 9. (t 2 ) - ,v
H
Chapter 1 Connections to Algebra
In Lesson 1.3 you will learn several rules for order of operations. One of those
tells us that 2x 3 is to be interpreted as 2(x 3 ).
Student McCp
► Reading Algebra
Notice that in part (a)
of Example 4, the
exponent applies to x,
while in part (b), the
exponent applies to lx.
I j
4 Exponents and Grouping Symbols
Evaluate the variable expression when x = 4.
a. 2x 3 b. (2x) 3
Solution
a. 2x 3 = 2(4 3 ) Substitute 4 for x.
= 2(64) Evaluate power.
= 128 Multiply.
b. (2x) 3 = (2 • 4) 3 Substitute 4 for x.
= 8 3 Multiply within parentheses.
= 512 Evaluate power.
Exponents can be used to find the area of a square and the volume of a cube.
s
s
s s
Area of square: A = s 2 Volume of cube: V = s 3
Units of area, such as square feet, ft 2 , can be written using a second power. Units
of volume, such as cubic feet, ft 3 , can be written using a third power.
5
Find the Volume of the Tank
FISH TANKS The fish tank has the shape
of a cube. Each inner edge s is 2 feet
long. Find the volume in cubic feet.
Solution
V = s 3 Write formula for volume of a cube.
= 2 3 Substitute 2 for s.
= 8 Evaluate power.
2ft
ANSWER ► The volume of the tank is 8 cubic feet.
Find Area and Volume
10. Use the formula for the area of a square to find the area of each side of the
fish tank in Example 5. Express your answer in square feet.
1.2 Exponents and Powers
Guided Practice
Vocabulary Check Complete the sentence.
1 . In the expression 3 7 , the 3 is the ? .
2. In the expression 5 4 , the 4 is the ? .
3- The expression 9 12 is called a ? .
4- Two kinds of grouping symbols are ? and ?
Skill Check Match the power with the words that describe it.
5. 3 7
A. four to the sixth power
6. 7 3
B. three to the seventh power
7. 4 6
C. seven to the third power
8. 6 4
D. six to the fourth power
Evaluate the variable expression when t = 3.
9. t 2 10. 1 + t 3 11.4 1 1 12.(4 1) 2
Practice and Applications
WRITING POWERS Write the expression in exponential form.
13, two cubed 14 . p squared 15. nine to the fifth
power
16- b to the eighth power 17. 3 • 3 • 3 • 3 18. 4x • 4x • 4x
Student MeCp
► Homework Help
Example 1: Exs. 13-19
Example 2: Exs. 19-39
Example 3: Exs. 40-45
Example 4: Exs. 46-51
Example 5: Exs. 52-57
^ _/
19. Geometry Link / A square painting measures
5 feet by 5 feet. Write the power that gives the
area of the painting. Then evaluate the power.
EVALUATING POWERS Evaluate the power.
20. 9 2 21. 2 4 22. 7 3
24. 5 4 25. I 8 26. 10 3
5 ft
23. 2 6
27. 0 6
EVALUATING POWERS WITH VARIABLES Evaluate the expression for the
given value of the variable.
28 . w 2 when w — 12 29 . b 3 when b — 9
31 . h 5 when h = 2 32 . n 2 when n = 11
30. c 4 when c — 3
33. x 3 when x = 5
Chapter 1 Connections to Algebra
( Student HeCp
^Keystroke Help
Your calculator may
have a f^ key or a
key that you can
use to evaluate
powers.
s_ )
P EVALUATING POWERS Use a calculator to evaluate the power.
34. 8 6 35. 13 5 36. 5 9
37. 12 7 38. 6 6 39. 3 12
EVALUATING EXPRESSIONS Evaluate the variable expression when
c = 4 and d = 5.
40. (c + d ) 2 41. (d 2 ) + c 42. (c 3 ) + d
43. {d 2 ) - (c 2 ) 44. (d - c) 1 45. (d 2 ) - d
EXPONENTIAL EXPRESSIONS Evaluate the expression for the given
value of the variable.
46. 2x 2 when x — 7 47. 6 1 4 when t — 1 48. lb 2 when b — 3
49. (5w) 3 when w = 5 50. (4x) 3 when v = 1 51. (5y) 5 when y — 2
52. INTERIOR DESIGN The floor of a room is 14 feet long by 14 feet wide.
How many square feet of carpet are needed to cover the floor?
ARTIST Jon Kuhn used
mathematics when creating
the cubic sculpture Crystal
Victory. The solid glass cube
is made of lead crystal and
colored glass powders.
53. VOLUME OF A SAFE A fireproof safe is designed in the shape of a cube.
The length of each edge of the cube is 2 meters. What is the volume of the
fireproof safe?
54. ARTISTS In 1997, the artist Jon Kuhn of North Carolina created a cubic
sculpture called Crystal Victory , shown at the left. Each edge of the solid
glass cube is 9.5 inches in length. What is the volume of the cubic structure?
CRITICAL THINKING Count the number of cubic units along the edges of
the cube. Write and evaluate the power that gives the volume of the
cube in cubic units.
55 - /7*7 r\
./
CHALLENGE You are making candles. You melt paraffin wax in the cubic
container shown below. Each edge of the container is 6 inches in length.
The container is half full.
58. What is the volume of the wax in this container?
59. Each edge of a second cubic container is 4 inches in
length. Can this second candle mold hold the same
amount of melted wax that is in the candle mold
shown at the right? Explain your answer.
6 in.
60. Design a third cubic candle mold different from the one given above that will
hold all the melted wax. Draw a diagram of the mold including the
measurements. Explain why your mold will hold all the melted wax.
1.2 Exponents and Powers
Standardized Test
Practice
61. MULTIPLE CHOICE Evaluate the expression 2x 2 when x = 5.
(A) 20 CD 40 CD 50 CD 100
62. MULTIPLE CHOICE One kiloliter is equal to 10 3 liters. How many liters are
in one kiloliter?
CD 10
CD 100
CD 1000
CD 10,000
Student MeCp
►Test Tip
Jotting down the
formula for the volume
of a cube will help you
answer Exercise 63.
63. MULTIPLE CHOICE Sondra bought
this trunk to store clothes. What is the
volume of the trunk?
(A) 9 ft
CD 9 ft 3
CD 9 ft 2
Co) None of these
3 ft
Mixed Review
Geometry Link/ Find the perimeter of the geometric figure when x = 3.
(Lesson 1.1)
64.
66 .
2x\ 2x
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.1)
67. 9/ when j = 5
68 . 6 + t when t = 21
b
69. — when b = 18
70. 25 — n when n — 3 71. c + 4 when c — 24 72. (7)(r) when r — 11
24
73. — when s = 8
74. 3/77 when m = 7
75. J — 13 when d = 22
Maintaining Skills SIMPLIFYING FRACTIONS Simplify. (Skills Review p. 763)
76 -f
77 ^
"■ 2
78 -t!
79 —
20
s°4
00
^ | oo
“■!
03. f
-ft
21
85. T
86 4
37. f
ESTIMATING Estimate the answer. Then evaluate the expression.
(Skills Review p. 774)
88 . 2.5 - 0.5
91. 3.71 + 1.054
89. 0.3 - 0.03
92. 2.1 - 0.2
90. 10.35 + 5.301
93.5.175 + 1.15
■
Chapter 1 Connections to Algebra
Order of Operations
Goal
Use the established order
of operations.
Key Words
• order of operations
• left-to-right rule
How many points ahead are you?
You are playing basketball. You make
8 field goals and 2 free throws. Your
friend makes half as many field goals
as you and no free throws. You will
find how many points ahead you are
in Example 5.
ORDER OF OPERATIONS In arithmetic and algebra there is an order of
operations to evaluate an expression involving more than one operation.
ORDER OF OPERATIONS
step O First do operations that occur within grouping symbols.
step © Then evaluate powers.
step © Then do multiplications and divisions from left to right.
step © Finally, do additions and subtractions from left to right.
1 Evaluate Without Grouping Symbols
Evaluate the expression 3x 2 + 1 when x = 4. Use the order of operations.
Solution
3x 2 + 1 = 3 • 4 2 + 1
= 3 • 16 + 1
= 48 + 1
= 49
Substitute 4 for x.
Evaluate power.
Multiply 3 times 16.
Add.
Evaluate Expressions Without Grouping Symbols
Evaluate the variable expression when x = 2. Use the order of operations.
1- 2x 2 + 5 2. 8 — v 2 3- 6 + 3x 3 4. 20 - 4x 2
1.3 Order of Operations
LEFT-TO-RIGHT RULE Some expressions have operations that have the same
priority, such as multiplication and division or addition and subtraction. The
left-to-right rule states that when operations have the same priority, you perform
them in order from left to right.
2 Use the Left-to-Right Rule
Student HeCp
Evaluate the expression using the left-to-right rule.
a. 24 — 8 — 6 = (24 — 8) — 6 Work from left to right.
= 16 — 6 Subtract 8 from 24.
= 10 Subtract 6 from 16.
b. 15 • 2 + 6 = (15 • 2) + 6 Work from left to right.
= 30 + 6 Multiply 15 times 2.
= 5 Divide 30 by 6.
^ Study Tip
you inui li piy iirsi in *****
part (c) of Example 2,
because multiplication
has a higher priority
than addition and
subtraction.
L. J
*
*
c. 16 + 4 • 2 — 3 = 16 + (4 • 2) — 3 Do multiplication first.
= 16 + 8 — 3 Multiply 4 times 2.
= (16 + 8) — 3 Work from left to right.
= 24 — 3 Add 16 and 8.
= 21 Subtract 3 from 24.
1 + 2
A fraction bar can act as a grouping symbol: (1+2) +(4— 1) = ^ _ y
Student HeCp
^Skills Review
For help with writing
fractions in simplest
form, see p. 763.
_ J
EZQQ1S 3 Expressions with Fraction Bars
Evaluate the expression. Then simplify the answer.
7*4 7-4
8 + 7 2 - 1 8 + 49 -1
28
8 + 49-1
28
57 - 1
28
56
1
2
Evaluate power.
Simplify the numerator.
Work from left to right.
Subtract.
Simplify.
U se the Order of Operations and Left-to-Right Rule
Evaluate the variable expression when x = 1.
5. 4x 2 + 5 - 3 6. 5 - x 3 - 1 7.
Chapter 1 Connections to Algebra
USING A CALCULATOR You need to know if your calculator uses the order
of operations or not. If it does not, you must input the operations in the proper
order yourself.
a Use a Calculator
Enter the following in your calculator. Does the calculator display 6 or 1 ?
10H6H2
ENTER
Solution
a. If your calculator uses the order of operations, it will display 6.
10 — 6-5-2—1 = 10 — (6-5-2) — 1
= (10 - 3) - 1
= 6
b. If your calculator does not use the order of operations and performs the
operations as they are entered, it will display 1.
[(10 - 6) -5- 2] - 1 = (4 -v- 2) - 1
= 2-1
= 1
Link to
Sports
BASKETBALL SCORES
A field goal is worth 2 points.
A free throw is worth 1 point.
5 Evaluate a Real-Life Expression
BASKETBALL SCORES You are playing basketball. You make 8 field goals
and 2 free throws. Your friend makes half as many field goals as you and no
free throws. How many points ahead of your friend are you?
Solution
8 • 2 + 2 * 1 -
8 • 2
16 + 2-
16
2
Multiply from left to right.
16 + 2 -
8
Divide.
18-8
Add.
10
Subtract.
ANSWER ► You are 10 points ahead of your friend.
Evaluate a Real-Life Expression
8. Your friend makes 4 field goals. You make three times as many field goals as
your friend plus one field goal. How many points do you have? Explain the
order of operations you followed.
9. Your friend makes 6 field goals and 2 free throws. You make twice as many
field goals as your friend and half the number of free throws. How many
points do you have? Explain the order of operations you followed.
1.3 Order of Operations
feW Exercises
Guided Practice
Vocabulary Check 1 . Place the operations in the order in which you should do them.
a. Multiply and divide from left to right.
b. Do operations within grouping symbols.
c. Add and subtract from left to right.
d. Evaluate powers.
2. What rule must be applied when evaluating an expression in which the
operations have the same priority?
Skill Check Evaluate the expression.
3. 5 • 6 • 2 4. 16 - 4 - 2 5. 4 + 9 - 1 6. 2 • 8 2
7. 15 + 6 - 3 8. 9 - 3 • 2 9. 2 • 3 2 + 5 1 0. 2 3 • 3 2
Evaluate the variable expression when x = 3.
11. 5 12. x 3 + 5x 13. x + 3x 4
14. —-2 + 16 15. — + 2 3 -10 16. — *5
V V V
Practice and Applications
NUMERICAL EXPRESSIONS Evaluate the expression.
17.13 + 3*7 18.7 + 8-2 19. 2 4 - 5 • 3
20. 6 2 + 4 21. 4 3 + 9 • 2 22. 3 • 2 + IN¬
VARIABLE EXPRESSIONS Evaluate the expression for the given value of
the variable.
23. 6 • 2 p 2 when p = 5 24. 2 g • 5 when g = 4
Student HeCp
^Homework Help
Example 1: Exs. 17-26
Example 2: Exs. 27-35
Example 3: Exs. 36-41
Example 4: Exs. 43-46
Example 5: Exs. 47-53
25. 14(n + 1) when n = 2
26. y + 16 when x =14
NUMERICAL EXPRESSIONS Evaluate the expression.
27. 2 3 + 5 - 2
30. 5 + 8 • 2 - 4
33. 10 - 3 + (2 + 5)
28. 4 • 2 + 15 - 3
31. 16 + 8 • 2 2
34. 7 + 18 - (6 - 3)
29. 6 - 3 + 2 • 7
32. 2 • 3 2 - 7
35. [(7 • 4) + 3] + 15
Chapter 1 Connections to Algebra
I
Student HeCp
^Homework Help
Extra help with
problem solving in
Exs. 36-41 is available at
www.mcdougallittell.com
EXPRESSIONS WITH FRACTION BARS Evaluate the expression. Then
simplify the answer.
36.
6 • 4
39.
4 + 3 2 - 1
21+9
5 2 + 40 - 5
37.
13-4
38.
18 - 4 2 + 1
40.
3 3 + 8 - 7
41.
2 • 7
5 2 • 2
1 + 6 2 - 12
4 • 2 5
16 - 4 2 + 1
42. LOGICAL REASONING Which is correct?
9 2 + 3 o
A. = 9 2 + 3 + 5
B.
9 2 + 3
= [9 2 + 3] + 5
CRITICAL THINKING In Exercises 43-46, two calculators were used
to evaluate the expression. Determine which calculator performed the
correct order of operations.
43. 15060304 W4M+M
Calculator A: 12 Calculator B: 7
44. 15
_ I903Q7 ESSm
Calculator A: 19 Calculator B: 9
45.150100504
Calculator A: 21 Calculator B: 9
46.4030602
Calculator A: 9 Calculator B: 15
FOOTBALL UNIFORMS
In Exercises 47 and 48, use
the table showing the costs
of parts of a football player's
uniform.
47. A sporting goods company
offers a $2000 discount for
orders of 30 or more complete
uniforms. Your school orders
35 complete uniforms. Write
an expression for the total cost.
48. Evaluate the expression you
wrote in Exercise 47.
Part
Jersey
Shoulder
Lower
Knee
Cleats
Helmet
of uniform
and pants
pads
body pads
pads
Cost
$230
$300
$40
$15
$100
$200
1.3 Order of Operations
Geo metry Link / In Exercises 49 and 50, refer to
the squares shown at the right.
49. Write an expression that represents the area of the
shaded region. HINT: Subtract the area of the inner
square from the area of the outer square.
50. If x = 8, what is the area of the shaded region?
x
x
Link
State fairs
ADMISSION PRICES Every
year nearly 1,000,000 people
attend the California State Fair.
ADMISSION PRICES In Exercises 51 and 52, use the table below. It
shows the admission prices for the California State Fair.
California State Fair Admission Prices
Age
Admission price
General Admission (13-61 years of age)
$7.00
Seniors (62 years and above)
$5.00
Children (5-12 years)
$4.00
Children (4 years and under)
Free
► Source: Sacramento Bee
51. Write an expression that represents the admission price for a group
consisting of 2 adults, 1 senior, and 3 children. The children’s ages are
12 years, 10 years, and 18 months.
Standardized Test
Practice
52. Evaluate the expression you wrote in Exercise 51. Then find the total cost
of admission for the group.
53. CHALLENGE At a concert you buy a hat for $10.00, a hot dog for $2.75, and
nachos for $3.50. There is a 6% sales tax on the hat. Your calculator follows
the established order of operations. Write a keystroke sequence for the
amount you owe. Then find the amount you owe. HINT: 6% = 0.06
54. MULTIPLE CHOICE Evaluate the expression 4 2 — 10 -h 2.
<3)3 CD 11 CD 13 CD 21
55. MULTIPLE CHOICE Evaluate the expression 32 - x 2 + 9 when x = 2.
CD 19 CD 21 CD 37 CD 39
56. MULTIPLE CHOICE Which expression has a value of 12?
®3+3X5-2
CD 18 -h 6 X 3 + 3
CD 7 + 14 7 X 4
CD 2 2 • 3 - 6 • 2
57. MULTIPLE CHOICE Evaluate the expression —yy
CD 1 CD 5 CD 7 CD 10
Chapter 1 Connections to Algebra
Mixed Review
Maintaining Skills
Quiz 1
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.1)
24
58- (8)(a) when a — 4 59- H when x — 3 60- c + 15 when c — 12
61. ^ • x when x = 18 62- 9 1 when t = 1 63- 25 — y when y = 14
WRITING POWERS Write the expression in exponential form. (Lesson 1.2)
64- twelve squared 65- z to the sixth power 66- 2b • 2b • 2b
EXPONENTIAL EXPRESSIONS Evaluate the expression for the given
value of the variable. (Lesson 1.2)
67. 9 1 2 when t — 3 68. (7/z) 3 when h — 1 69. (6 w) 2 when w = 5
FACTORS Determine whether the number is prime or composite. If it is
composite, list all of its factors. (Skills Review p. 761)
70.15 71.9 72.13 73.38
74. 46 75. 50 76. 64 77. 29
Evaluate the variable expression when x = 3. (Lessons 1.1, 1.2)
1 . 6x 2. 42 -T- x 3. x + 29
4. 12 - x 5. 5x - 10 6. 10 + 2x
7. x 2 — 3 8. 2x 3 9. (2x) 3
Find the distance traveled using d = rt. (Lesson 1.1)
10. A car travels at an average speed of 50 miles per hour for 4 hours.
11. A plane flies at 500 miles per hour for 4 hours.
12. A marathon runner keeps a steady pace of 10 miles per hour for 2 hours.
Write the expression in exponential form. (Lesson 1.2)
13. six cubed 14.4 # 4«4«4 # 4 15. 5y • 5y • 5y
16. 3 • 3 • 3 17. 2x • 2x • 2x • 2x 18. eight squared
19. PACKING BOXES A cubic packing box has dimensions of 4 feet on each
edge. What is the volume of the box? (Lesson 1.2)
Evaluate the expression. Then simplify the answer. (Lesson 1.3)
20 .
7 • 2 2
7 + (2 3 — 1)
21 .
(3 2 ~ 3)
2 • 9
22 .
6 2 - 11
2(17 + 2*4)
1.3 Order of Operations
DEVELOPING CONCEPTS
For use with
Lesson 1.4
Goal Questi on How can you use algebra to describe a pattern?
Use algebraic expressions
to describe patterns.
Materials Explot#
• graph paper
• toothpicks O Copy the first four figures on graph paper. Then draw the fifth and sixth
figures of the sequence.
Figure 1
1
Figure 2
i
: igure 3
Figure
4
© The table shows the mathematical pattern for the perimeters of the first four
figures. Copy and complete the table.
Figure
1
2
3
4
5
6
Perimeter
4
8
12
16
?
?
Pattern
4 • 1
4 • 2
4 • 3
4 • 4
4 • ?
4 • ?
© Observe that 4(1) = 4, 4(2), = 8, 4(3) = 12, and so on. This suggests that
the perimeter of the nth figure is 4 n, where n = 1, 2, 3, 4, ... . Find the
perimeter of the 10th figure.
Think About It
1- Copy the four figures below. Then draw the fifth and sixth figures.
Fig
lure 1
i
Figure 2
i
Figure 3
i
Fig
lure 4
2 . Calculate the perimeters of all six figures. Organize your results in a table.
3. What is the perimeter of the 10th figure? Can you guess a formula for the
nth figure?
Chapter 1 Connections to Algebra
Explore
Q Use toothpicks to model the perimeter of all six figures in Explore on
page 22. Notice that the perimeter of each figure is equal to the number of
toothpicks used to form the figure.
Q Change the shape of Figures 2-6 by moving toothpicks until the figures
consist of n unit squares. Figures 2 and 3 in the sequence are shown below.
Complete Figures 4, 5, and 6 on a separate sheet of paper.
Figure 2
© You should be able to conclude that if one square unit has a perimeter of
4 • 1, then n squares must have a perimeter of 4 n. This conclusion verifies
the pattern you found on page 22.
Think About It
1. Use toothpicks to model the perimeter of all six figures in Exercise 1 on
page 22.
2 . Change the shape of the figures modeled above in Exercise 1 until they
consist of n unit squares.
3. Do the number of unit squares verify the pattern found in Exercise 3 on
page 22? Explain your reasoning.
Developing Concepts
Equations and Inequalities
Goal
Check solutions of
equations and
inequalities.
Key Words
• equation
• solution
• inequality
How much do the ingredients cost?
You can use an equation to
solve a real-life problem. In
Example 3 you will use an
equation to estimate the cost
of ingredients for nachos.
Student HeCp
\
►Vocabulary Tip
Equation comes from
a Latin word that
means "to be equal".
\ _ J
An equation is a statement formed by placing an equal sign (=) between two
expressions. An equation has a left side and a right side.
Left side
Right side
4x + 1
Equation
When the variable in an equation is replaced by a number, the resulting statement
is either true or false. If the statement is true, the number is a solution of the
equation.
J i Check Possible Solutions
Check to see if 2 and 3 are solutions of the equation 4x + 1 = 9.
Solution
Substitute the x values 2 and 3 into the equation
are equal in value, then the number is a solution
X VALUE SUBSTITUTE SIMPLIFY
2 4(2) + U9 9 = 9
3 4(3) +119 13 A 9
t
is not equal to
ANSWER ► The number 2 is a solution of the equation 4x + 1 = 9, because the
statement is true. The number 3 is not a solution, because the
statement is false.
. If both sides of the equation
CONCLUSION
True, 2 is a solution.
False, 3 is not a solution.
Chapter 1 Connections to Algebra
SOLVING EQUATIONS Finding all the solutions of an equation is called solving
the equation. Some equations are simple enough to be solved with mental math.
Later in the book you will learn how to systematically solve more complex
equations.
Student tteCp
► More Examples
More examples
are ava j| a |}| e a t
www.mcdougallittell.com
2 Solve Equations with Mental Math
To solve equations with mental math, think of the equation as a question.
EQUATION
2x — 10
4 = x — 3
2 + x — 6
* = 1
3
QUESTION
2 times what number gives 10?
4 is equal to what number minus 3?
2 plus what number gives 6?
What number divided by 3 gives 1?
SOLUTION
2*5 = 10, so x = 5
4 = 7 — 3, sox = 7
2 + 4 = 6, so x = 4
\ = 1, so x = 3
Then check each solution by substituting the number in the original equation.
If the statement is true, the number is a solution.
Solve Equations and Check Solutions
Use mental math to solve the equation. Then check your solution.
1.2 = 6 — x 2. x + 3 = 11 3. 4 = 5 4. 14 = 2x
4
Hacbo*
%0 torVttta cKn*
\ \!%
\ 6U\>
tovnatoe*
Use Mental Math to Solve a Real-Life Equation
buying ingredients for nachos. At the market you find that
\/£ 6U\> grated tortilla chips cost $2.99, beans cost $.99, cheese costs $3.99, two
cheese tomatoes cost $1.00, and olives cost $1.49. There is no tax. You have
\!% 6U\> sViceA $10. About how much more money do you need?
o\W®*
Solution
a'
Ask: The total cost equals 10 plus what number of dollars? Let x
represent the extra money you need. Use rounding to estimate the
total cost.
3 + 1+ 4+1 + 1.5 — 10 + x
10.5 = 10 + x
ANSWER ^ The total cost is about 10.5 or $10.50, so you need
about $.50 more to purchase all the ingredients.
Use Mental Math to Solve a Real-Life Equation
5. Solve the equation in Example 3 if a large bag of chips costs $3.99. About
how much more money would you need to buy the nacho ingredients?
1.4 Equations and Inequalities
Student HeCp
^
► Study Tip
The "wide end" of the
inequality symbol
faces the greater
number. For help with
comparing numbers,
see p. 770.
Careers
VETERINARIANS specialize
in the health care of either
small animals, such as cats,
or large animals, such as
horses.
■
An inequality is a statement formed by placing an inequality symbol, such as <,
between two expressions.
INEQUALITY SYMBOL
MEANING
EXAMPLE
<
is less than
1 + 3<5
<
is less than or equal to
6 - 1 <5
>
is greater than
10 > 2(4)
>
is greater than or equal to
10 >9 - 1
For inequalities involving a single variable, a solution is a number that produces a
true statement when it is substituted for the variable in the inequality.
Msmm*
Check Solutions of Inequalities
Check to see if x
= 4 is or is not a solution of the inequality.
INEQUALITY
SUBSTITUTE
SIMPLIFY
CONCLUSION
x + 3 >9
4 + 3> 9
7X9
False, 4 is not a solution.
2x - 1 < 8
2(4) - 1 < 8
7 < 8
True, 4 is a solution.
Check Solutions of Inequalities
Check to see if the value of n is or is not a solution of 3n — 4 < 8.
6. n = 2 7. n = 3 8. n = 4 9. n = 5
5 Check Solutions in Real Life
VETERINARIANS Your vet tells you to restrict your cat’s caloric intake to
less than or equal to 500 calories a day. Two times a day, you give your cat a
serving of food that has x calories. Does 250 calories for each serving meet the
vet’s restriction?
Solution
O Write the inequality. 2x < 500
0 Substitute 250 for x. 2(250) < 500
© Simplify by multiplying. 500 < 500
ANSWER ► Yes, 250 calories per serving meets the vet’s restriction.
Check Solutions in Real Life
10, Check to see if 300 calories per serving meets the vet’s restriction in
Example 5.
Chapter 1 Connections to Algebra
Guided Practice
Vocabulary Check Explain if the following is an expression, an equation, or an inequality.
1.3*+ 1 = 14 2.1y — 6 3. 5(j 2 + 4) — 7
4. 5x — 1 = 3 + x 5. 3x + 2 < 8 6. 5x > 20
7. Complete: An x value of 4 is a ? of the equation x + 1 =5, because
4+1=5.
Skill Check
Check to see if a = 5 is or is not a solution of the equation.
8. a + 8 = 13 9. 27 = 36 — 2a 10. a — 0 = 5
11. 2a + 1 = 11 12. 6a - 5 = 15
14. 45 - a = 9 15. a 2 + 2 = 21
13. 5a + 4 = 26
16. —= 8
a
Check to see if b = 8 is or is not a solution of the inequality.
17. b + 10 > 19
20 . 8 > 64 + b
23. 60 > lb + 3
18. 14 — Z? < 3 19. 5Z> > 35
21. 3*-24 >0 22. 16 < b 2
24. 18- b< 10 25. 37 >4 b
Practice and Applications
CHECKING SOLUTIONS OF EQUATIONS Check to see if the given value
of the variable is or is not a solution of the equation.
26. 3b + 1 = 13;Z> = 4 27. 5r - 10 = 11; r = 5
28. 4c + 2 = 10; c = 2
30. 5 + x 2 = 17; x = 3
32. 9 + 2t = 15; t = 12
29. 6d- 5 = 31; d = 6
31.2/ + 3 = 5; y = 1
33. n 2 - 5 = 20; n = 5
r Student HeCp
^Homework Help
Example 1: Exs. 26-33
Example 2: Exs. 34-48
Example 3: Exs. 49, 50
Example 4: Exs. 51-56
Example 5: Exs. 57, 58
SOLVING WITH MENTAL MATH Use mental math to solve the equation.
34. x + 3 = 8
35. n + 6 = 11
36. p - 13 = 20
37. r - 1 = 7
38. 3j = 12
39. 4p = 36
40. z + 4 = 5
II
U4
42. 2b = 28
43. lit = 22
44. 29 - d = 10
45. 3 + y = 8
46. r + 30 = 70
42
47. — = 7
X
48. 1m = 49
1.4 Equations and Inequalities
49. TIME MANAGEMENT You have a hair appointment in 60 minutes. It takes
20 minutes to get to the gas station and fill your tank. It takes 15 minutes to
go from the gas station to the hair stylist. You wait x minutes before leaving
your house and arrive on time for your appointment. Use the diagram to help
decide which equation best models the situation.
A. 20 + 15 — x = 60 B. 60 + 20 + 15 = x
C. 60 — 20 + 15 + x = 60 D. x + 20 + 15 = 60
50. MENTAL MATH Solve the equation you chose in Exercise 49.
CHECKING SOLUTIONS OF INEQUALITIES Check to see if the given value
of the variable is or is not a solution of the inequality.
51. n — 2 <6;n = 3 52. a — 1 > 15; a = 22
53. 6 + y<8;y = 3
55. Ig > 47; g = 7
54. s , + 5>8;5' = 4
56. 72 -r- t > 6; t = 12
57. SELLING CARDS Your community center is selling cards. Your goal is to
sell $100 worth of cards. Each box sells for $3. Using mental math, solve the
inequality 3 b> 100 to determine at least how many boxes you must sell to
meet your goal.
CHUCK YEAGER in 1947
became the first person to
fly faster than the speed of
sound (Mach 1) or about
660 miles per hour.
58. BUYING A GUITAR You are budgeting money to buy a guitar that costs
$150 including tax. If you save $20 per month, will you have enough money
in 6 months? Use the inequality 20 n > 150 to model the situation, where n
represents the number of months.
59. Sc ience Link ^ Mach number is the maximum speed at which a plane can
fly divided by the speed of sound. Copy and complete the table. Use the
equation m = where m is the Mach number and v is the speed (in miles
per hour) of the aircraft, to find the Mach number for each type of aircraft.
Airplane type
Test aircraft
Supersonic
Jet
Speed v
4620
1320
660
Mach number m
?
?
?
Test aircraft Supersonic aircraft Jet aircraft
Chapter 1 Connections to Algebra
Standardized Test
Practice
Mixed Review
Maintaining Skills
60. Use mental math to fill in the missing number so that all the
equations have the number 6 as a solution.
a. ? + x = 18 b. ? x = 30 c. ■ = 6
61 - MULTIPLE CHOICE Which is a solution of the equation 5(8 — x) = 25?
(A) 2 QD 3 (g) 4 CD) 5
62. MULTIPLE CHOICE For which inequality is x = 238 a solution?
(T) 250 > x + 12 Cg) 250 < x + 12
(R) 250 > x + 12 CJ) 250 < x + 1
63. MULTIPLE CHOICE The width of a soccer field cannot be greater than
100 yards. The area cannot be greater than 13,000 square yards. Which of
the following would you use to find the possible length x of a soccer field?
(A) 100x > 13,000 CD 100x < 13,000
® 100 + x< 13,000 (D) lOOx = 13,000
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.1)
64. b — 12 when b = 43 65. 12 + x when x = 4
y
66. 12 n when n = 4 67. — when y = 30
15 y
WRITING POWERS Write the expression in exponential form.
(Lesson 1.2)
68. 3 • 3 • 3 • 3 • 3 69. seven squared 70. y • y • y • y
71. 9 • 9 • 9 • 9 • 9 • 9 72. twelve cubed 73. 8 d • 8 d • M
NUMERICAL EXPRESSIONS Evaluate the expression. Then simplify the
answer. (Lesson 1.3)
74. 9 + 12 - 4 75. 7 + 56 - 8 - 2 76. 63 - 3 • 3
77. 4 • 2 - 5 78. 3 + 13 - 6 79. 49 - 7 + 2
80. (28 -h 4) + 3 2 81. — + 2 82. 2[(2 + 3) 2 - 10]
ROUNDING Round the number to the underlined place value.
(Skills Review p. 774)
83.5.64 84.0.2625 85.0.45695
86. 15.295 87.758.949 88.32.6582
89.0.325 90.26.96 91.4.0965
1.4 Equations and Inequalities
Translating Words into
Mathematical Symbols
Goal
Translate words into
mathematical symbols.
How long were you on the phone?
Key Words
• translate
• phrase
• sentence
;D unt Summary
€hargeS $36.00
= service .; 6.oc
' berv ' . $ 6.00
^Charges
$ 42.00
In Example 6 you will translate words
into an algebraic equation to find the
length of a long distance phone call.
To solve real-life problems, you
often need to translate words into
mathematical symbols. To do this, look
for words, such as sum or difference ,
that indicate mathematical operations.
i Translate Addition Phrases
Write the phrase as a variable expression. Let x represent the number.
PHRASE TRANSLATION
The sum of 6 and a number 6 + x
8 more than a number x + 8
A number plus 5 x + 5
A number increased by 7 x + 7
Student HeCp
p Reading Algebra
Order is important for
subtraction. "4 less
than a number" means
y — 4, not 4 — y.
I 7
J 2 Translate Subtraction Phrases
Write the phrase as a variable expression. Let y represent the number.
PHRASE TRANSLATION
The difference between 5 and a number 5 — y
1 minus a number 7 — y
A number decreased by 9 y — 9
4 less than a number y — 4
Translate Addition and Subtraction Phrases
Write the phrase as a variable expression. Let x represent the number.
1.11 more than a number 2 . A number decreased by 10
H
Chapter 1 Connections to Algebra
Student HeCp
1 ^ -V
►Vocabulary Tip
Quotient comes from a
word meaning "how
many times." When
you divide you are
finding how many
times one quantity
goes into another.
\ _
Notice that order does not matter for addition and multiplication. “The sum of 6
and a number” can be written as either 6 + x or x + 6. Order is important for
n 4
subtraction and division. “The quotient of a number and 4” means not —.
3 Translate Multiplication and Division Phrases
Write the phrase as a variable expression. Let n represent the number.
PHRASE
TRANSLATION
The product of 9 and a number
9 n
10 times a number
1 On
A number multiplied by 3
3 n
One fourth of a number
1
4 n
The quotient of a number and 6
n
6
7 divided by a number
7
n
Translate Multiplication and Division Phrases
Write each phrase as a variable expression. Let x represent the number.
3. The quotient of 8 and a number 4_ The product of 2 and a number
TRANSLATING SENTENCES In English there is a difference between a phrase
and a sentence. Phrases are translated into variable expressions. Sentences are
translated into equations or inequalities.
PHRASE EXPRESSION
SENTENCE EQUATION OR INEQUALITY
Student Hedp
► Reading Algebra
The word is by itself
means "=."
a Translate Sentences
Write the sentence as an equation or an inequality.
SENTENCE
The sum of a number x and 12 is 16.
TRANSLATION
x H - 12 — 16
The words is less than
mean "<."
h j
The quotient of 15 and a number x is less than 3.
3
Translate Sentences
Write the sentence as an equation or an inequality.
5. The product of 5 and a number x is 25.
6. 10 times a number x is greater than or equal to 50.
1.5 Translating Words into Mathematical Symbols
Student HeQp
► Reading Algebra
In mathematics, the
word difference means
"subtraction."
V, _ j
5 Write and Solve an Equation
a. Translate into mathematical symbols: “The difference between 13 and a
number is 7.” Let x represent the number.
b. Use mental math to solve the equation.
c_ Check your solution.
Solution
a. The equation is 13 — x = 7.
b. Using mental math, you can find that the solution is x = 6.
c. CHECK /
13 — x = 7 Write original equation.
13 — 6 L 7 Substitute 6 for x.
1 — 1 S Solution checks.
Translating sentences into mathematical symbols is an important skill for solving
real-life problems. Try your skills in Example 6.
6 Translate and Solve a Real-Life Problem
You make a long distance telephone call. The rate is $.20 for each minute.
The total cost of the call is $6.00. How long was the call?
Student HeCp
-Y
► Skills Review
For help with decimal
operations, see p. 759.
k _/
Solution
Let x represent the length of the call in minutes.
Rate per Number Cost of
minute of minutes the call
\ i ^
0.20x = 6.00
Ask what number times 0.2 equals 6. Use mental math to find x = 30.
ANSWER ► Your call was 30 minutes long.
Translate and Solve a Real-Life Problem
7. You make a long distance telephone call. The rate is $.10 for each minute.
The total cost of the call is $5.00. How long was the call? Check to see if
your solution is reasonable.
8 . You make a long distance telephone call. The rate is $.20 for each minute.
The total cost of the call is $4.00. How long was the call? Check to see if
your solution is reasonable.
H
Exercises
Guided Practice
Vocabulary Check Consider the phrase seven decreased by a number n.
1 _ What operation does decreased by indicate?
2 . Translate the phrase into a variable expression.
Skill Check Match the phrase with its variable expression. Let x represent the
number.
3. A number increased by 11
A. x —
4. The product of 11 and a number
B. x +
5. The difference of a number and 11
C —
C - 11
6. The quotient of a number and 11
D. 1 lx
Write the sentence as an equation or an inequality.
7. A number x increased by 10 is 24.
8 . The product of 7 and a number y is 42.
9. 20 divided by a number n is less than or equal to 2.
Practice and Applications
TRANSLATING PHRASES Write the phrase as a variable expression. Let x
represent the number.
10. A number decreased by 3
12. The sum of 5 and a number
14. Product of 4 and a number
16. 15 increased by a number
18. 6 less than a number
11. Difference of 10 and a number
13. 9 more than a number
15. Quotient of a number and 50
17. A number plus 18
19. A number minus 7
Student Hedp
>
► Homework Help
Example 1: Exs. 10-19
Example 2: Exs. 10-19
Example 3: Exs. 10-19
Example 4: Exs. 20-31
Example 5: Exs. 32-35
Example 6: Exs. 36-39
v _ )
TRANSLATING SENTENCES Match the sentence with its equation. Let x
represent the number.
20. A number increased by 2 is 4. A. x — 4 = 2
21. The product of 2 and a number is 4. B. x + 2 = 4
x
22. A number decreased by 4 is 2. C. — = 2
23. A number divided by 4 is 2. D. 2x = 4
1.5 Translating Words into Mathematical Symbols
I
Student HeCp
► Homework Help
Extra ^ e 'p
problem solving in
Exs. 24-31 is available at
www.mcdougallittell.com
TRANSLATING SENTENCES Write the sentence as an equation or an
inequality. Let x represent the number.
24. The sum of 20 and a number is 30.
25. A number increased by 10 is greater than or equal to 44.
26. 18 decreased by a number is 6.
27. 35 is less than the difference of 21 and a number.
28. The product of 13 and a number is greater than 60.
29. 7 times a number is 56.
30. A number divided by 22 is less than 3.
31. The quotient of 35 and a number is 7.
WRITING AND SOLVING EQUATIONS Write the sentence as an equation.
Let x represent the number. Use mental math to solve the equation. Then
check your solution.
32. The sum of a number and 10 is 15.
33. 28 decreased by a number is 18.
34. The product of a number and 25 is 100.
35. The quotient of 49 and a number is 7.
Link ta
History
36. The area of the rectangle
is less than or equal to 50 square meters.
Write an inequality for the area using the
dimensions in the diagram.
MICHIGAN land patterns
result from the Ordinance of
1785. The Northwest Territory
became the states of Ohio,
Indiana, Illinois, Michigan,
Wisconsin, and part of
Minnesota.
37. PLANNING A TRIP You want to go to an amusement park. The distance
between your house and the amusement park is 110 miles. Your rate of travel
is 55 miles per hour. Use the formula d = rt to write an equation. Use mental
math to solve the equation for the time you spend traveling.
38. History Link 7 The Land Ordinance
of 1785 divided the Northwest Territory
into squares of land called townships.
Every township was divided into
36 square sections, 1 mile on each side.
How many square miles were in each
township? How many acres?
HINT: 1 mi 2 = 640 acres
CHALLENGE You want to hire a live band for a school dance. You have
$175 in your budget. The live band charges $75 per hour and each
student pays $2 admission.
39. If the band is to play for 3 hours, how much extra money do you need
to raise?
Chapter 1 Connections to Algebra
Standardized Test
Practice
Mixed Review
Maintaining Skills
Quiz 2
40. MULTIPLE CHOICE Translate into mathematical symbols “the difference of
a number and 4 is 10.” Let n represent the number.
(a) n — 4 = 10 (JD 4 — n = 10 Cep 10 — 4 = /? ((d) 10 — n = 4
41, MULTIPLE CHOICE Which is the correct algebraic translation of “Howard’s
hourly wage h is $2 greater than Marla’s hourly wage m?”
CE) h<m + 2 = m + 2 CH) m = h + 2 CD h> m + 2
42, Find the volume of a cube when each side x is 10 feet.
(Lesson 1.2)
CHECKING SOLUTIONS OF EQUATIONS Check to see if the given value
of the variable is or is not a solution of the equation. (Lesson 1.4)
43. Sk ~ 2 = 30; k = 4 44. 15 + 2c = 5c; c = 5
r 2
45. — = 40; r = 9 46. 50 = 3w; w = 15
PERCENTS AND DECIMALS Write the percent as a decimal.
(Skills Review p. 768)
47. 28% 48. 25% 49. 40% 50. 22%
51. 45% 52. 90% 53. 17.4% 54. 6.51%
Check to see if x = 4 is or is not a solution of the equation. (Lesson 1.4)
1.10c- 5 = 35 2. | = 0 3. x 2 + 5 = 21
Check to see if a = 20 is or is not a solution of the inequality. (Lesson 1.4)
4. 3 a> 50
5. 10 + a <30
6. 40 + 3 a> 50
7.f <5
8. — >5
9.f
- 2<5
5
a
5
10. Geoirofr]
f linky The rectangle shown at the
right has an area of 32 square units. Write an
equation to find the width x. Use mental math to
solve the equation. HINT: The area of a rectangle
equals length times width. (Lesson 1.4)
Write the sentence as an equation or an inequality. (Lesson 1.5)
11. A number x divided by 9 is less than 17.
12. The product of 10 and a number x is 50.
13. A number y plus 10 is greater than or equal to 57.
14. A number y minus 6 is 15.
1.5 Translating Words into Mathematical Symbols
A Problem Solving Plan
Using Models
Goal
Model and solve real-life
problems.
How much food did you order?
Key Words
• modeling
• verbal model
• algebraic model
In Chinese restaurants the bill is
sometimes totaled by counting
the number of plates ordered. In
Example 1 you will use an algebraic
model to find out how many plates of
food you ordered.
Writing algebraic expressions, equations, or inequalities that represent real-life
situations is called modeling. First you write a verbal model using words. Then
you translate the verbal model into an algebraic model.
Write a verbal model.
Assign labels.
Write an algebraic model.
Student HeCp
► Study Tip
Be sure you
understand the
problem before you
write a model. For
example, notice that
the tax is added after
the cost of the plates
is figured.
I ^
i Write an Algebraic Model
You and some friends are at a Chinese restaurant. You order several $2 plates
of wontons, egg rolls, and dumplings. Your bill is $25.20, which includes tax
of $1.20. Use modeling to find how many plates you ordered.
Solution
Verbal
Model
Cost per
Number
plate
of plates
Amount of bill
Tax
Labels
Cost per plate = 2
(dollars)
Number of plates = p
(plates)
Amount of bill = 25.20
(dollars)
Tax = 1.20
(dollars)
Algebraic
Model
2 p = 25.20 - 1.20
2 p = 24
p — 12
Write algebraic model.
Subtract.
Solve using mental math.
ANSWER ^ Your group ordered 12 plates of food costing $24.
Chapter 1 Connections to Algebra
A PROBLEM SOLVING PLAN USING MODELS
Verbal
Model
Ask yourself what you need to know to solve the problem. Then
write a verbal model that will give you what you need to know.
▼
Labels
Assign labels to each part of your verbal model.
Algebraic
Model
Use the labels to write an algebraic model based on your verbal
model.
Solve
Solve the algebraic model and answer the original question.
Check
..-
Check that your answer is reasonable.
Student HeCp
^
►Study Tip
Sometimes a diagram
can help you see what
you know and what
you need to find to
solve the problem.
v _ J
2 Write an Algebraic Model
A football field is about 53 yards wide and 120 yards long. A soccer field has
the same area, but is 60 yards wide. How long is the soccer field?
Solution
Write a verbal model showing that the area (width X length) of the soccer field
equals the area (width X length) of the football field.
xyd 120 yd
Verbal
Width of
Length of
Width of
Length of
Model
soccer field
soccer field
football field
football field
Labels Width of soccer field = 60 (yards)
Length of soccer field = x (yards)
Width of football field = 53 (yards)
Length of football field = 120 (yards)
Algebraic 60 a* = 53 • 120 Write algebraic model.
Model
60x = 6360 Simplify.
x — 106 Solve with mental math. (60 • 106 = 6360)
ANSWER ^ The soccer field is 106 yards long.
Write an Algebraic Model
1. You want two rectangular gardens to have equal areas. The first garden is
5 meters by 16 meters. The second garden is 8 meters wide. How long should
the second garden be? Apply the problem solving plan to find the answer.
1.6 A Problem Solving Plan Using Models
link to
Careers
JET PILOTS select a route,
an altitude, and a speed that
will provide the fastest and
safest flight.
More about jet pilots
available at
www.mcdougallittell.com
3 Write an Algebraic Model
JET PILOTS A jet pilot is flying from Los Angeles to Chicago at a speed of
500 miles per hour. When the plane is 600 miles from Chicago, an air traffic
controller tells the pilot that it will be 2 hours before the plane can get
clearance to land.
a. At what speed would the jet have to fly to arrive in Chicago in 2 hours?
b. The pilot knows that at her present altitude, the speed of the jet must be
greater than 322 miles per hour or the plane could stall. Is it reasonable for
the pilot to fly directly to Chicago at the reduced speed from part (a) or
should the pilot take some other action?
Solution
a, You can use the formula (rate)(time) = (distance) to write a verbal model.
Verbal
Speed of jet
• | Time
=
Distance to travel
Model
Labels
Speed of jet -
(miles per hour)
Time = 2
(hours)
Distance to travel = 600
(miles)
Algebraic
Model
2 x = 600 Write algebraic model.
x = 300 Solve using mental math.
ANSWER ► To arrive in 2 hours, the pilot would have to slow the jet down
to a speed of 300 miles per hour.
b. It is not reasonable for the pilot to fly at 300 miles per hour, because the
plane could stall. The pilot should take some other action, such as circling
in a holding pattern, to use some of the time.
You can ignore information that is not needed to solve a problem. To solve
Example 3, you do not need to know that the plane is traveling at 500 miles
per hour.
Write an Algebraic Model
Use the following information to write and solve an algebraic model.
You are running in a marathon. During the first 20 miles, your average speed is
8 miles per hour. During the last 6.2 miles, you increase your average speed by
2 miles per hour.
2 . How long will it take you to run the last 6.2 miles of the marathon? Use the
problem solving plan with models to answer the question.
3. A friend of yours completed the marathon in 3.2 hours. Did you finish ahead
of your friend or behind your friend? Explain.
H ~
Exercises
Guided Practice
Vocabulary Check In Exercises 1 and 2, complete the sentence.
1. Writing expressions, equations, or inequalities to represent real-life situations
is called ? .
2 . A ? model with labels is used to form an algebraic model.
3. Write the steps of the problem solving plan.
Skill Check 4. ADMISSION PRICES Your family and friends are going to an amusement
park. Adults pay $25 per ticket and children pay $15 per ticket. Your group
has 13 children and your total bill for tickets is $370. How many adults are in
your group? Choose the verbal model that represents this situation.
Cost per
adult
Cost per
adult
Number
of adults
-
Cost per
child
•
Number of
children
=
Total
cost
Number
of adults
+
Cost per
child
•
Number of
children
=
Total
cost
Cost per
adult
Number
Cost per
+
Number of
Total
of adults
child
children
cost
Practice and Applications
WALK OR TAKE THE SUBWAY? In Exercises 5-10, use the following
information.
You are one mile from your home. You can walk at a speed of 4 miles per hour.
The subway comes by every 15 minutes, and you heard one come by 3 minutes
ago. The subway ride takes 8 minutes.
5- How many minutes will it take to get home by subway if you take the next
train?
6- Write a verbal model that relates the time it would take to walk home, your
walking speed, and the distance to your home.
? X ? = ?
i
Student HeCp
► Homework Help
Example 1: Exs. 5-20
Example 2: Exs. 5-20
Example 3: Exs. 5-20
7. Assign labels to your verbal model. Use t to represent the unknown value.
8. Use the labels to translate your verbal model into an equation.
9. Use mental math to solve the equation.
10, Which will get you home faster, walking or taking the subway? Explain.
1.6 A Problem Solving Plan Using Models
Link to
Science
PLANT GROWTH Kudzu
was introduced to the United
States in 1876. Today kudzu
covers over 7 million acres of
the southeastern United
Science Link / In Exercises 11-15, use the following information.
Kudzu is a type of Japanese vine that grows at a rate of 1 foot per day during
the summer. On August 1, the length of one vine was 50 feet. What was the
length on July 1? HINT: July has 31 days.
11. Use the verbal phrases to complete the verbal model.
Total length
Original length
? -f ? X ? = ?
Number of days
Growth rate
12. Assign labels to the verbal model. Use x to represent the unknown value.
13. Choose the algebraic model that best represents the verbal model.
A. (x + 1) • 31 = 50 B. x + (1 • 31) = 50
C.x = 50- 1 D. x + (1 + 31) = 50
14. Use mental math to solve the algebraic model you chose in Exercise 13.
15. Check that your answer is reasonable.
Student MeCp
► Homework Help
Extra help with
problem solving in
Exs. 16-20 is available at
www.mcdougallittell.com
BUYING A STEREO In Exercises 16-20, use the following information.
An appliance store sells two stereo models. The model without a CD player is
$350. The model with a CD player is $480. Your summer job allows you to
save $50 a week for 8 weeks. At the end of the summer, you have enough to
buy the stereo without the CD player. How much would you have needed to
save each week to buy the other model?
16. Write a verbal model that relates the number of weeks worked, the amount
you would have needed to save each week, and the price of the stereo with
the CD player.
? x ? = ?
17. Assign labels to your verbal model. Use m to represent the unknown value.
18. Use the labels to translate your verbal model into an equation.
19. Use mental math to solve the equation.
20. Check that your answer is reasonable.
21. CHALLENGE You are running for class president. By two o’clock on
election day you have 95 votes and your opponent has 120 votes. Forty-five
more students will be voting. Let x represent the number of students (of the
45) who vote for you.
a. Write an inequality that shows the values of x that will allow you to win
the election.
b. What is the smallest value of x that is a solution of the inequality?
Chapter 1 Connections to Algebra
Standardized Test
Practice
Student HeCp
►Test Tip
In Exercise 22, you can
use unit analysis (p. 8)
to see which answer
choice is correct.
% _ ^
Mixed Review
Maintaining Skills
22. MULTIPLE CHOICE A jet
is flying from Baltimore to
Orlando at a speed r of
500 miles per hour. The
distance d between the two
cities is about 793 miles.
Which equation can be used
to find the time / it takes to
make the trip?
(A) 793 = 500/
CD
500
793
CD 793/ = 500
CD / = 793(500)
Orlando
23. MULTIPLE CHOICE Jim lives in a state in which speeders are fined $25 for a
speeding ticket plus $10 for each mile per hour over the speed limit. Jim was
given a ticket for $175 for speeding in a 45 mile per hour zone. Which
equation can be used to find how fast Jim was driving?
CD 45 + (jc - 45) • 10 = 175 - 25 CD x ~ 45 • 10 = 175 - 25
CD (x + 45) • 10 = 175 - 25 Q) (* ” 45) • 10 = 175 - 25
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.2)
24. x 2 — 2 when x — 1 25. (2x) 3 when x — 5 26. (10 — x) 2 when x = 6
NUMERICAL EXPRESSIONS Evaluate the expression. (Lesson 1.3)
27. 22 - 4 2 v 2 28. 4 + 8 • 4 - 1 29. 2 • 4 + (7 - 3)
CHECKING SOLUTIONS Check to see if the given value of the variable is
or is not a solution of the equation or the inequality. (Lesson 1.4)
30. 2x — 3 < 15; x = 9 31. 3x + 4 < 16; x = 4 32. 16 + x 2 -r- 4 = 17; x = 2
33. FUNDRAISING Your fundraising group earns 250 for each lemonade and
500 for each taco sold. One hundred tacos are sold. Your total profit is $100.
How many lemonades are sold? Write an equation that models this situation.
Solve the equation using mental math. (Lesson 1.5)
FRACTIONS AND MIXED NUMBERS Write the improper fraction as a
mixed number. (Skills Review p. 763)
34. f
35
39 - f
37 ^
6
33. f
39- f
40. f
„„ 15
41 t
43. f
43. M
44 51
36
4B.f
1.6 A Problem Solving Plan Using Models
Tables and Graphs
Goal
Organize data using a
table or graph.
Key Words
• data
• bar graph
• line graph
How much does it cost to
make a movie?
Almost every day you have the chance
to interpret data that describe real-life
situations. In Example 3 you will
interpret data about the average cost
of making a movie.
Data are information, facts, or numbers
that describe something. It is easier to
see patterns when you organize data in
a table.
Student HeCp
► Study Tip
To find how much dairy
was consumed in 1980,
you go across the row
labeled Dairy and stop
at the column for 1980.
k j
i Organize Data in a Table
The table shows the top three categories of food eaten by Americans.
Top Categories of Food Consumed by Americans (lb per person per year)
Year
1970
1975
1980
1985
1990
1995
2000
Dairy
563.8
539.1
543.2
593.7
568.4
584.4
590.0
Vegetables
335.4
337.0
336.4
358.1
382.8
405.0
410.0
Fruit
237.7
252.1
262.4
269.4
273.5
285.4
290.0
DATA UPDATE of U.S. Department of Agriculture at www.mcdougallittell.com;
2000 data are estimated by authors.
Make a table showing total dairy and vegetables consumed (pounds per person)
per year. In which year did Americans consume the least dairy and vegetables?
In which year did Americans consume the most dairy and vegetables?
Solution
To make the table, add the data for dairy and vegetables for the given year.
Year
1970
1975
1980
1985
1990
1995
2000
Total
899.2
876.1
879.6
951.8
951.2
989.4
1000.0
ANSWER ► The least consumption was in 1975 and the greatest in 2000.
Organize Data in a Table
1. Make a table showing the total dairy products, vegetables, and fruit
consumed (pounds per person) per year. Which year had the least
consumption? Which had the greatest consumption?
B “™
Student Hedp
►Vocabulary Tip
Horizontal bars go
across parallel with
the horizon. Vertical
bars go straight up
and down.
N _ J
BAR GRAPHS One way to represent the data in a table is with a bar graph. The
bars can be either vertical or horizontal. Example 2 shows a vertical bar graph of
the data from Example 1.
2 Interpret a Bar Graph
The bar graph shows the total amount of dairy products, vegetables, and fruit
consumed by the average American in a given year. It appears that Americans
ate about five times the amount of dairy products, vegetables, and fruit in 1995
as compared with 1970.
If you study the data in Example 1, you can see that the bar graph could be
misinterpreted. Explain why the graph could be misinterpreted.
Consumption of Dairy, Vegetables, and Fruit
o
(/)
Q)
O
E
3
GO
3
O
o
Solution
The bar graph could be misinterpreted because the vertical scale is not
consistent. The zigzag line shows a break where part of the scale is not shown.
Because of the break, the first tick mark on the vertical scale represents 1125
pounds of food consumed per person. The other tick marks on the vertical scale
represent 25 pounds of food consumed per person.
To make a bar graph that could not be misinterpreted, you must evenly space
the tick marks and make sure that each tick mark represents the same amount.
Make and Interpret a Bar Graph
Student HeCp
►Skills Review
For help with drawing
bar graphs, see p. 777.
i J
2_ The bar graph at the
right is set up so that it
is not misleading. The
first two bars are drawn
for you.
Copy and complete the
bar graph using the
data from Example 2.
Describe the pattern
from 1970 through 2000.
Total of Dairy, Vegetables,
and Fruit Consumed
1400
1200
~ 1000
o
f 1 800
S -I- 600
o £ 400
200
0
II
Year
1.7 Tables and Graphs
DIRECTORS OF
PHOTOGRAPHY decide the
type of film and equipment
used and the composition of
the movie.
More about movie
making is available at
www.mcdougallittell.com
LINE GRAPHS As an alternative to a vertical bar graph, data is sometimes
represented by a line graph. Here the vertical bars are replaced by a single point
located at the top of the bar. These points are then connected by line segments.
Line graphs are especially useful for showing changes in data over time.
3 Make and Interpret a Line Graph
MOVIE MAKING From 1983 to 1996, the average cost (in millions of dollars)
of making a movie is given in the table. Draw a line graph of the data. Then
determine in which three years did the cost decrease from the prior year.
Average Cost of Making a Movie
Year
1983
1984
1985
1986
1987
1988
1989
Cost (millions)
$ 11.8
$ 14.0
$ 16.7
$ 17.5
$ 20.0
$ 18.1
$ 23.3
Year
1990
1991
1992
1993
1994
1995
1996
Cost (millions)
$ 26.8
$ 26.1
$ 28.9
$ 29.9
$ 34.3
$ 36.4
$ 33.6
► Source: International Motion Picture Almanac
Solution
Draw the vertical scale from 0 to 40 million dollars. Mark the number of years
on the horizontal axis starting with 1983. For each average cost in the table,
draw a point on the graph. Then draw a line from each point to the next point.
Student HeCp
~
► Study Tip
This point represents
the year 1983 and the
cost$11.8 million.. .
\ __ /
Average Cost of Making a Movie
In 1988, 1991, and 1996 the average cost of making a movie decreased from
the prior year.
Make and Interpret a Line Graph
3. Make a line graph of the data above changing the tick marks on the vertical
scale to 0, 10, 20, 30, and 40. Which graph is easier to interpret? Why?
Chapter 1 Connections to Algebra
Exercises
Guided Practice
Vocabulary Check
1. Explain what data are. Give an example.
2 . Name two ways to display organized data.
Skill Check
WEATHER Use the graph to
classify the statement as
true or false.
3. Rainfall increases each month
over the previous month.
4. The amount of rainfall is the
same in May and July.
5. The greatest amount of rainfall
occurs in August.
► Source: National Oceanic and Atmospheric Administration
Practice and Applications
GOLF In Exercises 6 and 7, use the table showing scores for two rounds
of golf.
Player 1
Player 2
Player 3
Player 4
Round 1
90
88
79
78
Round 2
94
84
83
80
Student HeCp
► Homework Help
Example 1: Exs. 6-10
Example 2: Exs. 11-14
Example 3: Exs. 15-18
v _ J
6. Make a table showing the average score of each player. HINT: Find each
average by adding the two scores and dividing by the number of rounds.
7. Which player has the lowest average? Which one has the highest average?
8- SCHOOL ENROLLMENT The table shows the number of students (in
millions) enrolled in school in the United States by age. Make a table
showing the total number of students enrolled for each given year.
Age
1980
1985
1990
1995
2000
14-15 years old
7282
7362
6555
7651
8100
16-17 years old
7129
6654
6098
6997
7600
18-19 years old
3788
3716
4044
4274
4800
► Source: U.S. Bureau of the Census; 2000 data are estimated by authors.
9. Which year had the least number of students enrolled? Which had the
greatest number of students enrolled?
10, Did the total enrollment increase for each 5 year period? Explain.
1.7 Tables and Graphs
BRAKING DISTANCE is the
distance it takes for a vehicle
to come to a complete stop
after the brakes have been
activated. The length of a skid
mark indicates the speed at
which a vehicle was traveling.
BRAKING DISTANCE In Exercises 11-13, use the bar graph showing
average braking distances for medium sized cars.
11. Estimate the braking distance for
a car traveling 50 miles
per hour.
12, Does it take twice as far to stop
a car that is going twice as fast?
Explain.
13- Explain why it would be dangerous
to follow another car too closely
when driving at 70 miles
per hour.
Average Braking Distance
20
0 100 200 300 400
Braking distance (ft)
14- Scie nce Link y The table shows the number of gallons of water needed to
produce one pound of some foods. Make a bar graph of the data.
Food (1 lb)
Lettuce
Tomatoes
Melons
Broccoli
Corn
Water (gallons)
21
29
40
42
119
► Source: Water Education Foundation
MINIMUM WAGE
In Exercises 15-17, use the line
graph showing the minimum
wage for 1991-1999.
15. For how many years did the
minimum wage remain the
same as it was in 1991?
16. Estimate the minimum wage
during 1992.
17. In which year did the minimum
wage first increase to over $5?
1
-=■ 6
3
O r
= ederal Minimum Wage
3
| 4
1 3
€ 2
s. 1
1 0
In June of year
>
► Source: U.S. Bureau of Labor Statistics
Student Hedp
► Homework Help
Extra help with
problem solving
in Ex. 18 is available at
www.mcdougallittell.com
J
18. History Link/ The table shows the population (in thousands) of California
following the Gold Rush of 1849. Make a line graph of the data.
Year
1850
1860
1870
1880
1890
Population
93
380
560
865
1213
19. CRITICAL THINKING The table shows the average fuel efficiency for
passenger cars for different years. Organize the data into a graph. Explain
why you chose the type of graph you used.
Year
1980
1985
1990
1995
2000
Fuel efficiency (miles per gallon)
24.3
27.6
28.0
28.6
29.2
* DATA UPDATE of National Highway Traffic Safety Administration at www.mcdougallittell.com;
2000 data are estimated by authors.
Chapter 1 Connections to Algebra
Standardized Test
Practice
Mixed Review
Maintaining Skills
20. MULTIPLE CHOICE Which way of organizing data is useful for showing
changes in data over time?
(A) Table (ID Line graph (Cp Circle graph (TD None of these
MULTIPLE CHOICE In Exercises 21 and 22, use the bar graph showing
one household's monthly electricity usage in kilowatt-hours (kWh).
Monthly Electricity Use
J FMAMJJASOND
Month
21. Which month shows the greatest decrease in use from the prior month?
CD May CD October
CE) June GD November
22 . About how many total kilowatt-hours were used for the months of January
through April?
(A) 480 CJD 400 ® 550 C® 600
Geometry Lmu Find the perimeter and area of the geometric figure.
(Lesson 1.1)
23.
14 in.
7 in. 6m 6m 7ft
CHECKING SOLUTIONS Check to see if x = 5 is or is not a solution of
the equation or the inequality. (Lesson 1.4)
26. 17 - jc < 12 27. x + 3x > 18 28. 5v - 2 = 12.5
29. 2.5 > l.2x - 3 30. x 2 = 25 31. (3x) 2 < 255
32. 3x + 2x = 25 33. 19 — 2x > 10 34. 16 < 3x + 1
COMPARING DECIMALS Compare using <, > or =.
(Skills Review p. 770)
35. 71.717 @77.117
38. 1.666 ? 1.67
41.0.48 @0.479
36. 2.6 @ 2.65
39. 15.7 ? 15.700
42. 3.11 @3.09
37. 0.01 @ 0.0001
40.0.4321 ? 0.434
43.9.54 ? 9.540
1.7 Tables and Graphs
An Introduction to Functions
Goal
Use four different ways
to represent functions.
Key Words
• function
• input
• output
• input-output table
• domain
• range
What is the altitude of the balloon?
You are in a hot-air balloon. You
rise at a steady rate of 20 feet per
minute. In Example 2 you will
use the relationship between time
and height to find the altitude of
the balloon after a given number
of minutes.
A function is a rule that establishes a relationship between two quantities, called
the input and the output. For each input, there is exactly one output—even
though two different inputs may give the same output.
One way to describe a function is to make an input-output table.
i Make an Input-Output Table
GEOMETRY LINK The diagram shows the first six triangular numbers.
4
4
4
44
•
44
4 44
*
44
444
4444
4*
444
4444
44444
m
4 #
444
4444
44444
#44444
t
3
6
to
16
2t
Figius t
Fiijitf & 2
Figure 3
Figure 5
Fifjiue €
a. Using the first six figures, make an input-output table in which
the input is
the figure number n and the output is the triangular number T.
b. Does the table represent a function? Justify your answer.
Solution
a. Use the diagram to make an input-output table, as shown below.
Input #7
1
2
3
4
5
6
Output T
1
3
6
10
15
21
b. This is a function, because for each input there is exactly one output.
Chapter 1 Connections to Algebra
DOMAIN AND RANGE The collection of all input values is the domain of the
function and the collection of all output values is the range of the function. The
domain of the function in Example 1 is 1, 2, 3, 4, 5, 6; the range of the function is
1,3,6, 10, 15,21.
When you are given the rule for a function, you can prepare to graph the function
by making a table showing numbers in the domain and their corresponding
output values.
2 Use a Table to Graph a Function
BALLOONING You are at an altitude of 250 feet in a hot-air balloon. You turn
on the burner and rise at a rate of 20 feet per minute for 5 minutes. Your
altitude h in feet after you have risen for t minutes is given by
h = 250 + 20 1, where t > 0 and t < 5.
a, Use the function to find the output h in feet for several inputs. Then
organize the data into an input-output table.
b_ Use the data in the table to draw a graph that represents the function.
Solution
a. Find the outputs for t = 0,1, 2, 3, 4, and 5. Then make a table,
INPUT (MINUTES)
t = 0
t = 1
t = 2
t = 3
t = 4
t = 5
Input t
Output h
FUNCTION
h = 250 + 20(0)
h = 250 + 20(1)
h = 250 + 20(2)
h = 250 + 20(3)
h = 250 + 20(4)
h = 250 + 20(5)
OUTPUT (FEET)
h = 250
h = 270
h = 290
h = 310
h = 330
h = 350
0
1
2
3
4
5
250
270
290
310
330
350
Student HeCp
► Study Tip
To plot the first point
( t ; h) find t = 0 on the
horizontal axis. Then
find h = 250 on the
vertical axis. Mark the
point (0, 250). .
Altitude of Balloon
W
0
0 1 2 3 4 5
Time (min)
Let the horizontal axis represent the input
t (in minutes). Label the axis from 0 to 5.
Let the vertical axis represent the output
h (in feet). Label the axis from 0 to 400.
Plot the data points given in the table.
Finally, connect the points.
The graph shows that as the time increases,
the height of the balloon increases.
The graph represents the function
h = 250 + 20^, where t > 0 and t < 5.
v».
1.8 An Introduction to Functions
Student MeCp
r
► More Examples
More examples
are available at
www.mcdougallittell.com
3 Write an Equation to Represent a Function
SCUBA DIVING As you dive deeper and deeper into the ocean, the pressure of
the water on your body steadily increases. The pressure at the surface of the
water is 14.7 pounds per square inch (psi). The pressure increases at a rate of
0.445 psi for each foot you descend. Write an equation to represent the pressure
P as a function of the depth d for every 20 feet you descend until you reach a
depth of 60 feet.
Solution
Verbal
Pressure at
Pressure
+
Rate of change
Model
given depth
at surface
in pressure
Diving
depth
Labels
Algebraic
Model
Pressure at given depth = P (psi)
Pressure at surface = 14.7 (psi)
Rate of change in pressure = 0.445 (psi per foot of depth)
Diving depth = d (feet)
P = 14.7 + 0.445 d where d > 0 and d < 60
ANSWER ► The function can be represented by the equation
P = 14.7 + 0.445J, where d > 0 and d < 60.
Represent a Function
Use the algebraic model from Example 3.
1. Make an input-output table for the function. Use d = 0, 20, 40, and 60.
2 . Draw a graph that represents the function.
Four Ways to Represent Functions
INPUT-OUTPUT TABLE
Input 17
Output P
1
1
2
3
3
6
4
10
5
15
6
21
WORDS
You are in a hot-air
balloon at a height
of 250 feet. You
begin to rise higher
at a rate of 20 feet
per minute for a
period of 5 minutes.
EQUATION
h — 250 + 20 f,
where t > 0 and
t < 5
GRAPH
Altitu ds of Balloon
400
0 12 3 4 5
Timet min)
H
Chapter 1 Connections to Algebra
Exercises
Guided Practice
Vocabulary Check Complete the sentence.
1. A function is a relationship between two quantities, called the ? and
the ? .
2 . The collection of all input values is the ? of the function.
3- The collection of all output values is the ? of the function.
Skill Check CAMPING In Exercises 4-6, use the following information.
You are going camping. The cost for renting a cabin at Shady Knoll Campground
is $65.00 plus $12.00 per person. The cost in dollars is
C = 65 + 12/z, where n is the number of people.
4. Copy and complete the input-output table.
Input I?
1
2
3
4
5
6
Output C
?
?
?
?
?
?
5. Draw a graph that is made up of isolated points representing the cost of
renting a cabin.
6. Determine the range of the function from the given input values in the
input-output table.
Practice and Applications
INPUT-OUTPUT TABLES Make an input-output table for the function. Use
0, 1,2, 3, 4, and 5 as values for x.
7. y = 6x + 5 8. y = 26 — 2x 9. y = (x + 3) • 7
10 - y = 85 — 15x 1 1 - y = 5(15 — x) 12 . y = 2(6x + 10)
Student HeCp
► Homework Help
Example 1: Exs. 7-12,
15-20, 23, 24
Example 2: Exs. 13-15,
21, 23-25
Example 3: Exs. 23, 24
l _>
LINE GRAPHS Draw a line graph to represent the function given by the
input-output table.
13.
Input x
1
2
3
4
5
6
Output y
14
12
10
8
6
4
14.
Input x
1
2
3
4
5
6
Output y
8
11
14
17
20
23
1.8 An Introduction to Functions
Student MeCp
► Homework Help
Extra help with
v problem solving in
Exs. 16-19 is available at
www.mcdougallittell.com
15. Science Ls The distance d (in miles) that sound travels in air in time t
(in seconds) is represented by the function d = 0.2 1. Make a table of the
input t and the output d. Use t values of 0, 5, 10, 15, 20, 25, and 30. Use your
table to help you draw the graph of the function.
CRITICAL THINKING Determine whether the table represents a function.
Input
Output
1
3
2
4
3
5
Input
Output
2
2
3
4
4
6
Input
Output
1
2
3
3
3
4
Input
Output
1
3
1
4
2
5
Link
History
JESSE CHISHOLM, the
person for whom the
Chisholm Trail is named,
was a trader who was
part Cherokee.
CAR RACING In Exercises 20-22, use the following information.
The fastest winning speed in the Daytona 500 is about 178 miles per hour.
In the table below, calculate the distance traveled d (in miles) after time
t (in hours) using the equation d = 178 1.
20 . Copy and complete the input-output table.
Time (hours)
0.25
0.50
0.75
1.00
1.25
1.50
Distance traveled (miles)
?
?
?
?
?
?
21. Use the data to draw a graph.
22. For what values of t does the formula d = 178/ correspond to the situation
being modeled?
23. Hist ory Link / In 1866 Texas
cowhands used the Chisholm Trail
to drive cattle north to the railroads
in Kansas. The average rate r that
the cattle could be moved along the
trail was 11 miles per day.
a. Write an equation, where d is
distance and t is time in days.
b. Make a table of input t and
output d for / = 7, 14, and 28.
Then graph the data.
c. The distance d from San Antonio
to Abilene was about 1100 miles.
How long did it take to drive
cattle the entire length of the trail?
Chapter 1 Connections to Algebra
SCUBA DIVERS must take
an instructional class in order
to become certified.
Standardized Test
Practice
24. SCUBA DIVERS While you are on vacation, you want to rent scuba
equipment. It costs about $90 a day to rent the equipment. Find the cost of
renting equipment for 1, 2, 3, and 4 days.
a. Write an equation where R is the total rental cost and n is the number of
days. Make an input-output table.
b. Draw a graph that represents the function.
25. WATER TEMPERATURE The table below gives the temperature of water as
it cools. Using this table, draw a graph that estimates the temperature of the
water for t > 0 and t < 25.
Time (minutes)
0
5
10
15
20
25
Temperature (°C)
100
90
81
73
66
60
26. CHALLENGE The function y — x 2 has a U-shaped graph called a parabola.
If the domain of this function is given as v > 0 and x<4, find the range.
27. MULTIPLE CHOICE Which table does not represent a function?
(A)
CD
Input
Output
CD
Input
Output
1
3
1
2
2
3
2
4
3
3
3
6
4
3
4
8
CD
Input
Output
Input
Output
5
4
5
1
6
4
5
3
7
5
6
1
8
5
6
3
28. MULTIPLE CHOICE Which function has an output of j = 27 for an input
of a = 3?
CD j = 4a + 15 CD j = 15 a + 4 CED j =15 • 4 a Cj ) j = 27 a
29. MULTIPLE CHOICE Which function is best represented by the graph?
(a) F — 50 + 25 1
CD F = 25 + t
CD F = 25 + 50;
CD F = 25t
■
1.8 An Introduction to Functions
Mixed Review
Maintaining Skills
Quiz 3
H
EVALUATING EXPRESSIONS Evaluate the variable expression when
a = 3 and c = 5. (Lessons 1.1 and 1.2)
31 .(a + c) 2 3 4 5 6 32 .a 2 + c 2
30. a + c
33. ac
34. a • (c 2 )
35. (a 2 )
36. TRANSLATING PHRASES Write a variable expression for the phrase
9 decreased by a number n. (Lesson 1.5)
37. TRANSLATING SENTENCES Write the inequality for the sentence: The
quotient of 72 and a number x is greater than 7. (Lesson 1.5)
ADDING FRACTIONS Add. Write the answer as a fraction or a mixed
number in simplest form. (Skills Review p. 764)
38 ' 9 + 9
__ 5 , 1
39 -T2 + T2
12 , 7
40 -T5 + T5
41 ii+2
3 3
43.1 + i
..3,1
44.5 + 5
3.9
45. T + T
4 4
9.3
46 -T4 + T4
1. RECYCLING A recycling center pays 50 apiece for aluminum cans and
certain glass bottles. Jean has four cans and the total amount paid for her
collection of cans and bottles is 500. Use a verbal model to find how many
glass bottles are in Jean’s collection. (Lesson 1.6)
ARTS ACTIVITIES In Exercises 2 and 3, use the table showing the
percent of 18-to-24-year-olds that attended various arts activities at
least once a year. (Lesson 1.7)
Arts Activities Attended by 18-to-24-year-olds
Jazz
Musical play
Non-musical play
Art museum
Historic park
15%
26%
20%
38%
-,
46%
2 . Make a bar graph of the data.
3. What conclusions can you draw from the bar graph?
HOT-AIR BALLOONS You are at an altitude of 200 feet in a hot-air
balloon. You rise at a rate of 25 feet per minute for 4 minutes. Your
altitude h (in feet) after you have risen for t minutes is given by
h = 200 + 25f, where t > 0 and t < 4. (Lesson 1.8)
4. Make an input-output table using 0, 1,2, 3, and 4 as values for x.
5. Use your table to draw a graph that represents the function.
6 . Determine the range of the function.
Chapter 1 Connections to Algebra
Chapter Summary
and Review
• variable, p. 3
• variable expression, p. 3
• value, p. 3
• numerical expression, p. 3
• evaluate, p. 4
• power, p. 9
• exponent, p. 9
• base, p. 9
• grouping symbols, p. 10
• order of operations, p. 15
• left-to-right rule, p. 16
• equation, p. 24
• solution, p. 24
• inequality, p. 26
• modeling, p. 36
• verbal model, p. 36
• algebraic model, p. 36
• data, p. 42
—
• bar graph, p. 43
• line graph, p. 44
• function, p. 48
• input, p. 48
• output, p. 48
• input-output table, p. 48
• domain, p. 49
• range, p. 49
Variable s in Algebra
Examples on
pp. 3-5
Evaluate the variable expression when y = 4.
a. 10- y= 10-4 b. lly = 11(4)
= 6 =44 =4
16 16 ■ , n , i n
c. — = — d. y + 9 = 4 + 9
y 4 y
= 13
Evaluate the expression for the given value of the variable.
YYl
1 - a + 14 when a = 6 2. 18x when x = 2 3. — when m = 18
4_ when y = 3 5- p — 12 when p = 22 6- 5/? when b = 6
7. You are walking at a rate of 3 miles per hour. Find the distance you travel in
2 hours.
8_ You hike at a rate of 2 miles per hour. Find the distance you travel in 6 hours.
9_ A race car driver maintains an average speed of 175 miles per hour. How far
has she traveled in 3 hours?
Oeomefry Unkfa Find the perimeter of the geometric figure.
Chapter Summary and Review
Chapter Summary and Review continued
1.2 Exponents and Powers
Examples on
pp. 9-11
Evaluate the variable expression when b — 3.
b. (10 - b ) 3 = (10 - 3) 3
II
•
= 7 3
= 9
= 7 •'
= 343
c. 10(5*) = 10(5 3 )
= 10(5 ‘5*5)
= 10(125)
= 1250
Write the expression in exponential form.
12 . eight to the fourth power 13 . six cubed 14 . 5 • 5 • 5 • 5 • 5
Evaluate the expression for the given value of the variable.
15 . x 4 when x — 2 16 . (5v) 3 when x — 5 17 . 6 + (/? 3 ) when b — 3
1.3
Order of Operations
Examples on
pp. 15-17
Evaluate 550 - 4(3 + 5) 2 .
550 — 4(3 + 5) 2 = 550 — 4(8) 2 Add numbers within grouping symbols.
= 550 — 4 • 64 Evaluate the power.
= 550 — 256 Multiply.
= 294 Subtract.
Evaluate the numerical expression.
18.9 + (3 - 2) - 3 2
15-6
21 .
6 + 3 2 - 12
19. (14 - 7) 2 + 5
_28 + 4
22.-t—
20. 6 + 2 2 - (7 - 5)
3 3 + 7
23.
4 • 2
Equations and Inequalities
Examples on
pp. 24-26
Check to see if x = 4 is a solution of the equation 5x + 3 = 18 or the inequality
lx — 5 > 20.
Substitute: 5(4) + 3 2= 18 Simplify: 23 ^ 18 Conclusion: False, 4 is not a solution.
Substitute: 7(4) — 5 > 20 Simplify: 23 > 20 Conclusion: True, 4 is a solution.
Chapter 1 Connections to Algebra
Chapter^ Summary and Review continued
1.5
1.6
Check to see if the given value of the variable is or is not a solution of the
equation or the inequality.
24. 2a — 3 = 2; a = 4 25. x 2 — x = 2; x = 2
26. 9y — 3 > 24; y = 3 27. 5x + 2 < 27; x = 5
Use mental math to solve the equation.
28. w + 7 = 15 29. 10 - r = 7
31.J = 4
32. 16 + k = 20
30. 4/7 = 32
33. 10g = 100
Translating Words Into Mathematical Symbols
Examples on
pp. 30—32
Write the phrase or sentence as a variable expression, an equation, or an
inequality. Let v represent the number.
A number increased by 10
The difference of 15 and a number is 8.
x + 10
15 -x = 8
The quotient of a number and 7
The product of 5 and a number is less than or equal to 10.
x_
7
5x< 10
Write the phrase or sentence as a variable expression, an equation, or an
inequality. Let x represent the number.
34, 27 divided by a number is 3. 35, A number plus 30
36, A number times 8 is greater than 5. 37. A number decreased by 9
A Problem Solving Plan Using Models
Examples on
pp. 36-38
You can model problems like the following: If you can save $5.00 a week, how
many weeks must you save to buy a CD that costs $15.00?
Verbal
Model
Amount saved per week
•
Number of weeks
=
Cost of CD
Labels
Amount saved per week = 5
Number of weeks = w
Cost of CD = 15
5 w = 15 Write algebraic model.
w = 3 Solve with mental math.
ANSWER ► You must save for 3 weeks.
(dollars per week)
(weeks)
(dollars)
Algebraic
Model
Chapter Summary and Review
■
Chapter Summary and Review continued
38, You are given $75 to buy juice for the school dance. Each bottle of juice costs
$.75. Write a verbal and an algebraic model to find how many bottles of juice
you can buy. Then use mental math to solve the equation.
Tables and Graphs
Examples on
pp. 42-44
The table shows the number of tennis titles won by United States
women. Write an inequality to determine if the number of Wimbledon titles won
by United States women is greater than the number of Australian Open titles plus
the number of French Open titles.
Event
Number of Titles
Australian Open
14
French Open
25
Wimbledon
43
► Source: USA Today as of July 1999
Inequality 43 > 14 + 25
43 >39
ANSWER ► The number of Wimbledon titles won
is greater.
39. Make a bar graph of the data showing the percent of the voting-age population
that voted. Write an inequality to determine if the percent in 1996 plus the
percent in 1992 is less than the percent in 1976 plus the percent in 1984.
Percent of Voting-Age Population That Voted for President 1976
i-1996
Year
1976
1980
1984
1988
1992
1996
Percent
53.5
52.8
53.3
50.3
55.1
48.9
► Source: US Bureau of the Census
An Introduction to Functions
Examples on
pp. 48-50
Make an input-output table for the equation
C = 5n + 10
where n = 1, 2, 3, and 4. Then determine the range of the
function from the given input values in the table.
ANSWER ► The range for the input values in the table is 15, 20, 25, 30.
If
1
15
2
20
3
25
4
30
40. The perimeter P for rectangular picture frames with side lengths of 2 w and
3 w is given by the function P = 4w + 6w. Make an input-output table that
shows the perimeter when w = 1, 2, 3, 4, and 5. Then determine the range of
the function from the values in the table.
Chapter 1 Connections to Algebra
Chapter Test
Evaluate the variable expression when y = 3 and x = 5.
1-5 y + x 2 2. — x 3- 2y + 9x — 7 4. (5y + x) -r- 4
In Exercises 5-7, write the expression in exponential form.
5- 5y • 5y • 5y • 5y 6- nine cubed 7. six to the nth power
8. Insert grouping symbols in 5 • 4 + 6 so that the value of the expression is 50.
9- TRAVEL If you can travel only 35 miles per hour, is 3 hours enough time to
get to a concert that is 100 miles away? Give the expression you used to find
the answer.
Write the phrase or sentence as a variable expression, an equation,
or an inequality.
10. 7 times a number n 11. x is at least 90. 12. quotient of m and 2
13. y decreased by 3 14. 8 minus s is 4. 15. 9 is less than t.
In Exercises 16-21, decide whether the statement is true or false.
16. (2 • 3) 2 = 2 • 3 2 17. 8 — 6 = 6 — 8 18. The sum of 1 and 3 is 4.
19. x 3 = 8 when x = 2 20. 9x > x 3 when x = 3 21. 8 < y 2 when y — 3
22 . The senior class is planning a trip that will cost $35 per student. If $3920
has been collected, how many seniors have paid for the trip?
MARCHING BAND In Exercises 23 and 24, use the following information.
Members of the marching band are making their own color-guard flags. Each
rectangular flag requires 0.6 square yards of material. The material costs
$2 per square yard.
23. Write a verbal model that relates the number of flags and the total cost of
the material.
24. How much will it cost to make 20 flags?
PET OWNERS In Exercises 25
and 26, use the table showing
the number of pet owners in
your eighth-grade class.
25. Draw a bar graph of the data.
26. From the bar graph, what is the
most popular household pet?
Kind of pet
Number of pet owners
Hamster
7
Dog
12
Cat
15
Bird
4
Fish
5
Chapter Test
Chapter Standardized Test
^fastTip
C^c£jDCE>CjD
Avoid spending too much time on one question. Skip questions that are too
difficult for you, and spend no more than a few minutes on each question.
1. Which of the following numbers is a solution of the inequality 9 — x > 2?
CS) 7 (B) 8 CS 9 CS) 10
2_ What is the value of the expression [(5 • 9) -r- x] + 6 when x = 3?
CS) 5 GD 15 CD 18 CD 21
3. Which variable expression is a translation of “5 times the difference of 8 and
a number x”?
CS) 5(8 — x) Cb) x — 5 • 8 CD 5 • 8 — x CD 5 — 8x
4. The number of students on the football team is 2 more than 3 times the number
of students on the basketball team. If the basketball team has y students, write a
variable expression for the number of students on the football team.
CS) 3y Cb) 3y — 2 Cc) 2 + 3 y CD 2y + 3
5- Which of the following represent a function?
CS) All Cb) I and II Cc) I and III CS) II and III CD None of these
Use the graph to compare the amount of chocolate
eaten in different countries.
6 . About how many more pounds of chocolate per person
is consumed in Switzerland than in the United States?
CS) 7 ® 6 CD 9 CS 3
7. About how many more pounds of chocolate per person
is consumed in Norway than in the United States?
CS) 1.1 CD 5.7 CD 6.8 CD 7.8
► Source: Chocolate Manufacturers Association
Annual Consumption of Chocolate
Switzerland
Austria
Germany
Norway
United States
0 10 12 14 16 18 20 22
Pounds per person
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
i Add and Subtract Decimals
Add 3.52 and 12.698. Subtract 8.28 and 4.095.
Solution
Line up decimals.
Write a zero in the
thousandths place.
Add columns from
right to left.
Solution
1 17 10 Line up decimals.
8.2 8 0 ^ — Write a zero in the
—4 .0 9 5 thousandths place.
4.185 Borrow from the
left and subtract
from right to left.
1 1
3.5 2 0 —
+ 12.698
16.218
Try These
Add or Subtract.
1. 2.3 + 0.4
5. 3.006 + 2.8
9. 123.5 + 32.3
2 . 3.5 - 2.1
6 . 4.25 - 0.08
10 . 32.8 - 12.21
3. 8.75 + 3.35
7. 3.99 + 0.254
11. 0.09 + 0.9
4. 10.6 - 2.6
8 . 6.2 - 0.17
12 . 17.0 - 16.5
Student tteCp
p Extra Examples
More examples
and practice
exercises are available at
www.mcdougallittell.com
2 Use a Number Line
3
Plot 0.3 and -ona number line.
3
Solution Begin by writing the fraction as a decimal, ^ = 1.5. Draw a
number line. Mark equally spaced tick marks to represent a distance of 0.1 on
the number line. Label several numbers as shown.
- | I I I I 1 I I I I | I II I M I I I | »
0 0.5 1 1.5 2
3
Plot 0.3 and — by marking a solid dot on the number line.
3
0.3 2
♦ |. I- ,|. . iH ....). . t . ■! f . + + J . | l I I I ] »
0 0.5 1 1.5 2
Try These
Draw a number line and plot the numbers.
17. 0.2
18. 1.7
19. 0.4
16 -5
20. 1.9
Maintaining Skills
Why are helicopters able to take off
and land without runways?
Helicopters are capable of vertical flight —
flying straight up and straight down. Rotor blades
generate an upward force (lift) as they whirl through
the air.
Mathematics provides a useful way of distinguishing
between upward and downward motion. In this
chapter you will use positive numbers to measure the
velocity of upward motion and negative numbers to
measure the velocity of downward motion.
Think & Discuss
1. Describe some real-life situations that you
might represent with negative numbers. What
do positive and negative numbers represent in
each situation?
2 . Describe the average speed and direction of
each helicopter’s movement if it travels the
given distance in 15 seconds.
Not drawn to scale
Learn More About It
APPLICATION: Helicopters
You will calculate the speed and velocity of different
objects in Exercises 47-50 on page 75.
application link More about helicopters is available at
www.mcdougallittell.com
V
PREVIEW
PREPARE
STUDY TIP
What’s the chapter about ?
• Adding, subtracting, multiplying, and dividing real numbers
Key Words
>
• real number, p. 65
• closure property, p. 78
• coefficient, p. 107
• integer, p. 65
• term, p. 87
• like terms, p. 107
• opposite, p. 71
• absolute value, p. 71
• distributive property, p. 100
• reciprocal, p. 113
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to pp. 3, 24)
1- Identify the variable in the expression — 2r 3 — 8.
(A) -2 CD r CD 3 CD -8
2 . Complete: A(n) ? is a statement formed by placing an equal sign between
two expressions.
(A) equation CD solution CD inequality CD function
SKILL Check (refer to pp. 763, 765, 770)
5 2 11 5 25
3_ Write the numbers 2^, 2—, —, 2~, and — in order from least to greatest.
U 25
4 ’ 11
2 — 2 —
3’ 6
2 5 5
■ 2- 2- 2—
’ 3’ 6’ 8
_ 2 11 5 5 25
^ 3’ 4 ’ 6’ 8’ 11
; ? 5 11 25
>’ Z 8’ 4 ’ 11
^ 25 .5 .2 11 5
^ 11’ 8’ 3’ 4 ’ 6
1 3
4. Write the quotient 9~p ^ 1— as a mixed number.
O o
^ 605
® 20
GD 6f
29
® 12 §
Study a Lesson
Take notes. Add to your list of
vocabulary words, rules, and
properties in your notebook.
Lesson 2.3
Commutative Property of Addition
a+-b = b-Ha -4-H6 = 6-H (-4)
Associative Property of Addition
(a-hb) +- c = a-h (b+- c )
(-Z +- 5) +- 3 = -g -h ( 5 + 3)
Chapter 2 Properties of Real Numbers
The Real Number Line
Goal
Graph, compare, and
order real numbers.
Key Words
• real number
• real number line
• positive number
• negative number
• integer
• whole number
• graph of a number
What was the coldest temperature in Nome, Alaska?
Russia
Nome
Bering
Sea
Alaska
* (U.S.)
Canada
In meteorology, temperatures that
are above zero are represented by
positive numbers and temperatures
that are below zero are represented
by negative numbers. In Example 5
you will compare low temperatures
for Nome, Alaska.
The numbers used in this book are real numbers. Real numbers can be pictured
as points on a line called a real number line, or simply a number line.
Every real number is either positive, negative, or zero. Points to the left of zero
represent the negative real numbers. Points to the right of zero represent the
positive real numbers. Zero is neither positive nor negative.
REAL NUMBER LINE
Negative numbers Positive numbers
- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -►
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
The scale marks on the real number line are equally spaced and represent
integers. An integer is either negative, zero, or positive. Zero and the positive
integers are also called whole numbers.
-3,-2,-1, 0, 1,2,3,...
Negative integers Zero Positive integers
The point on a number line that corresponds to a number is the graph of the
number. Drawing the point is called graphing the number or plotting the point.
Student HeCp
-
► Reading Algebra
In Example 1, -2 is
read as "negative two,"
0 is read as "zero," and
3 is read as "three" or
as "positive three."
\ _ >
1 Graph Integers
Graph —2, 0, and 3 on a number line.
Solution
-2 is a negative number so it is plotted 2 units to the left of zero.
*
-2 0 3
-f-1-1-1-1-1-1-1-1-f-h
-5 -4 -3 -2 -1 0 1 2 3 4 5
t
3 is a positive number so it is plotted 3 units to the right of zero.
2.1 The Real Number Line
On a number line, numbers that are to the left are less than numbers to the right
and numbers that are to the right are greater than numbers to the left.
Student HeCp
^
► Skills Review
For help with comparing
and ordering numbers,
see pp. 770-771.
k _ J
S3ZQQ29 2 Compare Integers
Graph —4 and —5 on a number line. Then write two inequalities that compare
the numbers.
Solution
-5 -4
H-4-1-1-1-1- \ -1-f- \ -h
-8 -7 -6 -5 -4 -3-2-1 0 1 2
On the graph, —5 is to the left of —4, so —5 is less than —4. You can write this
using symbols:
—5 < —4
On the graph, —4 is to the right of —5, so —4 is greater than —5. You can
write this using symbols:
—4 > -5
Compare Integers
Graph the numbers on a number line. Then write two inequalities that
compare the numbers.
1. —6 and — 2 2. 2 and —3 3. 5 and 7
You can graph decimals and fractions, as well as integers, on a real number line.
The scale marks on a number line do not have to be integers. They can be in units
of 0.1, 0.5, 2, 5, or any other amount.
Student HeCp
► Study Tip
When you work with
fractions, sometimes it
is easier to first convert
the fraction to a
decimal. For example:
| = 4-7« 0.57
V _ )
ttZEEBi 3 Graph Real Numbers
4
Graph —0.8 and — on a number line.
Solution
4
Because —0.8 and — are not integers, use a number line that has scale marks in
smaller units.
-0.8 is 0.8 unit to the left of zero.
1 4
-0*8 7
-H—I—I—I—I—t—t—I—hH—I—I—I—I—I—I—t—I—I—I—h—
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
y, which is about 0.57, is 0.57 unit to the right of zero._f
Chapter 2 Properties of Real Numbers
Student Hedp
p More Examples
More examples
* are available at
www.mcdougallittell.com
■wmjia a Order Real Numbers
f i 3
Write the numbers —2, 4, 0, 1.5, —, and — — in increasing order.
Solution Graph the numbers on a number line. Remember that y = 0.5 and
that — = —1.5.
3 1
-2 2 0 2 1.5 4
-- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -►
-3 -2 -1 0 1 2 3 4 5 6
3 1
ANSWER ► From the graph, you can see that the order is: —2, — — , 0, — , 1.5, 4.
Order Real Numbers
Link to
Science
NOME, ALASKA The
coldest low temperature
on record for Nome, Alaska,
is -54°F.
data update of
National Oceanic
and Atmospheric
Administration data at
www.mcdougallittell.com
Write the numbers in increasing order.
4. -3, 0, 4, |, -1 5. -3, 3, 3.2, -j, -8, 4.5
EXAMPLE
Compare Real Numbers
NOME, ALASKA The table shows the low temperatures in Nome, Alaska, for
five days in December. Which low temperature was the coldest?
Date
Dec. 18
Dec. 19
Dec. 20
Dec. 21
Dec. 22
Low Temp.
— 10°F
— 11°F
16°F
3°F
2°F
Solution First graph the temperatures on a number line.
- 11 Tr ~ 10 2 ir 3
—I—i—i—i—i—i—i —h
-20 -15 -10 -5 0 5 10 15
+
20
ANSWER ^ The coldest low temperature was — 11°F.
Compare Real Numbers
6. The table shows the low temperatures in Nome, Alaska, for five days in
February. Which dates had low temperatures above 10°F?
Date
Feb. 22
Feb. 23
Feb. 24
Feb. 25
Feb. 26
Low Temp.
—20°F
— 11°F
20°F
17°F
— 15°F
2.1 The Real Number Line
lag BECSSSj ^
LA Exercises
Guided Practice
Vocabulary Check Complete the statement.
1. On a number line, the numbers to the left of zero are ? numbers, and the
numbers to the right of zero are ? numbers.
2_ Zero and the positive integers are also called ? numbers.
Skill Check Graph the numbers on a number line.
3. -5, -1, 4 4. -3, 0, 3 5. 6, -2, 0.5 6. -1, -2, -
Complete the statement using < or >. Use the number line shown.
2 3
-8 -6.7 -5 -4 0 3 2 6.7
—H—I—I-1—I—I—I—I—I—I—I—I—I—I—I—I—I—b—
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
7. -4 0-5 8. 0 0 -8 9.6.7 0-6.7 10. |||
Write the numbers in increasing order.
11. 2,-3,-8, 1,-2 12. 1.2,-4, 5,7,-6.1 13.-7,-9, 2, |,-|
Practice and Applications
GRAPHING INTEGERS Graph the numbers on a number line.
14. 0, 2, 6
17. -7, -4, -8
20. 1, -2, 3
15. 10, 9, 3
18. -1, -6, -7
21 . -3, 1,5
16.5,2,8
19. -2, -4, -6
22 . -4, 4, -5
Student HeCp
p Homework Help
Example 1: Exs. 14-22
Example 2: Exs. 23-30
Example 3: Exs. 31-43
Example 4: Exs. 44-51
Example 5: Exs. 52-59
COMPARING INTEGERS Graph the numbers on a number line. Then
write two inequalities that compare the numbers.
23 . -2, 3 24 . 4, -6 25 . -1, -6 26 . -7, -5
27 . 0,-4 28 . 8,-8 29 . 10,11 30 . 9,-12
REAL NUMBERS Match the number with its position on the number line.
A B C D
* 1*1-1-1-f-1-N—i-1-1-HM-1-1—I-1-1 > I >
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
31.8.1 32.-1.8 33. | 34. -y
Chapter 2 Properties of Real Numbers
<N | CO
GRAPHING REAL NUMBERS Graph the numbers on a number line.
35. 0.5, -1.5, 2.5 36. -5.6, -0.3, 2 37. 4.2, 4.4, 4.6
38. 0, -0.5
39. 4.3, -|, -2.8
3 7
40. f, ~j,~ 3
Student HeGp
1 ^ ---
► Homework Help
Extra help with
“4^ problem solving in
Exs. 44-49 is available at
www.mcdougallittell.com
41 — — — _1
2’ 3’ 2
42 I 3 n
3’ 2’ 4
43 ^
10’ 5’ 3
ORDERING REAL NUMBERS Write the numbers in increasing order.
44. 4.6, 0.7, -4, -1.8, 3, -0.6 45. -0.3, 0.2, 0, 2.0, -0.2, -3.0
46. 6.3, -6.8, -6.1, 6.1, -6.2, 6.7 47. |, 3.4, 4.1, -5.2, -5.1, —^
13 1
48 7 —— 2 —— -5 —
'5 2 ’ 6
49. 4.8, -2.6, 0, -|, -i
LOGICAL REASONING Complete the statement using < or >.
50. If x >—4, then—4 ? x. 51. If 3 <y, then y ? 3.
ELEVATION In Exercises 52-54, write a positive number, a negative
number, or zero to represent the elevation of the location.
Elevation is represented by comparing a location to sea level, which is given a
value of zero. A location above sea level has a positive elevation, and a location
below sea level has a negative elevation.
52. Granite Peak, Montana, 12,799 feet above sea level
53. New Orleans, Louisiana, 8 feet below sea level
54. Long Island Sound, Connecticut, sea level
ASTRONOMY A star may
appear dim because it is far
from Earth. It may actually be
brighter than a star that looks
very bright only because it is
closer to Earth.
S cience Link / In Exercises 55-59, use the table shown which gives the
apparent magnitude of several stars.
A star’s brightness as it appears to a person on Earth is measured by its apparent
magnitude. The lesser the apparent magnitude, the brighter the star.
55. Graph the apparent magnitudes on a number
line. Label each point with the name of
the star.
56. Which stars have an apparent magnitude
that is less than the apparent magnitude
of Altair?
57. Which stars have an apparent magnitude
that is greater than the apparent magnitude
of Procyon?
58. Which star has the least apparent magnitude
and so looks the brightest?
59. Which star has the greatest apparent
magnitude and so looks the dimmest?
Star
Apparent
magnitude
Canopus
-0.7
Procyon
0.4
Pollux
1.1
Altair
0.8
Spica
1.0
Regulus
1.4
Sirius
-1.5
Deneb
1.3
2.1 The Real Number Line
Standardized Test
Practice
Mixed Review
Maintaining Skills
60. MULTIPLE CHOICE Which inequality is true?
(a)-9>-5 CD 9 <5 CD 9 < -5 CD -9 <5
61. MULTIPLE CHOICE Which number is less than -0.1?
CD-io CD o CD o.ooi CD io
62. MULTIPLE CHOICE Which set of numbers is in increasing order?
(A) -1.9, 1.8, -0.5, 0,0.5 CD -1.9, -0.5, 0, 0.5, 1.8
CD 0, -0.5, 0.5, 1.8, -1.9 CD -0.5, 0,0.5, 1.8, -1.9
G eometry Link) . Find the area of the object. (Lesson 1.2)
63. The top of a computer desk measures 2 feet by 2 feet.
64. The cover of a children’s book is 4 inches long and 4 inches wide.
65. A square piece of construction paper has a side length of 9 centimeters.
MENTAL MATH Use mental math to solve the equation. (Lesson 1.4)
66. 9 - y = 1 67. t + 6 = 10 68. 2a = 8
69. 15 = r = 3 70-4 = 8 71. — = 9
2 n
72. BIRTHS The table shows the number of births (in thousands) in the United
States by month for 1997. Make a bar graph of the data. (Lesson 1.7)
Jan.
Feb.
Mar.
Apr.
May
June
July
Aug.
Sep.
Oct.
Nov.
Dec.
305
289
313
342
311
324
345
341
353
329
304
324
Sci ence Link When it is 70°F, the function T = 0.08 H + 64.3 gives the
apparent temperature T (in degrees Fahrenheit) based on the relative
humidity H (as a percent). (Lesson 1.8)
73 . Copy and complete the input-output table.
Input H
20%
40%
60%
70%
100 %
Output T
?
?
?
?
?
74. Use the table to draw a graph that represents the function.
75. Determine the range of the function.
FACTORS Write the prime factorization of the number if it is not a prime.
If the number is a prime, write prime. (Skills Review p. 761)
76 . 18 77 . 35 78 . 47 79 . 64
80.100 81.101 82.110 83.144
H
Chapter 2 Properties of Real Numbers
Absolute Value
Goal
Find the opposite and
the absolute value of a
number.
Key Words
• opposite
• absolute value
• counterexample
What is a launch pad elevator's velocity and speed?
Velocity and speed are different concepts.
Velocity tells you how fast an object is moving
and in what direction. It can be positive or
negative. Speed tells you only how fast an
object is moving. It can only be positive. In
Example 4 you will find the velocity and speed
of a launch pad elevator for a space shuttle.
Two numbers that are the same distance from 0 on a number line but on opposite
sides of 0 are opposites. The numbers — 3 and 3 are opposites because each is
3 units from 0.
3 3
r - rA ’- r - A - - \
-- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -
-5 -4 -3 -2 -1 0 1 2 3 4 5
i Find the Opposite of a Number
Use a number line to find the opposite of —4.
Solution
4 4
_ K _ _ A_
-—I-1-1-1-i-1-1-i-1-1-1—**
-5 -4 -3 -2 -1 0 1 2 3 4 5
You can see that —4 is 4 units to the left of 0. The opposite of —4 is 4 units to
the right of 0. So the opposite of —4 is 4.
ABSOLUTE VALUE The absolute value of a number is its distance from zero on
a number line. The symbol | a | represents the absolute value of a.
Student tfeCp
>
I ►Reading Algebra
The expression -a
can be read as
"negative a" or as "the
opposite of a."
K _ J
THE ABSOLUTE VALUE OF A NUMBER
• If a is a positive number, then |a| = a.
>
Example: | 3 | = 3
• If a is zero, then | a| = 0.
Example: 10 | = 0
• If a is a negative number, then |a| = -a.
Example: | —3 | =: -(-3) = 3
H
2.2 Absolute Value
2 Find Absolute Value
Evaluate the expression,
a. 15 | b. | -2.3 | c. -
-8
Solution
a. 15 | =5
b. |-2.3 | = -(-2.3)
= 2.3
1
2
d. — | —81 = -( 8 )
= -8
If ois positive, then \a\ = a.
If a is negative, then | a \ = -a.
Use definition of opposites.
The absolute value of ^ is
Use definition of opposites.
The absolute value of -8 is 8.
Use definition of opposites.
Find Absolute Value
Evaluate the expression.
1 . | —4 1 2 . |0
4. -| 1.7 |
Student HeCp
-V
p Look Back
For help with the
solution of an
equation, see p. 24.
\ _ /
3 Solve an Absolute Value Equation
Use mental math to solve the equation,
a. |x| =1 b. |x| = 5.1 c. |x| = —^
Solution
a. Ask, “What numbers are 7 units from 0?” Both 7 and —7 are 7 units from
0, so there are two solutions: 7 and —7.
b. Ask, “What numbers are 5.1 units from 0?” Both 5.1 and —5.1 are 5.1
units from 0, so there are two solutions: 5.1 and —5.1.
c. The absolute value of a number is never negative, so there is no solution.
Solve an Absolute Value Equation
Use mental math to solve the equation. If there is no solution, write
no solution.
5. | x | = -4 6. | x | = 1.5 7. | x | = 1
H
Chapter 2 Properties of Real Numbers
VELOCITY AND SPEED Velocity indicates both speed and direction (up is
positive and down is negative). The speed of an object is the absolute value of
its velocity.
J a Find Velocity and Speed
SCIENCE LINK A launch pad elevator for a space shuttle drops at a rate of
about 12 feet per second. What are its velocity and speed?
Solution
Velocity = —12 feet per second
Speed = | —12 | = 12 feet per second
Find Velocity and Speed
A parachutist descends at a rate of
about 17 feet per second.
8. What is the parachutist’s velocity?
9. What is the parachutist’s speed?
COUNTEREXAMPLE To prove that a statement is true, you need to show that it
is true for all examples. To prove that a statement is false, it is enough to show
that it is not true for a single example, called a counterexample.
Motion is downward.
Speed is never negative.
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" 40 ' are available at
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5 Use a Counterexample
Determine whether the statement is true or false. If it is false, give a
counterexample.
a. The opposite of a number is always negative.
b. The absolute value of a number is never negative.
Solution
a. False. Counterexample: The opposite of —5 is 5, which is positive.
b. True, by definition.
Use a Counterexample
Determine whether the statement is true or false. If it is false, give a
counterexample.
10, The expression — a is never positive.
11, The expression | a | is always greater than or equal to a.
12, The absolute value of a negative number is always negative.
2.2 Absolute Value
Guided Practice
Vocabulary Check
1 . What is the opposite of 2?
2. Complete: The absolute value of a number is its distance from ? on a
number line.
Skill Check Find the opposite of the number.
3. 1 4. -3 5. -2.4 6. ^
Evaluate the expression.
7. | —121 8. 161 9. — 15.1 | 10. -|
Use mental math to solve the equation. If there is no solution, write
no solution.
11. | x | = 8 12. | x | = -9 13. | x | = 5.5 14. Ul = |
Determine whether the statement is true or false. If it is false, give a
counterexample.
15, The opposite of a number is always less than the number.
16- The absolute value of a number is always positive or zero.
Practice and Applications
i
Student HeCp
► Homework Help
Example 1: Exs. 17-24
Example 2: Exs. 25-32,
41,42
Example 3: Exs. 33-40
Example 4: Exs. 43-50
Example 5: Exs. 51-53
FINDING OPPOSITES Find the opposite of the number.
17-8 18- -3 19-10 20-0
21 . -3.8
22 . 2.5
1
23 —t
24.
6
FINDING ABSOLUTE VALUE Evaluate the expression.
25. 1 7 I 26.1-41 27. — 1 3 I 28. — I —2 I
29. -0.8
30. -4.5
31.
32. -
SOLVING AN EQUATION Use mental math to solve the equation. If there
is no solution, write no solution.
33.
\x | = 4
34. \x = 0
35.
\x\ = —2
36.
37.
*
II
38. |x| = —9.6
39.
1 1 11
l x l = ~2
40.
H
Chapter 2 Properties of Real Numbers
Planet
Linktf^
Science
SATURN radiates more
energy into space than it
receives from the sun,
resulting in a constant
average surface temperature.
Student HeCp
► Homework Help
Extra help with
-jjgjy p ro bie m solving in
Exs. 51-53 is available at
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Scie nce Lank y In Exercises 41 and 42,
use the table at the right which shows
the average high and low surface
temperatures for the planets in our
solar system.
41 , The range of temperatures for
Mercury and Mars is the sum of the
absolute values of the high and low
temperatures. Find the range of
temperatures for these planets.
42 , The range of temperatures for the
other planets is the difference of
the absolute values of the high and
low temperatures. Find the range of
temperatures for these planets.
High (°F) Low (°F)
Mercury
800
-280
Venus
847
847
Earth
98
8
Mars
98
-190
Jupiter
-244
-244
Saturn
-301
-301
Uranus
-353
-353
Neptune
-373
-373
Pluto
-393
-393
VELOCITY Determine whether to use a positive or a negative number to
represent the velocity.
43 , The velocity of a descending hot-air balloon
44 , The velocity of a rising rocket
45 , The velocity of a kite as it lifts into the air
46 , The velocity of a falling meteorite
VELOCITY AND SPEED A helicopter is descending at a rate of 6 feet
per second.
47 . What is the helicopter’s velocity? 48 . What is the helicopter’s speed?
VELOCITY AND SPEED The elevator in the Washington Monument
in Washington, D.C., climbs at a rate of about 400 feet per minute.
49 . What is the elevator’s velocity? 50 . What is the elevator’s speed?
USING COUNTEREXAMPLES Determine whether the statement is true or
false. If it is false, give a counterexample.
51 . The opposite of —a is always positive.
52 . The opposite of | a | is never positive.
53 . The expression | — a | is never negative.
CHALLENGE Determine whether the statement is always , sometimes , or
never true. Explain.
54 . The absolute value of a number is the same as the absolute value of the
opposite number. In other words, |x| = \ —x\.
55 . The opposite of the absolute value of a number is the same as the absolute
value of the opposite of the number. In other words, — | x | = | —jc |.
2.2 Absolute Value
Standardized Test
Practice
Mixed Review
Maintaining Skills
H
56. MULTIPLE CHOICE What is the opposite of 5?
CD j CD -j CD 5 CD -5
57. MULTIPLE CHOICE What is the value of - 1 -2 |?
CD 2 CD -2 CD |2| CD I —2 1
58. MULTIPLE CHOICE What is the solution of |x| = 18 ?
CD 18 CD -18
Cc) 18 and — 18 (d) none of these
59- MULTIPLE CHOICE What is the velocity of a diver who descends to the
ocean floor at a rate of 3 meters per second?
CD I — 3 | m/sec CD I 3 | m/sec
CD -3 m/sec CD 3 m/sec
EVALUATING EXPRESSIONS
of the variable. (Lesson 1.1)
60. v + 3 when x = 2
62. 3 y when y — 0
64. ^ when z — 8
Evaluate the expression for the given value
61. a — 7 when a = 10
63. (0(5) when t = 15
9
65. — when p = 3
TRANSLATING SENTENCES Write the sentence as an equation or an
inequality. Let x represent the number. (Lesson 1.5)
66 . 5 less than a number is 8.
67. 8 more than a number is 17.
68 . The quotient of 15 and a number is greater than or equal to 3.
69. 9 times a number is less than 6.
COMPARING NUMBERS Graph the numbers on a number line. Then
write two inequalities that compare the numbers. (Lesson 2.1)
70. 7, -7 71. -2, -6 72. -10,-1
73. 0.4, -3 74. 2.2, -3.3 75. -10, —Ij
SUBTRACTING FRACTIONS Subtract. Write the answer in simplest form.
(Skills Review p. 764)
4 4
7 2
77 -9~9
78.
14 4
/V - 15 15
80. — — —
27 27
81.
1 _
10
41
44
10
19
44
Chapter 2 Properties of Real Numbers
DEVELOPING CONCEPTS
nun us Issiasjay.
For use with
Lesson 2.3
Goal
Use reasoning to find a
pattern for adding integers.
Question
How can you model the addition of integers with algebra tiles?
Materials
• algebra tiles
Each + represents positive 1 and each | represents negative 1. Combining a
+ tile and a — tile equals zero.
Explore
Use algebra tiles to find the sum of —8 and 3.
0 Model negative 8 and positive 3 using algebra tiles.
■■■■ ■■
■ ■■■ +
-8 3
© Group pairs of positive and negative tiles. Count the remaining tiles.
Each pair has
a sum of 0. —
© The remaining tiles show the sum of —8 and 3. Complete: — 8 + 3 = ? .
Think About It
Use algebra tiles to find the sum of the numbers given.
1- + + + 2 - 1 ■■■ 3 - + + ■■
■ ■■ ■■ ■ ■
2 4 -1 -5 3 -3
Use algebra tiles to find the sum. Sketch your solution.
4. 3 + 3 5. -4 + (-2) 6. -3 + 2 7. 5 + (-2)
LOGICAL REASONING Based on your results from Exercises 1-7,
complete the statement with always , sometimes , or never.
8. The sum of two positive integers is ? a positive integer.
9. The sum of two negative integers is ? a positive integer.
10, The sum of a positive integer and a negative integer is ? a
negative integer.
H
Developing Concepts
Adding Real Numbers
Goal
Add real numbers using a
number line or the rules
of addition.
Key Words
• closure property
• commutative property
• associative property
• identity property
• inverse property
What is the profit or loss of a company?
In business a profit can be
represented by a positive number
and a loss can be represented by a
negative number. In Example 4 you
will add several profits and losses to
find the overall profit of a summer
excursion company.
The sum of any two real numbers is itself a unique real number. We say that the
real numbers are closed under addition. This fact is called the closure property
of real number addition. Addition can be modeled with movements on a
number line.
• You add a positive number by moving to the right on the number line.
• You add a negative number by moving to the left on the number line.
i Add Using a Number Line
Use a number line to find the sum.
a. -2 + 5 b. 2 + (-6)
Solution
Start at -2.
Move 5 units to the right.
+
+
+
+
+
+
End at 3.
-4 -3-2-1 0 1 2
ANSWER ► The sum can be written as — 2 + 5 = 3.
b.
End at -4.
Move 6 units to the left.
■ Start at 2.
+
+
-6 -5 -4 -3-2-1 0 1 2
ANSWER ► The sum can be written as 2 + (—6) = —4.
Add Using a Number Line
Use a number line to find the sum.
1. -4 + 5 2. -1 + (-2) 3. 4 + (-5)
4. 0 + (-4)
Chapter 2 Properties of Real Numbers
RULES OF ADDITION
To add two numbers with the same sign :
STEP0 Add their absolute values.
step © Attach the common sign.
To add two numbers with opposite signs :
step 0 Subtract the smaller absolute value from the larger one.
step 0 Attach the sign of the number with larger absolute value.
r Student HeCp
p Look Back
For help with absolute
value, see p. 71.
\ _ >
2 Add Using Rules of Addition
a. Add —4 and —5, which have the same sign.
0 Add their absolute values. 4 + 5 = 9
0 Attach the common (negative) sign. —(9) = —9
ANSWER ^ -4 + (-5) = -9
b. Add 3 and —9, which have opposite signs.
0 Subtract their absolute values. 9 — 3 = 6
© Attach the sign of the number with larger absolute value. —(6) = —6
ANSWER ► 3 + (-9) = -6
Add Using Rules of Addition
Use the rules of addition to find the sum.
5. -3 +(-7) 6. -1 + 3 7. 8 +(-3) 8.2 + 3
The rules of addition are a consequence of the following properties of addition.
PROPERTIES OF ADDITION
closure property The sum of any two real numbers is a unique real number.
a + b is a unique real number Example: 4 + 2 = 6
commutative property The order in which two numbers are added does
not change the sum.
a + b = b + a Example: 3 + ( — 2) = —2 + 3
associative property The way three numbers are grouped when adding
does not change the sum.
(a + b) + c = a + (b + c) Example: (-5 + 6) + 2 = -5 + (6 + 2)
identity property The sum of a number and 0 is the number.
a + 0 = a Example: -4 + 0 = -4
inverse property The sum of a number and its opposite is 0.
a + (-a) = 0 Example: 5 + (-5) = 0
2.3 Adding Real Numbers
Student HeCp
► More Examples
More examples
are available at
www.mcdougallittell.com
3 Add Using Properties of Addition
a. 4 + (-6) + 9 = 4 + (-6 + 9)
= 4 + 3
= 7
Use associative property.
Add -6 and 9.
Add 4 and 3.
b. _ 0.5 + 3 + 0.5 — —0.5 + 0.5 + 3
= (-0.5 + 0.5) + 3
= 0 + 3
= 3
Use commutative property.
Use associative property.
Use inverse property.
Use identity property.
Add Using Properties of Addition
Use the properties of addition to find the sum.
9. -7 + 11+7 10. -5 + 1+2 11.3 + (—I) + (—|)
PROFIT AND LOSS A company has a profit if its income is greater than
its expenses. It has a loss if its income is less than its expenses. Income
and expenses are always positive, but business losses can be indicated by
negative numbers.
4 Use Addition in Real Life
8 BUSINESS A summer excursion company had the monthly profits and
losses shown. Add them to find the overall profit or loss of the company.
Student HeCp
► Keystroke Help
To enter -5 on a
calculator with a
key, enter 5 QEB.To
enter -5 on a calculator
with a ^2 key, enter
s_/
JANUARY
-$13,143
APRIL
$3,A2S
FEBRUARY
-$b,7fl3
HAY
$7,bl3
MARCH
-$4,735
JUNE
$12,133
Solution With this many large numbers, you may want to use a calculator.
13143 _) Of 6783 Qfl EM 4735 SM EU 3825 7613
EM 12933 0 [ -5901
ANSWER ► The company had an overall loss of $290.
Use Addition in Real Life
12. Find the total profit or loss of the company in Example 4 during the first
quarter (January through March).
13. Find the total profit or loss of the company in Example 4 during the spring
months (March through May).
Chapter 2 Properties of Real Numbers
Exercises
Guided Practice
Vocabulary Check
Skill Check
Match the property with the statement that illustrates it.
1- Commutative property A. — 8 + 0= — 8
2. Associative property B. 5 + (— 9) = — 9 + 5
3. Identity property C. — 8 + 8 = 0
4. Inverse property D. 5 + (4 + 9) = (5 + 4) + 9
5. Write an addition equation for the sum modeled on the number line.
-5
—*-
-6 -5 -4 -3 -2 -1 0
Use a number line to find the sum.
6. 7 + (-3) 7. 0 + (-10)
H-1-h
8 .
4
-+-
-7 + 3
Use the rules of addition to find the sum.
9. 12 + (-5) 10. -4 + 5 11. -7 + (-3)
Use the properties of addition to find the sum.
12. —4 + 3 + (—2) 13. 5 + (—5) + 7 14. —3 + 0 + 7
Practice and Applications
ADDING REAL NUMBERS Match the exercise with its answer.
15. -1 +(-2) 16. 3 +(-5) 17. -2 + 2
NUMBER LINES Use a number line to find the sum.
18. -6 + 2
19. 2 + (-8)
20. -3 + (-3)
21. -4 + (-7)
22. -4 + 5
23. 3 + (-7)
24. -10+1
25. 15 + (-9)
26. -12 + (-5)
Student HeCp
r ->
^ Homework Help
RULES OF ADDITION
Find the sum.
Example 1: Exs. 15-26
27. 9 + (-2)
28. -6 + (-11)
29. -7 + (-4)
Example 2: Exs. 27-35
Example 3: Exs. 36-49
30. -5 + 2
31. 8 + (-5)
32. -6 + (-3)
Example 4: Exs. 50-55
_>
33. -10 + (-21)
34. 49 + (-58)
35. -62 + 27
H
2.3 Adding Real Numbers
GOLF If you complete a
round of golf in 68 strokes
at a course with a par of
71 strokes, you have shot
"3 under par," or -3.
NAMING PROPERTIES Name the property shown by the statement.
36. —16 + 0 = -16 37. -3 + (-5) = -5 + (-3)
38. (-4 + 3) + 5 = -4 + (3 + 5) 39. 16 + (-16) = 0
40. There is only one real number that is the sum of 4 and 6.
PROPERTIES OF ADDITION Find the sum.
41. 6 + 10 + (-6) 42. 7 + (-2) + (-9) 43. 8 + (-4) + (-4)
44. -24.5 + 6 + 8 45. 5.4 + 2.6 + (-3) 46. 2.2 + (-2.2) + (2.2)
47 ' 4 + To + H)) 48.9 + <-4) + (-±) 49. i + (-2) + (-f)
FINDING SUMS Find the sum. Use a calculator if you wish.
50. -2.95 + 5.76 + (-88.6) 51. 10.97 + (-51.14) + (-40.97)
52. 20.37 + 190.8 + (-85.13) 53. 300.3 + (-22.24) + 78.713
54. PROFIT AND LOSS A company had the following profits and losses over
a 4-month period: April, $3,515; May, $5,674; June, —$8,993; July, —$907.
Did the company make an overall profit or loss? Explain.
55. GOLF In golf par is the expected number of strokes needed to finish a hole.
A bogey is a score of one stroke over par. A birdie is a score of one stroke
under par. An eagle is a score of two strokes under par. Using the table find
the number of strokes you are off from par at the end of a round of golf.
Hole
1
2
3
4
5
6
7
8
9
Score
Birdie
Par
Birdie
Par
Eagle
Bogey
Bogey
Bogey
Birdie
56. CHALLENGE Determine whether the following statement is true or false.
If it is true, give two examples. If it is false, give a counterexample.
The opposite of the sum of two numbers is equal to the sum of the
opposites of the numbers.
Standardized Test
Practice
In Exercises 57 and 58, use the financial data in the table.
57. MULTIPLE CHOICE In which
month was the most money saved?
(a) January CfT) March
CcT) May Cd) June
58. MULTIPLE CHOICE In which
month did the money spent most
exceed the money earned?
(Jp January CcT) February
(J±) April GD June
Month
$ Earned
$ Spent
$ Saved
Jan.
1676
1427
?
Feb.
1554
1771
?
Mar.
1851
1556
?
Apr.
1567
1874
?
May
1921
1602
?
June
1667
1989
?
Chapter 2 Properties of Real Numbers
Mixed Review
Maintaining Skills
Quiz 1
WRITING POWERS Write the expression in exponential form. (Lesson 1.2)
59. four squared 60. k to the ninth power 61. .v cubed
NUMERICAL EXPRESSIONS Evaluate the expression. (Lesson 1.3)
62. 15 - 5 + 5 2 63. 18 • 2 - 1 • 3 64. 1 + 3 • 5 - 8
65. 2(9 - 6 - 1) 66. 10 - (3 + 2) + 4 67. 2 • (6 + 10) - 8
CHECKING SOLUTIONS OF EQUATIONS Check to see if the given value
of the variable is or is not a solution of the equation. (Lesson 1.4)
68. x + 5 = 10; x = 7 69. ly - 15 = 6; y = 3 70. 17 - 3w = 2; w = 5
71. a 2 - 3 = 5; a = 4 72. 1 + p 3 = 9; p = 2 73. 2n 2 + 10 = 14; n = 1
ESTIMATING Round the values to the nearest hundred and estimate the
answer. (Skills Review p. 774)
74. 422 + 451 75. 8362 + 941 76. 27 + 159
77. 675 - 589 78. 1084 - 179 79. 3615 - 663
Graph the numbers on a number line. Then write two inequalities that
compare the numbers. (Lesson 2.1)
1.7, -2 2.-2,-3 3. 1,-6
Write the numbers in increasing order. (Lesson 2.1)
4. -8, 2, -10, -3,9 5. -5.2, 5,-7, 7.1, 3.3 6. -1, 2, 0,|gg
Evaluate the expression. (Lesson 2.2)
7. 15 | 8. |-13 | 9. -| 0.56 |
Use mental math to solve the equation. If there is no solution, write
no solution. (Lesson 2.2)
10. |x|=-10 11. |jc| = 2.7 12.|x| = |
Find the sum. (Lesson 2.3)
13. -6 +(-7) 14. 4+ (-10) 15. -5 + 9
16. —5 + 1 + (—3) 17. -6 + 2.9 + 1.1 18. | + 0 + (-jj
19. FOOTBALL Your high school football team needs 9 yards to score a
touchdown. The last four plays result in a 5 yard gain, a 2 yard gain, a
12 yard loss, and a 15 yard gain. Does your team score a touchdown?
If not, how many yards do they still need? (Lesson 2.3)
2.3 Adding Real Numbers
2 2A*«
DEVELOPING CONCEPTS
uiwin^iiun us Issiasjar:
For use with
Lesson 2.4
Coal
Use reasoning to find a
pattern for subtracting
integers.
Materials
• algebra tiles
Student HeCp
^ Look Back
For help with using
algebra tiles, see p. 77.
V _ }
Question
'' —»
How can you model the subtraction of positive integers
with algebra tiles?
Explore
Use algebra tiles to model 3 — 6.
© Use 3 yellow tiles to model +3.
■■■
© Before you can remove 6 yellow tiles you need to add three “zero pairs.”
+ + +
n
42
9
■
&
© To subtract 6 from 3, remove six of the yellow tiles.
(+ + + + ++)
Q The remaining tiles show the difference of 3 and 6.
Complete: 3 — 6 = ?
Think About It
u i — ■ -—
Use algebra tiles to find the difference. Sketch your solution.
1.7-2 2.2-3 3.4-7
4.-3 - 5 5.-5 - 8 6.-1 -2
Use algebra tiles to find the sum. Sketch your solution.
7. 7 + (-2) 8. 2 + (-3) 9. 4 + (-7)
10 . -3 + (-5) 11 . -5 + (-8) 12 . -1 + (-2)
LOGICAL REASONING Based on your results from Exercise 1-12,
determine whether the statement is true or false. Explain.
13. To subtract a positive integer, add the opposite of the positive integer.
14. When you subtract a positive integer, the difference is always negative.
Chapter 2 Properties of Real Numbers
Question
How can you model subtraction of negative integers with
algebra tiles?
Explore
Use algebra tiles to model —6 — (—2).
Q Use 6 red tiles to model —6.
Q To subtract —2 from —6, remove 2 red tiles.
© The remaining tiles show the difference of —6 and —2.
Complete: — 6 — (—2) = __?
Think About It
i i ■
Use algebra tiles to find
the difference. Sketch your solution.
1.4-(-2)
2- 8 — (—1)
3. 3 - (-4)
cn
1
r-*
1
5. -5 - (-1)
6. —6 — (~6)
Use algebra tiles to find
the sum. Sketch your solution.
7.4 + 2
8 8 + 1
9. 3 + 4
10. -7 + 3
11. -5 + 1
12. -6 + 6
LOGICAL REASONING Based on your results from Exercises 1-12,
determine whether the statement is true or false. Explain.
13. To subtract a negative integer, add the opposite of the negative integer.
14. When you subtract a negative integer, the difference is always negative.
Developing Concepts
Subtracting
Real Numbers
Goal
Subtract real numbers
using the subtraction
rule.
Key Words
• term
What is the change in a stock's value?
The daily change in the price
of a company’s stock can be
calculated by subtracting one
day’s closing price from the
previous day’s closing price.
In Example 5 you will see that
this change can be positive
or negative.
Some addition expressions can be evaluated using subtraction.
ADDITION PROBLEM EQUIVALENT SUBTRACTION PROBLEM
5 + (-3) = 2 5-3 = 2
-2 + (- 6 ) = -8 -2 - 6 = -8
Adding the opposite of a number is equivalent to subtracting the number.
SUBTRACTION RULE
To subtract b from a, add the opposite of b to a.
a — b = a + ( — b) Example: 3 — 5 = 3 + ( — 5)
The result is the difference of a and b.
Student HeCp
j ►Study Tip
In Example 1 notice that
10 - 11 =£11 - 10 .
Subtraction is not
commutative. The order
of the numbers affects
the answer.
\ _ /
J3SSE3 1 Use the Subtraction Rule
Find the difference.
a. 10-11 b. 11-10 c. -4-(-9)
Solution
a. 10 - 11 = 10 + (-11)
= -1
b. 11 - 10 = 11 + (-10)
= 1
c. -4 - (-9) = -4 + 9
= 5
Add the opposite of 11.
Use rules of addition.
Add the opposite of 10.
Use rules of addition.
Add the opposite of -9.
Use rules of addition.
Chapter 2 Properties of Real Numbers
You can change subtractions to additions by “adding the opposite” as a first step
in evaluating an expression.
■SEES* 2 Expressions with More than One Subtraction
Evaluate the expression 3 — (— 4 ) —
Solution
3 ~ (-4) - — = 3 + 4 + ( — “)
1
Add the opposites of -4 and ^ •
+
II
Add 3 and 4.
=
Add 7 and - j.
Use the Subtraction Rule
Use the subtraction rule to find the difference.
1.-3 - 5 2.12.7 - 10 3.1 -(-2)- 6 4. 7 - | - |
Student HeCp
1 >
p Look Back
For help with functions,
see p. 48.
\ _ J
3 Evaluate a Function
Evaluate the function y = —5 — x when x = — 2, — 1, 0, and 1.
Organize your results in a table.
Solution
Input
Function
Output
v = —2
y=-S- (-2)
y= -3
x = — 1
y = —5 — (-D
y = -4
x = 0
II
1
Ut
1
O
y=-5
,Y = 1
y = -5 - i
y= -6
Evaluate a Function
5. Evaluate the function y = 4 — x when x = — 3, — 1, 1, and 3. Organize your
results in a table.
TERMS OF AN EXPRESSION When an expression is written as a sum, the
parts that are added are the terms of the expression. For instance, you can write
— 5 — v as the sum —5 + (— x). The terms are —5 and —x. You can use the
subtraction rule to write an expression as a sum of terms.
2.4 Subtracting Real Numbers
4 F*wd the Terms of an Expression
Find the terms of —9 — 2x.
Solution Use the subtraction rule to rewrite the difference as a sum.
—9 — 2x = — 9 + (— 2x)
ANSWER ► The terms of the expression are —9 and — 2x.
Find the Terms of an Expression
Find the terms of the expression.
6. x — 3 7. —2 — 5x 8- —4 + 6x
9- lx + 2
Linkup
History
STOCK MARKET When
the New York Stock Exchange
opened in 1792, it reported
stock prices as fractions.
Stock prices began to be
reported as decimals in the
early 2000's.
5 Subtract Real Numbers
STOCK MARKET The daily closing prices for a company’s stock are given in
the table. Find the change in the closing price since the previous day.
Date
Aug. 23
Aug. 24
Aug. 25
Aug. 26
Aug. 27
Closing Price
21.38
21.25
21.38
20.69
20.06
Change
?
?
?
?
Solution Subtract the previous day’s closing price from the closing price for
the current day.
DATE
CLOSING PRICE
CHANGE
Aug. 23
21.38
Aug. 24
21.25
21.25 - 21.38 =
-0.13
Aug. 25
21.38
21.38 - 21.25 =
0.13
Aug. 26
20.69
20.69 - 21.38 =
-0.69
Aug. 27
20.06
20.06 - 20.69 =
-0.63
Subtract Real Numbers
10, The daily closing prices for a company’s stock are given in the table. Find
the change in the closing price since the previous day.
Date
Nov. 10
Nov. 11
Nov. 12
Nov. 13
Nov. 14
Closing Price
46.75
47.44
47.31
47.75
48.75
Change
?
?
?
?
Chapter 2 Properties of Real Numbers
L4 Exercises
Guided Practice
Vocabulary Check 1. Complete: In an expression that is written as a sum, the parts that are added
are called the ? of the expression.
2. Is lx a term of the expression Ay — lx — 9? Explain.
Skill Check 3. Use the number line to complete this statement: — 2 — 5 = ?
*-1-1-1-1-1-H-1- * -1-1-1-t-*
-9 -8 -7 -6 -5 -4 -3-2-1 0 1 2
Find the difference.
4. 4 - 5 5. 0 - (-7) 6. -2 - 8.7
Evaluate the expression.
7. 2 - (-3) - 6 8. -3 - 2 - (-5)
10- Evaluate the function y = 10 — x, when x =
results in a table.
Find the terms of the expression.
11- 12 — 5x 12- 5w — 8 13- — 12y + 6
9 . 6 - 2 -±
—5, —l, 1 and 5. Organize your
Practice and Applications
SUBTRACTION RULE Find the difference.
14. 4 - 9
18. -10 - 5
22. -3 - 1.7
4 7
26. -x -
15. 6-(-3) 16. -8-(-5)
19. 25 -(-14) 20. -10 -(-42)
23. 5.4 - (-3.8) 24. 9.6 - 6.5
27.
28 ' “I- "I
17. -2-9
21. 95 - 59
25. -2.2 - (-1)
29. -4 - i
■ Student HeCp
^ ->
► Homework Help
Example 1: Exs. 14-29
Example 2: Exs. 30-41
Example 3: Exs. 42-47
Example 4: Exs. 48-53
Example 5: Exs. 54-57
l _ )
EVALUATING EXPRESSIONS Evaluate the expression.
30. -1 - 5 - 8
33. 46 - 17 - (-2)
36. -8 - 3.1 - 6.2
31. 2 - (-4) - 7
34. -15 - 16 - 81
37. 2.3 - (-9.5) - 1.6
„ 4 2 5
40.-
9 9 9
32. 4-(-3)-(-5)
35. 11 - (-23) - 77
38. 8.4 - 5.2 - (-4.7)
2.4 Subtracting Real Numbers
EVALUATING FUNCTIONS Evaluate the function when x = -2, -1, 0,
and 1. Organize your results in a table.
42. y = x — 8 43. y = 12 — x 44. y = — x — ( — 5)
45. y = —8.5 — x 46. y = -x - 12.1 47. y = x - ~
FINDING TERMS Find the terms of the expression.
48. -4 - y 49. — jc — 7 50. — 3jc + 6
51. 9 - 28v 52. -10 + 4/? 53. a - 5
54. STOCK MARKET The daily closing prices for a company’s stock are given
in the table. Find the change in the closing price since the previous day.
Date
Sept. 11
Sept. 12
Sept. 13
Sept. 14
Sept. 15
Closing Price
101.31
103.19
105.75
104.44
102.19
Change
?
?
?
?
55. SUBMARINE DEPTH A submarine is at a depth of 725 feet below sea level.
Five minutes later it is at a depth of 450 feet below sea level. What is the
change in depth of the submarine? Did it go up or down?
P
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 56-57 is available at
www.mcdougallittell.com
Science Link j . In Exercises 56 and 57, use the diagram below which
shows the journey of a water molecule from A to B.
56. Find the change in elevation from each point to the next point.
57. Using your answers from Exercise 56, write an expression using addition and
subtraction that models the change in elevation of the water molecule during
its journey. Then evaluate the expression.
CHALLENGE Determine whether the statement is true or false. Use the
subtraction rule or a number line to support your answer.
58. If you subtract a negative number from a positive number, the result is
always a positive number.
59. If you subtract a positive number from a negative number, the result is
always a negative number.
Chapter 2 Properties of Real Numbers
Standardized Test
Practice
Mixed Review
Maintaining Skills
60. MULTIPLE CHOICE What does 5 - (-|j + | equal?
(A) 4 CD 4| CD 5j CD 6
61. MULTIPLE CHOICE What does -jc - 7 equal when x = -1?
CD -8 CD -6 CD 6 CD 8
62. MULTIPLE CHOICE Which of the following is not a term of the expression
— 12x - 2y + 1 ?
Ca) -12x CD 2y CD -2y CD 1
63. MULTIPLE CHOICE For a correct answer on a game show, a positive amount
is added to a player’s score. For an incorrect answer, a negative amount is
added. If a player has a score of — 100 and incorrectly answers a 300 point
question, what is the player’s new score?
CD -400 CD -200 CD 200 CD 400
NUMERICAL EXPRESSIONS Evaluate the expression. (Lesson 1.3)
64. 9 - 2 • 2 - 3 65. 1 • 10 + 5 • 5 66. 8 2 + 6 - 7
67. 4 • 2 3 + 9
68 . 4 - (12 h- 6) — 5
69. (10 - 2) • 7 + 8
SPORTS The table below gives the number of male and female
participants in high school sports for three school years. Based on
the table, explain whether the statement is true or false. (Lesson 1.7)
High School Sports Participants (millions)
Year
1994-95
1995-96
1996-97
Male
3.54
3.63
3.71
Female
2.24
2.37
2.24
DATA UPDATE of Statistical Abstract of the United States data at
www.mcdougallittell.com
70. There were more than six million total participants during 1994-1995.
71. There were about six million participants in the 1996-1997 school year.
GRAPHING Graph the numbers on a number line. (Lesson 2.1)
72. -1, 9, 3 73. -8, 4, -2 74. 6, -5, 0
75. 6.5, 2, -4.3 76. 7, 0.5, -9.1 77. |, 1
MULTIPLYING DECIMALS Multiply. (Skills Review p. 759)
78. 5 X 0.25 79. 0.1 X 0.4 80. 0.004 X 4.2
81.1.69 X 0.02 82.3.6 X 0.3 83.9.4 X 2.04
2.4 Subtracting Real Numbers
EVELOPING CONCEPTS
JJJjril'ij-DIJ US
For use with
Lesson 2.5
Goal
Use reasoning to find a
pattern for multiplying
integers.
Materials
• paper
• pencil
Question
i i .
How can you use addition to find the product of integers?
Explore
. HIT*
O Copy and complete the table. Use repeated addition to find the product.
Product
Equivalent sum
Solution
3(—3)
-3 + (-3) + (-3)
-9
2(-5)
-5 + (-5)
?
4(—2)
?
?
Q Copy and complete the table. Use the definition of opposites and your results
from Step 1 to find the product.
Product
Use definition
of opposites
Use result
from Step 1
Solution
— 3 (— 3 )
—( 3 )(— 3 )
-(-9)
9
— 2 (— 5 )
-( 2 )(- 5 )
-(?)
?
- 4 (— 2 )
?
?
?
Think About It
Use repeated addition to find the product.
1.3(2) 2.4(5) 3. 2(—6) 4. 5(-3)
Use the definition of opposites and repeated addition to find the product.
5. -2(6) 6. -3(4) 7. — 5(—5) 8 . -4(-3)
LOGICAL REASONING Based on your results from Exercises 1-8,
complete the statement with always , sometimes , or never.
9. The product of two positive integers is ? positive.
10. The product of a positive and a negative integer is ? positive.
11. The product of two negative integers is ? negative.
Chapter 2 Properties of Real Numbers
Multiplying Real Numbers
Goal
Multiply real numbers
using the rule for the
sign of a product.
Key Words
• closure property
• commutative property
• associative property
• identity property
• property of zero
• property of negative
one
How far did a flying squirrel drop?
An object’s change in position
when it drops can be found by
multiplying its velocity by the
time it drops. In Example 4 you
will find the change in position
of a squirrel.
The product of any two real numbers is itself a unique real number. We say that
the real numbers are closed under multiplication. This fact is called the closure
property of real number multiplication. Multiplication by a positive integer
can be modeled as repeated addition. For example:
3(—2) = (-2) + (-2) + (-2) = -6
This suggests that the product of a positive number and a negative number is
negative. Using the definition of opposites you can see that:
3( 2) = —(3)(—2) = -(-6) = 6
This suggests that the product of two negative numbers is positive. The general
rules for the sign of a product are given below.
Student MeCp
->
►Study Tip
In Example 1 note that:
(- 2) 4 = (— 2 )(— 2 ) (— 2 ) (— 2 )
= 16
is not the same as:
- 2 4 = -( 2 4 )
= (— 1 ){ 2 ){ 2 )( 2 )( 2 )
= -16
RULES FOR THE SIGN OF A PRODUCT OF NONZERO NUMBERS
[ • A product is negative if it has an odd number of negative factors.
• A product is positive if it has an even number of negative factors.
i Multiply Real Numbers
a. ~4(5) = — 20 One negative factor, so product is negative.
b. — 2(5)(—3) = 30 Two negative factors, so product is positive.
c. — 10 (— 0 . 2 )(— 4 ) = -8
d. (—2) 4 = 16
Three negative factors, so product is negative.
Four negative factors, so product is positive.
Multiply Real Numbers
Find the product.
1 . 3(—5) 2 . — 2(4)(5)
3. —j(—3)(—2) 4. (—2) 3
2.5 Multiplying Real Numbers
The rules for the sign of a product are a consequence of the following properties
of multiplication.
Student HeCp
►Study Tip
Notice the similarities
in the properties of
multiplication and the
properties of addition
(p. 79).
I J
PROPERTIES OF MULTIPLICATION
closure property The product of any two real numbers is a
unique real number.
ab is a unique real number Example: 4*2 = 8
commutative property The order in which two numbers are
multiplied does not change the product.
ab = ba Example: 3( — 2) = (-2)3
associative property The way you group three numbers
when multiplying does not change the product.
(ab)c = a(bc) Example: (-6 • 2)3 = -6(2 • 3)
identity property The product of a number and 1 is the number.
1 • a = a Example: 1 • (-4) = -4
property of zero The product of a number and 0 is 0.
0 • a = 0 Example: 0 • (-2) = 0
property of negative one The product of a number and -1 is the
opposite of the number.
-1 • a = —a Example: -1 • (-3) = 3
Student HeCp
->
►Writing Algebra
Example 2 shows an
efficient method based
on the Multiplication
Property of Negative
One. Fully written out
Example 2(a) is as
follows:
—2(—*) = ( —1)(2)( —1)(x)
= ( —1)( —1)(2x)
= lx
L J
Products with Variable Factors
Simplify the expression.
a. — 2 (— x) b. 3 (—ri)(—ri)(—ri) c.
l(-a) 2
Solution
a. — 2 (— x) — 2x
b. 3 (—n)(—n)(—n) = 3( — n 3 )
= —3 n 3
c. — 1 (~a) 2 = (—1 )(—a)(—a)
= (-i x« 2 )
Two minus signs, so product has
no minus sign.
Three minus signs, so product has
a minus sign.
One minus sign, so product has a
minus sign.
Write the power as a product.
Two minus signs, so product has
no minus sign.
Property of negative one
Products with Variable Factors
Simplify the expression.
5. 8(— t) 6- — x(— jc)(— x)(— x) 7. — 7(— b) 3
Chapter 2 Properties of Real Numbers
Student HeQp
^ More Examples
More examples
-^pv gre ava j| a b| e at
www.mcdougallittell.com
3 Evaluate a Variable Expression
Evaluate —4(—1)(—x) when* = —5.
Solution You can simplify the expression first, or substitute for x first.
4?
1
II
I
Simplify expression first.
= — 4(— 5)
Substitute -5 for x.
= 20
Two negative factors, so product is positive.
—4(—1)(— *) = —4(—1)[— (—5)]
Substitute -5 for x first.
= —4(—1)(5)
Use definition of opposites.
= 20
Two negative factors, so product is positive.
Evaluate a Variable Expression
Evaluate the expression when x = -2.
8. — 9(x)(—2) 9. 3(4)(—x) 10. 3(-x) 3
11. 7(x 2 )(—5)
FLYING SQUIRRELS glide
through the air using "gliding
membranes/' which are flaps
of skin that extend from their
wrists to their ankles.
FLYING SQUIRRELS A flying squirrel drops from a tree with a velocity of
—6 feet per second. Find the displacement , which is the change in position, of
the squirrel after 3.5 seconds.
Solution
Verbal
Model
Labels
Algebraic
Model
Displacement
=
Velocity
•
Time
Displacement = d (feet)
Velocity = —6 (feet per second)
Time = 3.5 (seconds)
d = -6 • 3.5
d= -21
ANSWER ^ The squirrel’s change in position is —21 feet. The negative sign
indicates downward motion.
12. A helicopter is descending at a velocity of —15 feet per second. Find the
displacement of the helicopter after 4.5 seconds.
■
2.5 Multiplying Real Numbers
==^il Exercises
Guided Practice
Vocabulary Check
Match the property with the statement that illustrates it.
1. Commutative property
2_ Associative property
3. Identity property
4. Property of zero
5. Property of negative one
A. —1 • 9 = —9
B. 4(—2) = (-2)4
C. 0 • 8 = 0
D. 1 • (-15) = -15
E. -7(5 • 2) = (-7 • 5)2
Skill Check Find the product.
6. 9(—1) 7.-5(7)
8. —4(—6) 9. (— l) 5
Simplify the expression.
10. —3(—6)(a) 11.5 (-t)(-t)(-t)(-t) 12. 6(-x) 3
Evaluate the expression for the given value of the variable.
13. 2(—5)(— x) whenx = 4 14. 6(—2)(x) whenx = —3
Practice and Applications
CLOSURE PROPERTY Tell whether the set is closed under the operation
by deciding if the combination of any two numbers in the set of numbers
is itself in the set.
15, even integers under multiplication 16, odd integers under addition
p
Student HeCp
I
► Homework Help
Example 1: Exs. 15-31
Example 2: Exs. 32-40
Example 3: Exs. 41-49
Example 4: Exs. 50-55
H
MULTIPLYING REAL NUMBERS Find the product.
17.-7(4) 18. 5(-5) 19.-6.3(2)
20 . — 7 (— 1 . 2 )
23. (—6) 3
26. —2(—5)(7)
21 .
I
' 2 \3
22 . -12
1
24. (—4) 4
27. 6(9)(-l)
29. 2.7(—6)(—6)
30. —3.3(—1)(—1.5)
25. ~{lf
28. —5(—4)(—8)
2 V3
31.15 -
15 )\Aj
PRODUCTS WITH VARIABLE FACTORS Simplify the expression.
32. — 3(->-) 33. l{-x) 34. ~2{k)
35. 5(— a){—a){—a) 36. — 8(z)(z) 37. — 2(5)(—r)(— r)
38. {-bf 39. —2(—x) 2 40. -(-j) 4
Chapter 2 Properties of Real Numbers
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable.
41. —8(d) when d = 6
43. —3(—a)(—a) when a = —1
45. —4.1(—5 )(h) when h = 2
42. 3(—4 )(n) when n = —2
44. 9(—2)(—r) 3 when r = 2
46. — 2rffiJ(0 when t = —3
Student HeCp COUNTEREXAMPLES Determine whether the statement is true or false.
If it is false, give a counterexample.
47. (~a) • (-b) = (~b) • (~a)
48. The product (—a ) • (— 1) is always positive.
49. If a > b, then a • 0 is greater than b • 0.
► Look Back
For help with
counterexamples, see
p. 73
link to
Careers
MOUNTAIN GUIDES plan
climbing expeditions. The
guides also instruct students
on basic climbing techniques,
such as rappelling.
|Y| ore a k out mountain
guides is available at
www.mcdougallittell.com
MOUNTAIN RAPPELLING You rappel down the side of a mountain at a
rate of 2 feet per second.
50. Write an algebraic model for your displacement d (in feet) after t seconds.
51. What is your change in position after rappelling for 10 seconds?
52. If the mountain is 40 feet high, how much farther must you rappel before you
reach the ground?
Science Link / Scientists estimate that a peregrine falcon can dive for its
prey at a rate of about 300 feet per second.
53. Write an algebraic model for the displacement d (in feet) of a peregrine
falcon after t seconds.
54. What is a peregrine falcon’s change in position after diving for 2 seconds?
55. If the peregrine falcon spotted its prey 750 feet below, how much farther
must it dive to reach its prey?
A multiplication magic square is a square in which the
product of the numbers in every horizontal, vertical, and main diagonal
line is constant.
56. Find the constant of the
magic square shown by
multiplying the numbers in
the first row of the square.
57. Copy and complete the magic
square by finding the missing
number in each column.
58. Check your answer by
finding the product of
each main diagonal.
vertical
columns
C if)
o ^
N O
O
_C
ill
main diagonals
2.5 Multiplying Real Numbers
Standardized Test
Practice
Mixed Review
Maintaining Skills
59. [MULTIPLE CHOICE What does -3(6)(-|j equal?
(A) -6 Cb) -2 C© 2 C© 6
60. MULTIPLE CHOICE Which of the following statements is not true?
CD The product of any number and zero is zero.
C© The order in which two numbers are multiplied does not matter.
G±) The product of any number and — 1 is a negative number.
GD The product of any number and — 1 is the opposite of the number.
61. MULTIPLE CHOICE Simplify the expression 2(-4)(-x)(-x)(~x).
(a) — 24x QD — 8x 3 C© 8x 3 C© 24x
62. MULTIPLE CHOICE Evaluate 9(— jc) 2 ( —2) when jc = 3.
CD -162 C© -108 CH) 108 C© 162
MENTAL MATH Use mental math to solve the equation. (Lesson 1.4)
63. 6 + c = 8 64. x — 1 — 4 65. 8 - a = 4
66. 3z = 15 67. (ra)(2) = 24 68. r + 6 = 2
LINE GRAPHS Draw a line graph to represent the function given by the
input-output table. (Lesson 1.8)
FINDING ABSOLUTE VALUE Evaluate the expression. (Lesson 2.2)
71. 121 72.|-6| 73. — 191 74. — | —7 |
75. | -7.2 1 76.-| 6.8 1 77. 1 10.43 | 78.-| -0.05
FINDING TERMS Identify the terms of the expression. (Lesson 2.4)
79.12 -z 80. —t + 5 81. 4w — 11
82. 31 - 15w 83. -lx + 4x 84. -3 c ~ 4
LEAST COMMON MULTIPLE Find the least common multiple of
the numbers. (Skills Review p. 761)
85. 4 and 5 86. 24 and 36 87. 30 and 25
H
88 . Ill and 55
Chapter 2 Properties of Real Numbers
89. 312 and 210
90. 176 and 264
EVELOPING CONCEPTS
For use with
Lesson 2.6
Goal
Use reasoning to discover
how to use the distributive
property to write
equivalent expressions.
Question
How can you model equivalent expressions using algebra tiles?
Each represents 1 and each
represents v.
Materials
• algebra tiles
Explore
O Model 3(2 + 4). Q Model 6 + 12.
Student HeCp
-\
^ Look Back
For help with using
algebra tiles, see p. 77.
\ _>
( )
( )
(■■■■■■)
Make 3 groups each consisting
of two plus four, or six, 1-tiles.
Make a group of six 1-tiles
and a group of twelve 1-tiles.
Complete: The models show that 3(2 + 4) = ? .
This is an example of the distributive property.
© Model 3(x + 4).
c™
)
(H
)
(s
D
Make 3 groups each consisting
of one x-tile and four 1-tiles.
Q Model 3x + 12.
Make a group of three x-tiles
and a group of twelve 1-tiles.
Complete: The models show that 3(x + 4) = ? .
This is another example of the distributive property.
Think About It
Each equation illustrates the distributive property. Use algebra tiles to
model the equation. Draw a sketch of your models.
1. 5(1 + 2) = 5 + 10 2. 2(4 + 3) = 8 + 6 3. 7(1 + 1) = 7 + 7
4. 6(x + 2) = 6x + 12 5- 4(x + 4) = 4x + 16 6. 3(x + 5) = 3x + 15
7. 2(jc + 3) = 2x + 6 8- 5 (jc + 1) = 5x + 5 9, 9(x + 2) = 9x + 18
10- LOGICAL REASONING Use your own words to explain the distributive
property. Then use a, b, and c to represent the distributive property
algebraically.
Developing Concepts
The Distributive
Property
Goal
Use the distributive
property.
Key Words
• distributive property
How much will you pay for six CDs?
When you go shopping, you can
use estimation or mental math to
determine the total cost. In Example 5
you will learn how to use the
distributive property to calculate the
total cost of six CDs —without using
a calculator.
The distributive property is an important algebraic property. Example 1 uses
geometry to illustrate why the property is true for a single case. On the following
page the property is formally defined.
Student HeCp
\
►Vocabulary Tip
To distribute means to
give something to each
member of a group.
In Example 1 you can
think of the 3 as being
distributed to each
term in (x+ 2).
\ _ /
J 1 Use an Area Model
Find the area of a rectangle whose width is 3 and whose length is x + 2.
Solution You can find the area in two ways. Remember that the area of a
rectangle is the product of the length times the width.
Area of One Rectangle Area of Two Rectangles
x+2
Area = 3(x + 2) Area = 3(x) + 3(2)
x 2
ANSWER ^ Because both expressions represent the same area, the following
statement is true.
Area = 3(x + 2) = 3(x) + 3(2) = 3x + 6
Use an Area Model
1 - Write two expressions for the area of
the rectangle.
I
2. Write an algebraic statement that l _
shows that the two expressions |- 3 -|- x
from Exercise 1 are equal.
Chapter 2 Properties of Real Numbers
THE DISTRIBUTIVE PROPERTY
Student HeCp
►Study Tip
Although four versions
of the distributive
property are listed, the
last three versions can
be derived from the first:
a(b + c) = ab + ac
Student HeCp
► More Examples
More examples
^5^ are available at
www.mcdougallittell.com
V
The product of a and (b + c):
a(b + c) = ab + ac
Example:
5(x + 2) = 5x+ 10
(b + c)a = ba + ca
Example:
(x + 4)8 = 8x + 32
The product of a and (b - c):
a(b - c) = ab - ac
Example:
4(x - 7) = 4x - 28
(b - c)a = ba - ca
L._
Example:
(x- 5)9 = 9x- 45
E222SB 2 Use the Distributive Property with Addition
Use the distributive property to rewrite the expression without parentheses.
a. 2(x + 5)
Solution
a. 2(5 + 5 ) = 2(x) + 2 ( 5 )
b. (1 + 2n)8
= 2x + 10
Distribute 2 to each term of (x + 5).
Multiply.
b. (1 + 2n )8 = (1)8 + (2n)8
= 8 + 16n
Distribute 8 to each term of (1 + 2n).
Multiply.
3 Use the Distributive Property with Subtraction
Use the distributive property to rewrite the expression without parentheses.
a. 3(1 - y)
Solution
b. (2x - 4)j
a. 3(1 - y) = 3(1) - 3(y)
= 3 - 3y
Distribute 3 to each term of (1 - y).
Multiply.
i i
b. (2x - 4)j = (2x)j - (4)^
1
Distribute y to each term of (2x - 4).
Multiply.
Use the Distributive Property
Use the distributive property to rewrite the expression without
parentheses.
3, 5(ft + 3) 4. (2 p + 6)3
6 . (3y - 9)f
5. 2(x - 5)
2.6 The Distributive Property
Student HeCp
-5
► Study Tip
Be careful when using
the distributive property
with negative factors.
Forgetting to distribute
the negative sign is a
common error.
4 Use the Distributive Property
Use the distributive property to rewrite the expression without parentheses,
a. -3{x + 4) b. (y + 5)(-4) c. -(6 - 3x) d. (1 - t)(- 9)
Solution
a. —3(x + 4) = — 3(jc) + ( — 3)(4)
= -3jc - 12
Use distributive property.
Multiply.
b- (y + 5)(— 4) = (y)(— 4) + (5)(— 4) Use distributive property.
= -4 y - 20
Multiply.
c. —(6 — 3x) = —1(6) — (—l)(3x) Use distributive property.
= —6 + 3x
Multiply.
d. (1 - 0(-9) = (IX — 9) - (tX-9)
L
= -9 + 9t
Use distributive property.
Multiply.
Use the Distributive Property
Linknoj^
History
COMPACT DISCS CDs
were released in the United
States in 1984. That year
the average cost of a CD
was $17.81.
Use the distributive property to rewrite the expression without
parentheses.
7. —5(a + 2) 8. (x + 7)(-3) 9. -(4 - 2x) 10. (4 - m)(-2)
5 Mental Math Calculations
COMPACT DISCS You are shopping for compact discs. You want to buy
6 compact discs for $11.95 each including tax. Use the distributive property
to mentally calculate the total cost of the compact discs.
Solution If you think of $11.95 as $12.00 — $.05, the mental math is easier.
6(11.95) = 6(12 — 0.05) Write 11.95 as a difference.
= 6(12) — 6(0.05) Use distributive property.
= 72 — 0.30 Find products mentally.
= 71.70 Find difference mentally.
ANSWER ^ The total cost of 6 compact discs at $11.95 each is $71.70.
Mental Math Calculations
Use the distributive property to mentally calculate the total cost.
11. You are buying birthday cards for 3 of your friends. Each card costs $1.25.
What is the total cost of the cards?
Chapter 2 Properties of Real Numbers
H Exercises
Guided Practice
Vocabulary Check Explain how you would use the distributive property to simplify
the expression.
1- 2(x + 3) 2 . (x + 4)5 3 - 7(x — 3) 4 . (x - 6)4
Skill Check Use the area model shown.
5 - Write two expressions for the area
of the rectangle.
6 . Write an algebraic statement that
shows that the two expressions
from Exercise 5 are equal.
Match the expression with its simplified form.
7. 3(x + 2) 8. (x + 3)(—2) 9. ~3(x - 2) 10- (3 - x)2
A- 6 — 2x B. — 3x + 6 C- — 2x — 6 D. 3x + 6
Use the distributive property and mental math to simplify the expression.
11. 4(1.15) = 4(1 + 0.15) 12. 9(1.95) = 9( ? - ?)
= »:(■)+!(■) =?(?)- ? ( 7 )
= ■ + ? = ? - ?
-m = ?
Practice and Applications
Student HeCp
► Homework Help
Example 1: Exs. 13-16,
71-73
Example 2: Exs. 17-28
Example 3: Exs. 29-40
Example 4: Exs. 41-54
Example 5: Exs. 55-70
L _ )
AREA MODEL Use the area model to find two expressions for the area of
the rectangle. Then write an algebraic statement that shows the two
expressions are equal.
X
2
I
X
1
T
2 x
_L
\—\
2.6 The Distributive Property
DISTRIBUTIVE PROPERTY WITH ADDITION Use the distributive property
to rewrite the expression without parentheses.
17. 3(x + 4)
18. 5(w + 6)
19. 7(1 + 0
20. (y + 4)5
21. (2 + u) 6
22. (x + 8)7
23. 2(2 y + 1)
24. (3x + 7)4
25. 3(4 + 6a)
26. (9 + 3«)2
27. (x + 2)1.3
28. |(10 + 15r)
DISTRIBUTIVE PROPERTY WITH SUBTRACTION Use the distributive
property to rewrite the expression without parentheses.
29. 5 (y - 2)
30. 2(x - 3)
31. 9(7 - a)
32. (x - 2)2
33. (7 — m) 4
34. (n - 7)2
35. 10(1 - 3f)
36. 7(6w — 1)
37. (3x - 3)6
38. (9 - 5a)4
39. (-3.1 u - 0.8)3
40 - 5 (w ~ i)
DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the
expression without parentheses.
41. —3 (r + 8)
42. — 2(x - 6)
43. -(1 + s)
44. (2 + 0(—2)
45. (j + 9)(-l)
46. (x - 4)(—3)
47. —6(4 a + 3)
48. (9x + l)(-7)
49. -(6 y - 5)
50. (3 d - 8)(—5)
51. (2.3 - 7w)(—6)
52. -|6 + 24)
ERROR ANALYSIS In Exercises 53 and 54, find and correct the error.
Student HeCp
I
► Homework Help
Extra help with
problem solving in
Exs. 55-66 is available at
www.mcdougallittell.com
MENTAL MATH Use the distributive property and mental math to
simplify the expression.
55. 4(6.11)
56. 10(7.25)
57. 3(9.20)
58. 7(5.98)
59. 2(2.90)
60. 6(8.75)
61. -3(4.10)
62. -9(1.02)
63. -2(11.05)
64. -8(2.80)
65. -5(10.99)
66. -4(5.95)
67. CONTACT LENS SUPPLIES The saline solution that you use to clean your
contact lenses is on sale for $4.99 a bottle. You decide to stock up and buy
4 bottles. Use the distributive property to mentally calculate the total cost of
the bottles of saline.
Chapter 2 Properties of Real Numbers
Link to
Careers
AGRONOMISTS use soil
and plant science to help
farmers increase crop
production and maintain
soil fertility.
Standardized Test
Practice
68. SCHOOL SUPPLIES You are shopping for school supplies. You want to buy
7 notebooks for $1.05 each. Use the distributive property to mentally
calculate the total cost of the notebooks.
69. DECORATIONS You are volunteering for a charity that is operating a
haunted house. You are sent to the store to buy 5 bags of cotton balls that will
be used to make spider web decorations. Each bag costs $2.09. Use the
distributive property to mentally calculate the total cost of the cotton balls.
70. GROCERIES You see a sign at the grocery store that reads, “Buy
2 half-gallons of frozen yogurt and get one free.” Each half-gallon of frozen
yogurt costs $4.95. You decide to get three half-gallons. Use the distributive
property to mentally calculate the total cost of the frozen yogurt.
FARMING You are trying to determine the size of a cornfield, so you will
know how many rows of corn to plant. Let x be the width of the
cornfield. Use the diagram of the farming field shown.
71 . Use the diagram to find two
expressions for the area of the
entire field.
72. Write an algebraic statement that
shows the two expressions from
Exercise 71 are equal.
73. You decide to plant the cornfield
so that x = 75 yards. What is the
area of the entire field? Use one
expression from Exercise 71 to
find the solution and the other to
check your solution.
200 yd
74. MULTIPLE CHOICE Which expression is equivalent to (x + 7)3?
(A) x + 21 (Ip 3x + 7
Cep 3x + 10 Cd) 3x + 21
75. MULTIPLE CHOICE Which expression is equivalent to 6(x — 2)?
(F) 6x — 2 Cep 6x — 12
Ch) 6x + 2 GD 6x + 12
76. MULTIPLE CHOICE Which expression is equivalent to (5 — x)(—17)?
(A) 5 — 17x (ID 5 + 17x
Cep —85 — 17x Cd) —85 + 17x
77. MULTIPLE CHOICE You are buying 5 new shirts to wear to school,
the shirts are on sale for $19.99. What expression would you use to
mentally calculate the total cost of the shirts?
CE) 5(20) - 0.01 (ID 5(20 - 0.01) (E) 5(20) + 0.01 GD 5(20 +
2.6 The Distributive Property
All of
0 . 01 )
Mixed Review
EXPRESSIONS WITH FRACTION BARS Evaluate the expression. Then
simplify the answer. (Lesson 1.3)
78.
10 • 8
79.
6 2 - 12
3 2 + 1
80.
75 - 5 2
13 + 3 • 4
81.
3 • 7 + 9
2 4 + 5 - 11
82.
4 ♦ 2 + 5 3
3 2 - 2
83.
6 + 7 2
3 3 - 9 - 7
NAMING PROPERTIES Name the property shown by the statement.
(Lesson 2.3)
84. -10 + (-25) = -25 + (-10) 85. -19 + 0 = -19
86. 32 + (-32) = 0 87. (-13 + 8) + 7 = -13 + (8 + 7)
EVALUATING EXPRESSIONS Evaluate the expression. (Lesson 2.4)
88. 6 - 7 89. 9 - (-3) 90. 4 - 8 - 3
91. 6 -(-8)- 11 92.7.2 - 9 - 8.5 93. j - | - 1
Maintaining Skills DECIMALS AND FRACTIONS Write the decimal as a fraction in simplest
form. (Skills Review p. 767)
94.0.14 95.0.25 96.0.34 97.0.50
98.0.75 99.0.82 1 00.0.90 1 01.0.96
Quiz 2
Evaluate the function when x= -3, -1, 1, and 3. Organize your results in
a table. (Lesson 2.4)
1.y = x~12 2. y = 21 — x 3. y = x - ^
Find the terms of the expression. (Lesson 2.4)
4. 2x — 9 5- 8 — x 6. — l(k + 4
7. STOCK MARKET The daily closing prices for a company’s stock are
$19.63, $19.88, $20.00, $19.88, and $19.75. Find the day-to-day change in
the closing price. (Lesson 2.4)
Find the product. (Lesson 2.5)
8. -7(9) 9. —5(—6) 10. 35(—80)
11. —1.8(—6) 12. —15(|) 13. —10(—3)(9)
Rewrite the expression without parentheses. (Lesson 2.6)
14. (x + 2)11 15. 5(12- y) 16. -4(3a - 4)
17. SHOPPING You want to buy 2 pairs of jeans for $24.95 each. Use the
distributive property to mentally calculate the total cost of the jeans.
(Lesson 2.6)
Chapter 2 Properties of Real Numbers
Combining Like Terms
Goal
Simplify an expression by
combining like terms.
Key Words
• coefficient
• like terms
• simplified expression
How far is it to the National Air and Space Museum?
An algebraic expression is easier
to evaluate when it is simplified.
In Example 4 you will use the
distributive property to simplify
an algebraic expression that
represents the distance to the
National Air and Space Museum.
Student MaCp
^ — - \
► Reading Algebra
Note that —x has a
coefficient of -1 even
though the 1 isn't
written. Similarly, x has
a coefficient of 1.
^ j
In a term that is the product of a number and a variable, the number is called the
coefficient of the variable.
-1 is the coefficient of x.
—x + 3x 2
_j t_
3 is the coefficient of x 2 .
Like terms are terms in an expression that have the same variable raised to
the same power. For example, 8x and 3x are like terms. Numbers are considered
to be like terms. The terms x 2 and x, however, are not like terms. They have the
same variable, but it is not to the same power.
} ;l£ Identify Like Terms
Identify the like terms in the expression — x 2 + 5x — 4 — 3x + 2.
Solution
Begin by writing the expression as a sum:
—x 2 + 5x + (—4) + (— 3x) + 2
ANSWER ► The terms 5x and — 3x are like terms. The terms —4 and 2 are also
like terms.
Identify Like Terms
Identify the like terms in the expression.
1. — 5x 2 — x + 8 + 6x — 10 2 . — 3x 2 + 2x + x 2 — 4 + lx
2.7 Combining Like Terms
SIMPLIFIED EXPRESSIONS The distributive property allows you to combine
like terms by adding their coefficients. An expression is simplified if it has no
grouping symbols and if all the like terms have been combined.
student HeCp ■mmw 2 Combine Like Terms
Simplify the expression.
a. 8x + 3x b. 2 y 2 + ly 2 — y 2 + 2
Solution
a. 8x + 3x = (8 + 3)x Use distributive property.
= 1 lx Add coefficients.
V ->
► Study Tip
In Example 2 the
distributive property
has been extended to
three terms:
(b + c + d)a =
ba + ca + da
L _ j
b. 2 y 2 + ly 2 - y 2 + 2 = 2 y 2 + ly 2 - ly 2 + 2
= (2 + 7 - l)j 2 + 2
= 8/ + 2
Coefficient of -y 2 is -1.
Use distributive property.
Add coefficients.
Student HeCp
► More Examples
^ ore exam P' es
are available at
www.mcdougallittell.com
H21SS19 3 Simplify Expressions with Grouping Symbols
Simplify the expression.
a. 8 - 2(x + 4) b. 2(x + 3) + 3(5 - jc)
Solution
a_ 8 — 2(x + 4)
= 8 - 2(x) + (-2)(4)
= 8 - 2x - 8
= — 2x + 8 — 8
= —2x
b. 2(x + 3) + 3(5 - x)
= 2(x) + 2(3) + 3(5) + 3(-jc)
= 2x + 6 + 15 — 3x
= 2x — 3x + 6 + 15
= ~x + 21
Use distributive property.
Multiply.
Group like terms.
Combine like terms.
Use distributive property.
Multiply.
Group like terms.
Combine like terms.
Simplify Expressions
Simplify the expression.
3 . 5x — 2x
4 . 8 m — m — 3m + 5
5 . —x 2 + 5x + x 2
6 . 3(j + 2) -
7 . 9x - 4(2x - 1)
8 . -(z + 2) - 2(1 - z)
Chapter 2 Properties of Real Numbers
SUBWAY The Metrorail,
in Washington, D.C., has
over 90 miles of rail line and
serves an area of about
1500 square miles.
Student tfedp
► Study Tip
Notice that the total
time is 50 minutes and
that the time riding the
subway is t minutes.
So the time walking is
(50 - t) minutes.
4 Simplify a Function
SUBWAY It takes you 50 minutes to get to the National Air and Space
Museum. You spend t minutes riding the subway at an average speed of 0.5 mile
per minute. The rest of the time is spent walking at 0.05 mile per minute.
a. Write and simplify a function that gives the total distance you travel.
b. If you spend 40 minutes on the subway, how far is it to the museum?
Solution
a. Verbal
Model
Labels
Distancp
Subway
Time
• -f-
Walking
Time
•
speed
riding
speed
walking
Algebraic
Model
Distance = d
Subway speed = 0.5
Time riding = t
Walking speed = 0.05
Time walking =50 — t
d = 0.5 t + 0.05 (50 — t)
(miles)
(mile per minute)
(minutes)
(mile per minute)
(minutes)
You can use the distributive property to simplify the function.
d = 0.5 1 + 0.05(50 — t) Write original function.
= 0.5 1 + 0.05(50) — 0.05(0 Use distributive property.
= 0.5^ + 2.5 — 0.05/^ Multiply.
= 0A5t + 2.5 Combine like terms.
ANSWER ^ The total distance you travel is given by d = 0A5t + 2.5, where t
represents 0 to 50 minutes.
b. To find the total distance, evaluate the function for a time of / = 40.
d = 0A5t + 2.5 Write simplified function.
= 0.45(40) + 2.5 Substitute 40 for t.
= 20.5 Multiply and add.
ANSWER ► It is about 21 miles to the museum.
Simplify a Function
It takes you 45 minutes to get to school. You spend t minutes riding the
bus at an average speed of 0.4 mile per minute. The rest of the time is
spent walking at 0.06 mile per minute.
9. Write and simplify a function that gives the total distance you travel.
10. If you ride the bus for 30 minutes, how far away is your school?
2.7 Combining Like Terms
Guided Practice
Vocabulary Check
1. In the expression lx 2 — 5x + 10, what is the coefficient of the x 2 -term?
What is the coefficient of the x-term?
2 . Identify the like terms in the expression — 6 — 3x 2 + 3x —4x + 9x 2 .
Skill Check
Simplify the expression by combining like terms if possible. If not
possible, write already simplified.
3-5 r + r 4 . w — 3w 5 - —4k — 8 + 4 k
6- 12 — 10m + m
7 . 2 a 2 + 3 a + 2 a 2
8-8 — 4 1 + 61 2
Simplify the expression.
9. 14/ + 4(f + 1) 10. 21# — 2(g — 4) 11. -5(2m + 4) - m
12. 7(3 a + 2) + 5 13. 5(x - 7) + 4{x + 2) 14. 2(4; - 1) - 4(1 - ;)
Practice and Applications
IDENTIFY LIKE TERMS Identify the like terms in the expression.
15 . 3 a + 5 a 16 . 5 s 2 — 10s 2
17. m + 8 + 6m
19 . — 6w —12 — 3 w + 2w 2
18. 2p + 1 + 2p + 5
20 . 3x 2 + 4x + 8x — lx 2
COMBINING LIKE TERMS Simplify the expression by combining like
terms if possible. If not possible, write already simplified.
21 . —12m + 5m 22 . 4 y — 3 y 23 . 3c — 5 — c
24 . 5 - h + 2 25 . r + 2r + 3r - 7 26 . 8 + 2z + 4 + 3z
27 . 2rc
28 . 6a — 2a 2 + 4a
29 . p 2 + 4p + 5p 2 - 2
Student HeCp
f
^Homework Help
Example 1: Exs. 15-20
Example 2: Exs. 21-29,
39, 41,42
Example 3: Exs. 30-38,
40, 43, 44
Example 4: Exs. 45-52
SIMPLIFYING EXPRESSIONS Simplify the expression.
30 . -10O — 1) + 4b 31 . 9 - 4(9 + y) 32 . 6(4 +/) - 8/
33 . 1 - 2(6 + 3 r) 34 . -5(2 + lx) - 3x 35 . 5(2 m + 5) - 6
36 . 3(4 p + 3) + 4 {p - 1 ) 37 . 9 (c + 3) - 7(c - 3) 38 . 4(x + 2) - (x + 2)
ERROR ANALYSIS In Exercises 39 and 40, find and correct the error.
39 . ^ ^ 40 .
?x = 16
Chapter 2 Properties of Real Numbers
Student MeCp
► Skills Review
For help with perimeter,
see p. 772.
Then simplify the expression.
Write an expression for the perimeter of the figure.
41.
x —
x
7 x —
x
7
2x + 3
G eometry Link / Write an expression for the perimeter of the figure.
Then simplify the expression.
43.
2 (*+ 2 )
44.
x+ 4
x+2
9(x-2)
x+ 4
2(x-2)
2 (*+ 2 )
Link to
Transportation
FREIGHT TRAINS A 150-car
freight train is so heavy that
it takes 1.5 miles to come to
a complete stop if traveling
50 miles per hour.
FREIGHT TRAINS A train with 150 freight cars is fully loaded with two
types of grain. Each freight car can haul 90.25 tons of barley or 114 tons
of corn. Let n represent the number of freight cars containing corn.
45. Which function correctly represents the total weight W the train can haul?
A. W = 90.25(150 - n) + 114n B.W= 90.25 n +114(150- n)
46. Simplify the correct function.
47. If 90 freight cars contain corn, what is the total weight the train is hauling?
MOVING You have 8 moving boxes that you can use to pack for college.
Each box can hold 15 pounds of clothing or 60 pounds of books. Let c be
the number of boxes that contain clothing.
48. Write a function that gives the total pounds T of the boxes in terms of the
number of boxes that contain clothing.
49. Simplify the function.
50. If 5 boxes contain clothing, how many pounds will you be moving?
SHOPPING You have $58 and you want to buy a pair of jeans and a
$20 T-shirt. There is a 6% sales tax. Let x represent the cost of the jeans.
The following inequality models how much you can spend on the jeans.
x + 20 + 0.06(x + 20) < 58
51. Simplify the left side of the inequality.
52. If the jeans cost $35, can you buy both the T-shirt and the jeans? Explain.
CHALLENGE Evaluate the expression for the given value of x. Then
simplify the expression first and evaluate the expression again. Which
way is easier? Explain.
53. —x(8 — x) + 2x when x = 2 54. 6 (—x — 3) — x(9 + x) when x = 4
2.7 Combining Like Terms
Standardized Test
Practice
Mixed Review
Maintaining Skills
55. MULTIPLE CHOICE In the expression 5 + 12 d — 3d 2 , what is the
coefficient of the c/Merm?
(A) -3 CD 3 (©2 CD d
56. MULTIPLE CHOICE Which expression is simplified?
CD 7 + 5k — 5k CD 3x — 9 + 2x 2
CD -8g + 5-8 g CD llz - 4z
57. MULTIPLE CHOICE Simplify the expression 2x - 3x 2 - x.
(A) 3x — 3x 2 CD — 2x 2 CD x — 3x 2 CD 0
58. MULTIPLE CHOICE Simplify the expression -4(y + 2) - 5y.
CD -9 y - 8 CD — 9y - 6 CD -9y + 2 CD -9y + 8
INTERNET Use the bar graph at the
right which shows the percent of schools
in the United States with access to the
Internet. (Lesson 1.7)
59. What percent of schools had access
to the Internet in 1994?
60. What is the difference in the percent
of schools that had access to the
Internet in 1996 and 1997?
Schools with
Internet Access
100 %
80%
60%
40%
20 %
0 %
1994 1995 1996 1997
RULES OF ADDITION Find the sum. (Lesson 2.3)
61. -1 + 10 62. 8 + (-4) 63. -3 + (-3)
64. 6.5 + (-3.4)
65. -9.7 + (-4.4) 66. ~ + J
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 2.5)
67. 9(—4)(jc) whenx = 5 68. — 5(6)(a) when a = —2
69. —3(0(0 when t = — 1 70. 7(2)(—w)(— w) when w = 6
71. — 8.3(—1.2)(/0 when p — 3
72. when d — —4
ORDERING FRACTIONS Write the fractions in order from least to greatest.
(Skills Review p. 770)
73.
2 ^ 9 _ _ 5 _ 6 _
10 ’ 10 ’ 10 ’ 10 ’ 10
74.
3 3 3 3 3
5’ 2’ 3’ T 8
75.
1 3 3 4 7
4’ 8’ 4’ 8’ 8
4 1 3_ i 7
5’ 2’ 10’5’ 10
77.
4 5 3 2 2
6’ 2’ 4’ 6’ 2
7 q 1 2 1 1 I
7’ 5’ 10’ 14’ 2
79 2 I 12 3 2 8
8 ’ 8 ’ 8 ’ 8 ’ 8
14 2 12
80. 5- 4- 5- 5- 4-
9 ’ ^ ’ 3’ 2 ’ 9
74 2115
81 9—— — 9— —— —
10’ 5’ 3’ 15’ 6
Chapter 2 Properties of Real Numbers
Dividing Real Numbers
Goal
Divide real numbers and
use division to simplify
algebraic expressions.
Key Words
• reciprocal
How fast does a hot-air balloon descend?
In Lesson 2.5 you learned that
a downward displacement is
measured by a negative
number. In the example on
page 117 you will divide a
negative displacement by time
to find the velocity of a hot-air
balloon that is descending.
2 5
Two numbers whose product is 1 are called reciprocals. For instance, and ——
are reciprocals because ^-V—-0= 1.
INVERSE PROPERTY OF MULTIPLICATION
1
For every nonzero number a , there is a unique number — such that:
a
1 1
a • — = 1 and — • a = 1
a a
You can use a reciprocal to write a division expression as a product.
DIVISION RULE
To divide a number a by a nonzero number b, multiply a by the
reciprocal of b. The result is the quotient of a and b.
1 11
a -r- b = a • ^ Example: — 1-^3=— 1 •
i Divide Real Numbers
Student HeCp
a. 10 + (-2) = 10 • (--) = -5
► Study Tip
When you divide by a
\ /
b. 0-S- — = 0» — = 0
mixed number, it is
usually easiest to first
rewrite the mixed
c. -39 - (—4-) = -39 4- (-—) = -39
number as an
V 3/ V /
improper fraction.
A
V
= 9
2.8 Dividing Real Numbers
Student HeCp
p Study Tip
A quotient is defined as
quotient =
dividend
divisor
So you can check your
solution by showing:
quotient • divisor =
dividend.
2 Simplify Complex Fractions
Find the quotient.
1
3
a. 4
b - A
Solution
l
4
— - 1 • A- 1 1 _ 1
a. 4 3 • 4 3 * 12
4
CHECK / -^( 4 ) =
CHECK /-f(-f)=l
Divide Real Numbers
Find the quotient.
1.8-5- (—4) 2. -5 -(-2^)
3
4
3- 3
In Examples 1 and 2 notice that applying the division rule suggests the following
rule for finding the sign of a quotient.
THE SIGN OF A QUOTIENT RULE
• The quotient of two numbers with the same sign is positive.
— a 4- { — b) = a -# b = 4 - Examples: -20 4- (-5) = 4
b 20 + 5=4
• The quotient of two numbers with opposite signs is negative.
— a 4 - b = a -J| b) = Examples: -20 4 - 5 = -4
20 - (-5) = -4
L J
3 Evaluate an Expression
Evaluate —when a = —2 and b = — 3.
a + b
Solution
-2a = — 2 (— 2 )
a + b —2 + (— 3 )
. _4_
-5
_4
5
Substitute -2 for a and -3 for b.
Simplify numerator and denominator.
Quotient of two numbers with opposite signs is negative.
Chapter 2 Properties of Real Numbers
I CO
mwvjum a Simplify an Expression
0 . ,. r 32x - 8
Simplify 4
Solution
32x 4 ~ 8 = (32x - 8) t 4
Rewrite fraction as division expression
= (32x - 8) • -
Multiply by reciprocal.
= - (8)(j)
Use distributive property.
II
OO
X
1
ro
Multiply.
Evaluate and Simplify Expressions
-y 24 - 8x
5. Evaluate ^ — - when x = 2 and y = —5. 6. Simplify ———.
When a function is defined by an equation, its domain is restricted to real
numbers for which the function can be evaluated. Division by zero does not result
in a unique number. Thus, input values that result in a denominator equal to zero
are numbers that are not in the domain.
5 Find the Domain of a Function
—X
To find the domain of the function y = ^ _ , input some sample values of x.
Student HeCp
^ More Examples
More examples
are available at
www.mcdougallittell.com
INPUT
X = —1
x = 0
x = 1
SUBSTITUTE
= -(- 1 )
1 -(“
y
-o
1 - 0
OUTPUT
y = 0
Undefined
x — 2
y = 2
ANSWER ► From the list you can see that x = 1 is not in the domain of the
function because you cannot divide by zero. All other real numbers
are in the domain. The domain is all real numbers except x = 1.
Find the Domain of a Function
Find the domain of the function.
7.y =
2x
x — 2
8- y
1
8 — x
9- y =
5x
2
10. y
10
x
2.8 Dividing Real Numbers
Exercises
Guided Practice
Vocabulary Check Complete the statement.
1. The product of a number and its ? is 1.
2 . The result of a -r- b is the ? of a and b.
Skill Check Find the reciprocal of the number.
3.32 4. -7 5. - j 6. 4j
Find the quotient.
7 - -12-3 «--7+-5 9 -M40 1 0--8 + 2f
11. Evaluate a ^ — when a — — 2 and b — — 3. 12. Simplify ^ _^ x
Find the domain of the function.
i3 -' j., i4 ->-=f i5 -^=i
16.
X + 1
x + 2
Practice and Applications
ERROR ANALYSIS In Exercises 17 and 18, find and correct the error.
DIVIDING REAL NUMBERS Find the quotient.
19. 9 - 5 - (-3) 20. —10 -(-5) 21. -4-5-4 22. 8 -s- (-2)
23. -45 4- 9 24. -24 4- 4 25. -50 - (-25) 26. -51 4- 17
27.6+|-|
28. -9 + l-J
29. -7 +
30. 54 - ( —2y
Student HeCp
► Homework Help
Example 1: Exs. 17-30
Example 2: Exs. 31-38
Example 3: Exs. 39-42
Example 4: Exs. 43-48
Example 5: Exs. 49-52
^ j
SIMPLIFYING COMPLEX FRACTIONS Find the quotient.
8
34.
-20
3
5
_8 _ 21 _ _12
9 2 5
36. TI 2 37. — 38. Jfs
Chapter 2 Properties of Real Numbers
Student HcCp
► Homework Help
Extra help w 'th
" 4 ^ 0 ^ problem solving in
Exs. 39-42 is available at
www.mcdougallittell.com
EVALUATING EXPRESSIONS Evaluate the expression for the given
value(s) of the variable(s).
39- x ^ - when x = 3
6
40- - when r — —10
41. —— — when a — — 3 and b — 3 42- ——— when x — 2 and y =
a y y 2
SIMPLIFYING EXPRESSIONS Simplify the expression.
43. lix ~ 9
MM 22r + 10
-2
45.
46. 45 ~ —
47. _44 “ 81
48.
-56 + h
60y - 108
12
FINDING THE DOMAIN Find the domain of the function.
__ 1 __ 1 _ _ x + 6 __
* 9 -y = x + 2 50 ^ = ^ *'-y = — * 2 -y
10 — x
1 - x
B222GE3 Find a Velocity
HOT-AIR BALLOONING You are descending in a hot-air balloon at the rate of
500 feet every 40 seconds. What is your velocity?
Solution
Displacement
Verbal
Velocity =- _—
Model
| Time|
Labels
Velocity = v
(feet per second)
Displacement = -500
(feet)
Time = 40
(seconds)
-500 10C
Algebraic
v = = 12.5
Model
ANSWER t
Your velocity is —12.5 feet per
second.
Find the velocity of the object.
53- A submarine descends 21 meters in 2 seconds.
54- An airplane descends 20,000 feet in 25 minutes.
Standardized Test
Practice
55- MULTIPLE CHOICE Which of the following statements is false!
(A) The reciprocal of any negative number is a negative number.
(ID The reciprocal of any positive number is a positive number.
Cep Dividing by a nonzero number is the same as multiplying by
its reciprocal.
(D) The reciprocal of any number is greater than zero and less than 1.
2.8 Dividing Real Numbers
Mixed Review
MENTAL MATH Use mental math to solve the equation. (Lesson 1.4)
56. x + 17 = 25 57. a - 5 = 19 58. 34 - n = 17
59. 2b = 10 60. _y 4 = 6 61. —= 6
TRANSLATING SENTENCES Write the sentence as an equation or an
inequality. Let x represent the number. (Lesson 1.5)
62. 9 is equal to a number decreased by 21.
63. The product of 2 and a number is greater than or equal to 7.
64. 3 is the quotient of a number and —6.
EVALUATING EXPRESSIONS Evaluate the expression. (Lesson 2.4)
65. -8 - 4 - 9 66. 12 - (-8) - 5 67. -6.3 - 4.1 - 9.5
68.1.4-6.2-9.1 69. 5 - \ |
70.
Maintaining Skills
COMPARING NUMBERS Complete the statement using <, >, or =.
(Skills Review p. 770)
71. -3 ? 3
75. 0 ? —2
72.5 ? -6
76. -1 Q -1
73. -8 ? 9
77. -6 ? 2
74. -7 ? -4
78. -4 0 -5
Quiz 3
Identify the like terms in the expression. (Lesson 2.7)
1 . 3x — lx + 4 2. 6(3 — 5 + 9a + 10 3. —5 p + Ip 2 — p
Simplify the expression. (Lesson 2.7)
4. —17/ - 9/ 5. 5 + 3d - d + 2
7. 3 (a + 1) - 7
6 . 6 g 2 -8 g - 5 g 2
8 . —2(4 — p) + p — 1 9 - — (w — 7) — 2(1 + w)
Find the quotient. (Lesson 2.8)
10. 15 4- (-3)
11. -144 h-
„„ ( „4\
-36
13. -28 -h 1-2 yj
14. 2
3
Simplify the expression.
(Lesson 2.8)
16.
20 - 8x
17.
9x + 1
-3
Find the domain of the function. (Lesson 2.8)
19. y =
2 + x
20.y = $
12.-12 + ^
4
15. #2
18.
- I5x + 10
-5
21- y —
3x+ 1
Chapter 2 Properties of Real Numbers
CHAPTER
Extension
Inductive and Deductive
Reasoning
Goal
Identify and use
inductive and deductive
reasoning.
Key Words
• inductive reasoning
• deductive reasoning
• if-then statement
• hypothesis
• conclusion
Student HeCp
► Look Back
For help with
counterexamples,
see p. 73.
i J
INDUCTIVE REASONING Reasoning is used in mathematics, science, and
everyday life. When you make a general statement based on several observations,
you are using inductive reasoning. Such a statement is not always true. If you
can find just one counterexample, then you have proved the statement to be false.
1 Inductive Reasoning in Everyday Life
Your math teacher has given your class a homework assignment every Monday
for the last three weeks.
a. Using inductive reasoning, what could you conclude?
b. What counterexample would show that your conclusion is false?
Solution
a. From your observations, you conclude that your math teacher will give
your class a homework assignment every Monday.
b. A counterexample would be for your teacher not to give a homework
assignment one Monday.
2 Inductive Reasoning and Sequences
Observe the following sequence of numbers. Find the pattern. Then predict the
next three numbers.
1,5,9, 13, 17,21,1, | ,|?|
Solution
Notice that each number is 4 more than the previous number.
You can conclude that the next three numbers would be 21 + 4 = 25,
25 + 4 = 29, and 29 + 4 = 33.
Use Inductive Reasoning
Use inductive reasoning to predict the next three numbers in the sequence.
1. 0,3, 6, 9, 12, 15, ? , ? , ?
2 . 1 , 4, 9, 16, 25, 36, |, | , ■
Inductive and Deductive Reasoning
DEDUCTIVE REASONING When you use facts, definitions, rules, or properties
to reach a conclusion, you are using deductive reasoning. A conclusion reached
in this way is always true.
Student MeCp
3 Deductive Reasoning in Mathematics
p Look Back
For help with the
properties of addition,
see p. 79.
Prove that (a + b) + c = (c + b) + a is true when a , b , and c are real numbers.
Justify each step using the properties of addition.
(a + b) + c = c + (a + b) Commutative property of addition
j
— c + (b + a) Commutative property of addition
= (c + b) + a Associative property of addition
IF-THEN STATEMENTS Deductive reasoning often uses if-then statements.
The if part is called the hypothesis and the then part is called the conclusion.
When deductive reasoning has been used to prove an if-then statement, then the
fact that the hypothesis is true implies that the conclusion is true.
Use of If-Then Statements
Your teacher tells you the fact that if you receive an A on the final exam, then
you will earn a final grade of A in the course. You receive an A on the final
exam. Draw a conclusion about your final grade.
Solution The hypothesis of the if-then statement is “you receive an A on the
final exam.” The conclusion is “you will earn a final grade of A in the course.”
The hypothesis is true, so you can conclude that your final grade will be an A.
Exercises
In Exercises 1-3, tell whether the conclusion is based on inductive
reasoning, deductive reasoning, or an if-then statement. Explain.
1. You have observed that in your neighborhood the mail is not delivered on
Sunday. It is Sunday, so you conclude that the mail will not be delivered.
2 . If the last digit of a number is 2, then the number is divisible by 2. You
conclude that 765,432 is divisible by 2.
3. You notice that for several values of x, the value of x 2 is greater than x. You
conclude that the square of a number is greater than the number itself.
4. Find a counterexample to show that the conclusion in Exercise 3 is false.
5. Use inductive reasoning to predict the next three numbers in the sequence:
6. Use deductive reasoning to prove that (x + 2) + (—2) = x is true when x is a
real number. Write each step and justify it using the properties of addition.
7. Give an example of inductive reasoning and an example of deductive
reasoning.
Chapter 2 Properties of Real Numbers
2 Chapter Summary
" and Review
• real number, p. 65
• real number line, p. 65
• positive number, p. 65
• negative number, p. 65
• integer, p. 65
• whole number, p. 65
• graph of a number, p. 65
• opposite, p. 71
• absolute value, p. 71
• counterexample, P■ 73
• closure property of real
number addition, p. 78
• commutative property
of addition, p. 79
• associative property
of addition, p. 79
• identity property
of addition, p. 79
• inverse property of addition,
p. 79
• term, p. 87
• closure property of real
number multiplication, p. 93
• commutative property
of multiplication, p. 94
• associative property
of multiplication, p. 94
• identity property
of multiplication, p. 94
• multiplicative property
of zero, p. 94
• multiplicative property
of negative one, p. 94
• distributive property, p. 100
• coefficient, p. 107
• like terms, p. 107
• simplified expression,
p. 108
• reciprocal, p. 113
The Real Number Line
Examples on
pp. 65-67
Write the numbers 2, ——, 0.8,
-4
-2, and —0.8 in increasing order.
-3
0.8
—h
1
2
A
2
+
3
From the graph, you can see that the order is:
-2, -l|, -0.8, 0.8, and 2.
Write the numbers in increasing order.
1. -3, 5, -4, -6, 2, 1 2 . 3.1, -1.9, 5,4.6, 5.3, -2
3. 4, 6, -2, |, -1, 1
Absolute Value
Examples on
pp. 71-73
You can find the absolute value of any number,
a. | 6.7 | = 6.7 If o is positive, then |o| = o
b.
7 \ 7
— I = — If o is negative, then \a\= -o; use definition of opposites.
Chapter Summary and Review
Chapter Summary and Review continued
Evaluate the expression.
4.
31
5. -5
6. -
100
7.
-1-
8.
-3.21
9. — | —9.11
10. -
1
9
11.
3—
2
2.3 Adding Real Numbers
Examples on
pp. 78—80
To add real numbers, use the rules and properties of addition.
a. 4 + (-8) + (-6) = 4 + [-8 + (-6)]
= 4 + (-14)
= -10
b. 4.3 + (-7) + 5.7 = 4.3 + 5.7 + (-7)
= 10 + (-7)
= 3
Use associative property of addition.
Add -8 and -6.
Add 4 and -14.
Use commutative property of addition.
Add 4.3 and 5.7.
Add 10 and -7.
Find the sum.
12.9 + (-10) + (-3)
15. 2.4 + (-3.4) + 6
13. -35 + 41 + (-18)
1
14. —2.5 + 6 + (—3)
16. 9 + (-3) +
17.i + (-8)+ -j
2.4 Subtracting Real Numbers
Examples on
pp. 86—88
To subtract real numbers, add their opposites.
a. 10 - (-8) - 16 = 10 + 8 + (-16)
= 18 + (-16)
= 2
b. 9.6 - 6 - 3.5 = 9.6 + (-6) + (-3.5)
= 3.6 + (-3.5)
= 0.1
Add opposites of -8 and 16.
Add 10 and 8.
Add 18 and -16.
Add opposites of 6 and 3.5.
Add 9.6 and -6.
Add 3.6 and -3.5.
Evaluate the expression.
18. -2-1 - (- 8 )
21. -5.7 - (-3.1) - 8.6
19. 18 - 14 - (-15)
22 .
-7-|-13
20 . 2 - 1.5 - 4
23. -3
\
2
Chapter 2 Properties of Real Numbers
Chapter Summary and Review continue of
Multiplying Real Numbers
Examples on
pp. 93-95
Use the rules for the sign of a product to find products and simplify expressions.
a. —3(6) = —18 One minus sign, so product has a minus sign.
b. — 6(—2) = 12 Two minus signs, so product has no minus sign.
c. — 9(—4)(—x) = — 36x Three minus signs, so product has a minus sign.
d. (— x) 4 = x 4 Four minus signs, so product has no minus sign.
Find the product.
24. -3(12)
27. —14(—0.3)
25. —40(—15)
28. —3.2(10)(—2)
26. —7(—6)(—2)
7
29. 24
12
Simplify the expression.
30. — 5(— x)
33. —6(2 )(*)(*)
31-3 (-/)
34. (—y) 3
32. 10(— a)(—a)(—a)
35. —81 {-bj 1
The Distributive Property
Examples on
pp. 100-102
Use the distributive property to rewrite expressions without parentheses.
a. 8(x + 3) = 8(jc) + 8(3)
= 8x + 24
b. (a - 6)4 = (a)(4) - (6)(4)
= 4a - 24
c. — 7(y - 5) = — 7(y) - (-7)(5)
= -7y + 35
d. (2 + x)(— 2) = (2)(— 2) + (x)(— 2)
= -4 - 2x
Use distributive property.
Multiply.
Use distributive property.
Multiply.
Use distributive property.
Multiply.
Use distributive property.
Multiply.
Use the distributive property to rewrite the expression without parentheses.
36. 5(jc + 12) 37. (y + 6)9 38. 10(z - 1)
39. (3 - w) 2 40. —2(x + 13) 41. (t + 11)(—3)
42. -8 (m - 7) 43. (x - 10)(-6) 44. -2.5 (s - 5)
Chapter Summary and Review
Chapter Summary and Review continued
Combining Like Terms
Examples on
pp. 107-109
J5ECH328 To combine like terms, add their coefficients,
a. lx — 6x + x = (7 — 6 + l)x Use distributive property.
= 2x
Add coefficients.
b. 3 - 4(y + 4) = 3 - 4(y) + (—4)(4)
Use distributive property.
'sO
1
1
CO
II
Multiply.
= -Ay + 3 - 16
Group like terms.
II
1
1
oo
Combine like terms.
Simplify the expression.
45. 3a + 6a 46. 2x 2
+ 9x 2 + 4
47. 4 +/- 1
48.3(rf+ 1) — 2 49. 6^ -
' 2 it - 1)
50. 2(x + 3) + 3(2x — 5)
Dividing Real Numbers
Examples on
pp. 113-115
a. 9 4- (-3) = 9 •
= -3
To divide real numbers, multiply by their reciprocals.
1 ^
b. —7 -r — = —7 • —
= -6
Multiply by reciprocal of -3.
Simplify.
Multiply by reciprocal of
Simplify.
c. —4 -T- ( — It) — — 4 -r (— —I Rewrite mixed number as improper fraction.
= - 4 * --
20
7
Multiply by reciprocal of
Simplify.
Find the quotient.
51.8 - (-2)
52. -7-7
53.
-5 l-f
54. -10 -h
55.
1
5e - * H*
57. 12 - |-lj
58. -63 h- 4^
Chapter 2 Properties of Real Numbers
A
Chapter Test
Write the numbers in increasing order.
1. 4, -9, -5, 9, -2, 3 2 . 8, -2.7, -6.4, 3.1, -4, 5
Find the opposite of the number.
4. 5 5. -4 6. 9.2
Evaluate the expression.
8. | 8 | 9. | -17 | 10. - 1 4.5 |
Find the sum.
3. 3, -5,
5
5 3
'r 4
,2
7. -
5
6
11 . -
]_
4
12.4 + (-9)
13. -25 + 31
14. 9 + (-10) + 2 15. 7 + 6.5 + (-3.5)
16. PROFIT AND LOSS A company had the following profits and losses:
first quarter, $2,190; second quarter, $1,527; third quarter, —$2,502;
fourth quarter, —$267. What was the company’s profit or loss for the year?
Find the difference.
17. -6 - 8 18. 15 - (-15) 19. 6 - (-4) - (-3) 20. -2.47 - (-3.97) - 2
Find the product.
21. -6(4) 22. —8(—100) 23. -9(8)(-5) 24. -3(15)
Simplify the expression.
25. — 8(— x) 26. 5(—w)(—w) 27. 8(—4 )(a)(a)(a) 28. —15(— z) 2
29. EAGLES An eagle dives down from its nest with a velocity of —44 feet per
second. Find the displacement of the eagle after 4.5 seconds.
Use the distributive property to rewrite the expression without parentheses.
30. (a + 11)9 31. 8(4 - jc) 32. (6 + y)(-12) 33. (-5)(3 - z)
Simplify the expression.
34. t 2 — 9 + t 2 35. 14 p + 2(5 — p) 36. — 9(y + 11) + 6 37. 2 (a + 3) — 5 (a — 4)
38. MOVIE THEATER It takes you 17 minutes to get to the movie theater. You
spend t minutes riding the bus at an average speed of 0.5 mile per minute.
The rest of the time is spent walking at 0.06 mile per minute. If you spend
10 minutes on the bus, how far is it to the movie theater?
41. -
3
8
\
2
42. 39 -1 —ly^
Find the quotient.
39. -36 - (-4)
40. -56 -
Chapter Test
Chapter Standardized Test
Tip
If you can, check your answer using a method that is different from the
one you used originally to avoid making the same mistake twice.
1. Which inequality is true?
_ 1 1 _ 1 1
®“4 > 3 ®4 <_ 3
©H ®H
2 . What is the opposite of 3?
(A) -3 CD 3
©4 © 1
8 . What is the value of the expression
® 4 ® 4
?
9. Evaluate (—2)(4)(— n) 3 when n — 3.
CD -216 ® -90
(©90 C© 216
3, What is the solution of | x \ — 10?
CD -io CD 10
CD 1 10 | CD -10 and 10
4. What is the value of — 9 + 3 + (—4)?
CD -16 CD -10
(© -8 CD -2
5, What is the value of —4 — 6 — (—10)?
® -20 CD 0
CD 8 CD 12
6. Evaluate — x — 13 whenx = 9.
CD -22 CD -4
CD 4 CD 22
10, Rewrite the expression (4 — a)(— 3) without
using parentheses.
CD 4 — 3 a CD 4 + 3 a
CD —12 + 3 a CD 12 — 3 a
11, You are buying 3 new pairs of slacks. All of the
slacks are on sale for $24.99. What expression
would you use to mentally find the total cost of
the slacks?
CD 3(25) + 0.01 CD 3(25 + 0.01)
CD 3(25) - 0.01 CD 3(25 - 0.01)
12, Simplify the expression 6(x + 3) — 2(4 — x).
CD 4x + 10 CD 5x — 5
CD 5x + 11 CD + 10
CD none of these
7. On Monday the closing price for a company’s
stock was $26.81. On Tuesday it was $26.75.
What was the change in the closing price?
CD -26.78 CD -0.06
CD 0.06 CD 26.78
Chapter 2 Properties of Real Numbers
4p + q
13- Evaluate ——— when p = — 2 and q = — 3.
CD
9
2
CD
n
2
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
J| 11 Find the Area of a Figure
Find the area of the geometric figure,
a. square b. rectangle
7
Solution
a. A = s 2
b. A = i X w
= 4 2
= 7X3
= 16
= 21
c. triangle
6
c. A = jbh
= j(6)(5)
= 15
Try These
Find the area of the geometric figure.
1 - A square with side length 5 2 - A rectangle with length 8 and width 4
3- A square with side length 10 4- A triangle with base 4 and height 4
Student HeCp
► Extra Examples
More examples
anc j p ract j ce
exercises are available at
www.mcdougallittell.com
BJEEHEB 2 Draw a Circle Graph
The table shows the number of pets a
local pet store sold in one year. Draw
a circle graph to display the data.
Solution
Pet
Dog
Cat
Bird
Number
312
270
46
First find the total number of pets:
312 + 270 + 46 = 628
To find the degree measure of each sector, write
a fraction comparing the number of pets to the
total. Then multiply by 360°. For example:
Dog: HI • 360° - 179°
Cat
270 Bird
Dog
312
Try These
5. The table shows the
number of books a local
bookstore sold in one
year. Draw a circle graph
to display the data.
Book type
Fiction
Nonfiction
Other
Number
549
348
103
Maintaining Skills
r
APPLICATION: Bald Eagles
The bald eagle can fly at speeds up to 30 miles
per hour and dive at speeds up to 100 miles per hour.
Think & Discuss
1. Use the formula d = rt to find the distance a bald
eagle can fly for the given flying rate and time.
Convert the flying time from hours to minutes.
2. 1 hour = ? minutes
3. 2 hour =
4. \ hour =
o
? minutes
? minutes
5. How many minutes will it take an eagle flying at a
rate of 30 miles per hour to fly 1 mile?
Learn More About It
You will solve equations to find flying rates of bald
eagles in Exercises 40 and 41 on p. 181.
a**"**
APPLICATION LINK More about bald eagles is available at
www.mcdougallittell.com
Flying rate
(miles per hour)
Time
(hours)
Distance
(miles)
30
1
?
30
l
?
2
30
]_
?
6
PREVIEW
What’s the chapter about?
• Solving a linear equation systematically
• Using ratios, rates, and percents
Key Words
• equivalent equations,
p. 132
• inverse operations,
p. 133
• linear equation, p. 134
• properties of equality,
p . 140
• identity, p. 153
• rounding error, p. 164
• formula, p. 171
• ratio, p. 177
• rate, p. 177
• unit rate, p. 177
• unit analysis, p. 178
• percent, p. 183
• base number, p. 183
PREPARE
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to pp. 71,113)
1. What is the opposite of —3?
(A) -3
2_ Which number is the reciprocal of —?
®"3
CD 1
CD 3
CDf
Skill Check (refer to pp. 24,102,108)
3. Which of the following is a solution of the equation —11 = — 4y + 1?
(A) -3 CD 2 CD 3
4. Which expression is equivalent to — 3(x — 4)?
Ca) 3x — 12 CD — 3x + 12 CD — 3x — 4
5. Simplify the expression 4x 2 — 5x — x 2 + 3x.
Ca) 5x 2 — 2x CD 3x 2 + 8x CD 3x 2 — 2x
CD 4
CD — 3x - 12
CD x 2
STUDY TIP
Make Formula Cards
Write a formula and a sample
problem on each card. Make
sure you know what each
algebraic symbol represents
in a formula.
T
Chapter 3 Solving Linear Equations
DEVELOPING CONCEPTS
For use with
Lesson 3.1
Goal
Use algebra tiles to solve
one-step equations.
Question
How can you use algebra tiles to solve a one-step equation?
Materials
• algebra tiles
Explore
Student HeCp
^ Look Back
For help with algebra
tiles, see pp. 77, 84,
and 99.
V _>
© Model the equation x + 5 = — 2.
■+■ ■ ■ ■ = ■
■ +1 ■
© To find the value of x, get the x-tile by itself on one side of the equation. Take
away five 1-tiles from the left side. Be sure to take away five 1-tiles from the
right side to keep the two sides equal.
First add 5
■ L+
+
+
■
m
zero pairs
so you can
subtract5.
© The remaining tiles show the value of x. So, x = ? . -7
m
What operation did you use to solve this addition equation?
Think A bout It
Use algebra tiles to model and solve the equation. Sketch each step.
1. x + 4 = 6 2. x + 3 = 8 3. x + 7 = — 1 4. x + 2 = — 7
5- Use algebra tiles to model and solve x — 3 = 2. Start with the model below.
Use algebra tiles to model and solve the equation. Sketch each step.
6. x — 1 = 5 7. x — 7 = 1 8. x — 2 = —6 9. x — 4 = —3
10. A student solved the equation x + 3 = —4 by subtracting 3 on the left side
of the equation and got x = —4. Is this the correct solution? Explain.
+
Developing Concepts
Solving Equations Using
Addition and Subtraction
Goal
Solve linear equations
using addition and
subtraction.
Key Words
• equivalent equations
• transforming equations
• inverse operations
• linear equation
What size is a city park?
Griffith Park in Los Angeles is
one of the largest city parks in the
United States. It has miles of hiking
trails, a theater, and an observatory.
In Exercises 58 and 59 you will
solve equations to find the sizes of
some city parks.
You can solve an equation by writing an equivalent equation that has the variable
isolated on one side. Linear equations are equivalent equations if they have the
same solution(s). To change, or transform , an equation into an equivalent equation,
think of an equation as having two sides that need to be “in balance.”
Student HeCp
—i-
► Study Tip
When you subtract 3
from each side of the
equation, the equation
stays in balance.
Original equation:
x + 3 = 5
11
Subtract 3 from each side
to isolate xon the left.
Simplify both sides.
Equivalent equation: x=2
TRANSFORMING EQUATIONS
OPERATION
ORIGINAL
EQUATION
EQUIVALENT
EQUATION
Add the same
number to
each side.
x- 3 = 5
Add 3. mm
x — 8
Subtract the
same number
from each side.
x + 6 = 10
Subtract©!^*
x = 4
Simplify one or
both sides.
x= 8 - 3
SimplifpHH
x — 5
Chapter 3 Solving Linear Equations
INVERSE OPERATIONS Two operations that undo each other, such as addition
and subtraction, are called inverse operations. Inverse operations can help you to
isolate the variable on one side of an equation.
Student HaCp
► More Examples
More examples
are available at
www.mcdougallittell.com
J 1 Add to Each Side of an Equation
Solve X - 5 = -13.
Solution
This is a subtraction equation. Use the inverse operation of addition to undo the
subtraction.
x 5 — 13
x — 5 + 5 = —13 + 5
x = —8
CHECK /
x - 5 = -13
-8-51 -13
-13 = -13 /
Write original equation.
Add 5 to each side to undo the subtraction.
Simplify both sides.
Write original equation.
Substitute -8 for x.
Solution is correct.
Student HeCp
EZEB9 2 Simplify First
l F -V
► Study Tip
To subtract -4 from n ,
add the opposite of -4
to n. To review the
subtraction rule, see
p. 86. .*.
v j
Solve —8 — n — (—4).
Solution
-8 = n- (-4)
.► -8 = n + 4
-8-4=rc+4-4
— 12 = n
Write original equation.
Use subtraction rule to simplify.
Subtract 4 from each side to undo the addition.
Simplify both sides.
CHECK /
To check the solution, substitute —12 for n in the original equation
— 8 = n — (—4), not in the simplified equation — 8 = n + 4.
— 8 = n — (—4) Write original equation.
— 8 2 = —12 — (—4) Substitute -12 for n.
— 8 2=—12 + 4 Use subtraction rule to simplify.
— 8 = — 8 y Solution is correct.
Solve an Equation
Solve the equation. Check your solution in the original equation.
1- —2 = x — 4 2.x -(-9) = 6 3. y + 5 = -1
4. t — 7 = 30 5. -8 = x + 14 6- 3 = x - (-11)
3.1
Solving Equations Using Addition and Subtraction
LINEAR EQUATIONS The equations in this chapter are called linear equations.
In a linear equation the exponent of the variable(s) is one.
LINEAR EQUATION NOT A LINEAR EQUATION
x + 5 = 9 x 2 + 5 = 9
y = 3x - 8 3x 2 — 8 = 0
In Chapter 4 you will see how linear equations get their names from graphs.
PROBLEM SOLVING PLAN You can write linear equations to model many real-
life situations. Example 3 shows how to model temperature change using the
problem solving plan that you learned in Lesson 1.6.
Link
Qeograp&y
SPEARFISH is located in the
Black Hills of South Dakota.
Freezing and thawing can
loosen the rock walls of
Spearfish Canyon, causing
landslides.
3 Model Temperature Change
SPEARFISH, SOUTH DAKOTA On January 22, 1943, the temperature in
Spearfish fell from 54°F at 9:00 A.M. to —4°F at 9:27 A.M. Write and solve a
linear equation to find how many degrees the temperature fell.
Solution
Verbal
Model
Labels
Algebraic
Model
Temperature
at 9:27 A.M.
Temperature
at 9:00 A.M.
Degrees fallen
Temperature at 9:27 A.M. = —4
Temperature at 9:00 A.M. = 54
Degrees Fallen = T
-4 = 54-|
-4 - 54 = 54 - T - 54
-58 = -T
58 = T
ANSWER ► The temperature fell by 58°.
(degrees Fahrenheit)
(degrees Fahrenheit)
(degrees Fahrenheit)
Write linear equation.
Subtract 54 from each side.
Simplify both sides.
T is the opposite of -58.
A record 24-hour temperature change occurred in Browning, Montana, on
January 23-24, 1916. The temperature fell from 44°F to — 56°F.
7. Write a verbal model that can be solved to find how many degrees the
temperature fell.
8 . Rewrite the verbal model as a linear equation.
9. Solve the linear equation to find the record temperature fall in degrees.
Chapter 3 Solving Linear Equations
Exercises
Guided Practice
Vocabulary Check Complete the sentence.
1 _ Linear equations with the same solution(s) are called ? equations.
2 . You can use ? operations, such as addition and subtraction, to help you
isolate a variable on one side of an equation.
Tell whether each equation is iinear or not iinear. Explain your answer.
3. a 2 +1=9 4. y 4- 16 = 5 5. 4 + 2r=-10 6. 3x 2 = 8
Skill Check Solve the equation.
7. r + 3 = 2 8 . 9 = x — 4 9. 7 + c=-10
10 . — 1 =? — 6 11 . 4 + x = 8 12 . x + 4 — 3 = 9
13. r-(-2) = 5 14. -1 = d- (-12) 15. 6 - (-y) = 3
SPENDING MONEY You put some money in your pocket. You spend $4.50 on
lunch. You have $7.50 in your pocket after buying lunch.
16, Write an equation to find how much money you had before lunch.
17, Which inverse operation will you use to solve the equation?
18, Solve the equation. What does the solution mean?
Practice and Applications
STATING THE INVERSE
19. Add 28.
22 . Subtract 15.
State the inverse operation.
20. Add 17. 21. Subtract 3.
23. Add—12. 24. Subtract—2.
Student HeCp
^Homework Help
Example 1: Exs. 25-42
Example 2: Exs. 43-51
Example 3: Exs. 54-60
A j
SOLVING EQUATIONS
25.x + 9 = 18
28. 4 + x = 7
31. f- 2 = 6
34. y 4- 12 = -12
37. t - 5 = -20
Solve the equation.
26. m- 20 = 45
29. x + 5 = 15
32. -9 = 2 + y
35. y — 12 = 12
38.x + 7 = -14
27.x - 8 = -13
30. 11 = r - 4
33. n - 5 = -9
36. a — 3 = —2
39. 34 + x = 10
42.r + | =
4
3.1
Solving Equations Using Addition and Subtraction
Student He dp
^Homework Help
Extra help with
problem solving in
Exs. 43-51 is available at
www.mcdougallittell.com
SOLVING EQUATIONS Solve the equation by simplifying first.
43. t - (-4) = 4 44. 6 = y — (-11) 45. x - (-8) = 13
46. r - (-7) = -16 47. 19 - (~y) = 25 48. 2 - (-b) = ~6
49. x + 5 — 2 = 6 50. 12 — 5 = n + 7 51. -3 = a + (-4)
Geometry Lk Find the length of the side marked x.
52. The perimeter is 12 feet. 53. The perimeter is 43 centimeters.
MATCHING AN EQUATION In Exercises 54-56, match the real-life
problem with an equation. Then solve the problem.
A. x + 15 = 7 B. 15 — jc = 7 C. 15 + 7 = x D. x + 15 = — 7
54. You own 15 CDs. You buy 7 more. How many CDs do you own now?
55. There are 15 members of a high school band brass section. After graduation
there are only 7 members. How many members graduated?
56. The temperature rose 15 degrees to 7°F. What was the original temperature?
57. BASEBALL STADIUMS Turner Field in Atlanta, Georgia, has 49,831 seats.
Jacobs Field in Cleveland, Ohio, has 43,368 seats. How many seats need to
be added to Jacobs Field for it to have as many seats as Turner Field?
LinkJ to|.
Careers
CITY PARKS In Exercises 58 and 59, use the table that shows the sizes
(in acres) of the largest city parks in the United States.
Park (location)
Size (acres)
Cullen Park (Houston, TX)
?
Fairmount Park (Philadelphia, PA)
8700
Griffith Park (Los Angeles, CA)
4218
Eagle Creek Park (Indianapolis, IN)
?
Pelham Bay Park (Bronx, NY)
2764
► Source: The Trust for Public Land
PARK RANGERS guide
tours, provide information,
manage resources, and
maintain safety.
More about park
' rangers is available at
www.mcdougallittell.com
58. Griffith Park is 418 acres larger than Eagle Creek Park. Write an equation
that models the size of Eagle Creek Park. Solve the equation to find the size
of Eagle Creek Park.
59. Cullen Park is 248 acres smaller than the sum of the sizes of Griffith Park,
Eagle Creek Park, and Pelham Bay Park. Write and solve an equation to find
the size of Cullen Park. HINT: Use your answer from Exercise 58.
Chapter 3 Solving Linear Equations
Standardized Test
Practice
Mixed Review
Maintaining Skills
60. CHECKBOOK BALANCE You thought the balance in your checkbook was
$53, but when your bank statement arrived, you realized that you forgot to
record a check. The bank statement lists your balance as $47. Let x represent
the value of the check that you forgot to record. Which equation is a correct
model for the situation? Solve the correct equation.
A. 53 - x = 47
B. x - 47 = 53
61. LOGICAL REASONING Copy
the solution steps shown. Then
write an explanation for each step
in the right-hand column.
Solution Step
-7 = x-(-2)
-7 = x +- 2
-<7 = x
Explanation
Original Equation
?
?
62. CHALLENGE You decide to try to ride the elevator to street level (Floor 0)
without pushing any buttons. The elevator takes you up 4 floors, down 6
floors, up 1 floor, down 8 floors, down 3 floors, up 1 floor, and then down 6
floors to street level. Write and solve an equation to find your starting floor.
63. IV1ULTIPLE CHOICE The selling price of a certain video is $7 more than the
price the store paid. If the selling price is $24, find the equation that
determines the price the store paid.
(a) x + 7 = 24 Cb) x = 1 + 24 Cep 7 — 24 = x (S) x = 24
64. MULTIPLE CHOICE Which equation is not linear?
CD 7 + x = 15 Cg) x 2 = 10 (TT) 3x — x = 1 GD x = 6 2
TRANSLATING SENTENCES Write the sentence as an equation.
(Lesson 1.5)
65. The product of 5 and a number x is 160.
66 . A number t divided by 6 is 48.
67. 36 decreased by a number k is 15.
68 . The quotient of a number y and 3 is 12.
DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the
expression without parentheses. (Lesson 2.6)
69. 4(x + 2) 70. 7(3 - 2 y) 71. -5 (y + 4)
72. (3x + 8)( —2) 73. -2(x - 6) 74. 3(8 - lx)
MULTIPLYING FRACTIONS Multiply. Write the answer as a fraction or as
a mixed number in simplest form. (Skills Review p. 765)
6
7
16
21
76.
80.
4
9
3
5
77.
10
81.
22
11
16
22
7
78.
82.
16
2
4
• —
4
3.1 Solving Equations Using Addition and Subtraction
Solving Equations Using
Multiplication and Division
Goal
Solve linear equations
using multiplication and
division.
Key Words
• inverse operations
• transforming equations
• reciprocal
• properties of equality
How heavy is a pile of newspapers?
Paper is the most recycled product in
the United States. In Exercise 50 you
will solve an equation to find the total
weight of a pile of newspapers after it
has been divided into smaller bundles.
Multiplication and division are inverse operations that can help you to isolate the
variable on one side of an equation. You can use multiplication to undo division
and use division to undo multiplication.
TRANSFORMING EQUATIONS
ORIGINAL
EQUIVALENT
OPERATION
EQUATION
EQUATION
Multiply each side of
the equation by the
same nonzero number.
- = 3
2 J
Multiply by 21
x = 6
Divide each side of
the equation by the
same nonzero number.
4x = 12
Divide by HR
x = 3
1 Divide Each Side of an Equation
Solve —4x— 1.
Student HeCp
1 ^ --^
► Study Tip
When you multiply
or divide each side
of an equation by a
negative number, be
careful with the signs
of the numbers.
" _ J
Solution
The operation is multiplication. Use the inverse operation of division to isolate
the variable v.
— 4x — 1 Write original equation.
— 4x 1
- = - Divide each side by -4 to undo the multiplication.
x — — ^ Simplify.
ANSWER^ The solution is —Check this in the original equation.
Chapter 3 Solving Linear Equations
2 Multiply Each Side of an Equation
Solve | = -30.
Solution
The operation is division. Use the inverse operation of multiplication to isolate
the variable x.
x
— =—30 Write original equation.
5 (^j = 5( — 30) Multiply each side by 5 to undo the division.
x = —150 Simplify.
ANSWER ^ The solutionis -150.
Solve the equation. Check your solution in the original equation.
1. 60 = 5x 2 . -^ = 11 3. ^- = —2 4- — 3x = —9
RECIPROCAL To solve an equation with a fractional coefficient, such as
2
10 = ——m, multiply each side of the equation by the reciprocal of the fraction.
This will isolate the variable because the product of a nonzero number and its
reciprocal is 1 .
3 Multiply Each Side by a Reciprocal
Solve 10 = — jm.
Solution
2 2 3
The fractional coefficient is ——. The reciprocal of — — is ——.
2
10 = — —m Write original equation.
3 3 / 2 \ 3
- (10) = - (——ml Multiply each side by the reciprocal, -j.
— 15 — m Simplify.
ANSWER ► The solution is -15.
Student HeCp
^More Examples
More examples
are available at
www.mcdougallittell.com
Multiply Each Side by a Reciprocal
Solve the equation. Check your solution in the original equation.
5. 6 = \x 6. 12 = -|>> 7. |* = 24 8. -6 =
3.2 Solving Equations Using Multiplication and Division
PROPERTIES OF EQUALITY The ways you have learned to transform an
equation into an equivalent equation are based on rules of algebra called
properties of equality.
Properties of Equality
ADDITION PROPERTY OF EQUALITY If 3 = £>, then 3 + C = b + C.
SUBTRACTION PROPERTY OF EQUALITY If 3 = b, then 3 - C = b ~ C.
MULTIPLICATION PROPERTY OF EQUALITY If 3 = b, then C3 = Cb.
O jb
division property of equality If a = b and c ^ 0, then — = —.
c c
MOVIE FRAMES A motion
picture camera takes separate
pictures as frames. These are
projected rapidly when the
movie is shown.
Model a Real-Life Problem
MOVIE FRAMES A single picture on a roll of movie film is called a frame.
The usual rate for taking and projecting professional movies is 24 frames per
second. Find the total number of frames in a movie that is 90 minutes long.
Solution
Let x = the total number of frames in the movie. To find the total number of
seconds in the movie, multiply 90 • 60 because each minute is 60 seconds.
Total number of frames in the movie
Total number of seconds in the movie
Number of frames per second
x
5400
24
Write equation.
= 5400(24)
Use multiplication property of equality.
x = 129,600 Simplify.
ANSWER ► A 90-minute movie has a total of 129,600 frames.
L
Model a Real-Life Problem
Motion picture studios try to save older films from decay by restoring the film
frame by frame. Suppose that a worker can restore 8 frames per hour. Let y = the
number of hours of work needed to restore all the frames in a 90-minute movie.
9- Use the information from Example 4 and the verbal model shown below to
write a linear equation.
Number of frames
restored per hour
•
Number of
hours of work
=
Total number of
frames in the movie
10, Use the division property of equality to solve the linear equation. How many
hours of work are needed to restore all of the frames in the movie?
Chapter 3 Solving Linear Equations
§3 Exercises
Guided Practice
Vocabulary Check
Skill Check
1 . Name two pairs of inverse operations.
Match the property of equality with its description.
2. Addition Property of Equality A- If a — b , then ca — cb.
3. Multiplication Property of Equality B. If a = b , then a — c = b — c.
4. Division Property of Equality C. If a — b , then a + c — b + c.
cl b
5- Subtraction Property of Equality D. If a = b and c ^ 0, then — = — .
Solve the equation. Check your solution in the original equation.
b
-7
15- CAR TRIP Suppose you drive 630 miles from St. Louis, Missouri, to
Dallas, Texas, in 10.5 hours. Solve the equation 630 = r(10.5) for r to
find your average speed.
6. 3x = 18
7. 19m = -19
8.
9.j = 8
o
<N
II
^ 1 '
6
11.
4
12. = 6
13. 4 = -\x
14.
Practice and Applications
STATING INVERSES State the inverse operation.
16. Divide by 6.
17. Multiply by 5.
18. Multiply by |\
19. Multiply by -4.
20. Divide by —3.
21. Divide by 7.
SOLVING EQUATIONS
Use division to solve the equation.
22. 3 r = 21
23. ly = —56
24. 18 = —2a
25. -An = 24
26. 8x = 3
27.lOx = 110
28. 30 b = 5
29. — lOx = -9
30. 288 = 16 u
Student HcCp
SOLVING EQUATIONS
► Homework Help
Use multiplication to solve the equation.
Example 1: Exs. 22-30
Example 2: Exs. 31-36
Example 3: Exs. 37-45
31.f =-5
32- j=-4
^3|m
II
'sO
00
00
Example 4: Exs. 48-52
34.^=-!
-4 4
y
35. j = 12
36.-f= -16
3.2 Solving Equations Using Multiplication and Division
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 37-45 is available at
www.mcdougallittell.com
SOLVING EQUATIONS Multiply by a reciprocal to solve the equation.
37. |k = 1
38. y = 4
39. 0 = |x
40. ~y = 6
41. 10 = fx
0
42. = -20
O
43. 12 = |x
44. ~x = 6
45. —= 36
ERROR ANALYSIS In Exercises 46 and 47, find and correct the error.
MODELING REAL-LIFE PROBLEMS In Exercises 48 and 49, use the verbal
model to write a linear equation. Then use the multiplication property of
equality to solve the equation.
48. It takes 45 peanuts to make one ounce of peanut butter. How many peanuts
will be needed to make a 12-ounce jar of peanut butter?
Number of peanuts
Number of ounces |
Number of peanuts per ounce
Link to
Science
THUNDERSTORMS You see
lightning almost at the instant
it flashes since light travels
so quickly. You hear the
thunder later because sound
takes about 5 seconds to
travel a mile near the ground.
49. You ate 3 of the 8 slices of a pizza. You paid $3.30 as your share of the total
cost of the pizza. How much did the whole pizza cost?
Number of pieces you ate
Cost of the
Your share
•
Total number of pieces
whole pizza
of the cost
50. BUNDLING NEWSPAPERS You are loading a large pile of newspapers onto
a truck. You divide the pile into four equal-size bundles. One bundle weighs
37 pounds. You want to know the weight x of the original pile. Which
equation represents this situation? Solve the correct equation.
A. | = 37 B. 4x = 37 C. 37x = 4
51. MAIL DELIVERY Each household in the United States receives about
676 pieces of junk mail per year. If there are 52 weeks in a year, then
about how many pieces of junk mail does a household receive per week?
HINT: Let x = the number of pieces of junk mail received per week. Solve
the equation 52x = 676.
52. Science You can tell about how many miles you are from a
thunderstorm by counting the seconds between seeing the lightning and
hearing the thunder, and then dividing by five. How many seconds would you
count for a thunderstorm that is nine miles away?
Chapter 3 Solving Linear Equations
Standardized Test
Practice
S Student UeCp
Test Tip
Use mental math to
solve each equation
to help you answer
Exercise 56.
V _>
Mixed Review
Maintaining Skills
53. CHALLENGE A homeowner is
installing a fence around the garden at
the right. The garden has a perimeter
of 220 feet. Write and solve an
equation to find the garden’s
dimensions.
54. 1V1ULTIPLE CHOICE Which operation would you use to solve ^ = — 8 x?
(A) Divide by 4. dp Divide by — 8 .
(Cf) Multiply by — 8 . (g) Multiply by 4.
55. MULTIPLE CHOICE Solve -|x = -2.
56. MULTIPLE CHOICE Which equations are equivalent?
(A) I and II
CD II and IV
CD I, II, and III
CD I, III, and IV
. 3 a
I. —x = 3
"f = 2
III. 2x = 10
IV. -x = -5
SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 2.7)
57. 15 — 8 x 4- 12 58. 4y — 9 + 3y 59. 5x + 6 — 7x
60 . -2(x + 8 ) + 36 61 . 5(y + 3) + ly 62 . 3(y - 10) - 5y
SOLVING EQUATIONS Solve the equation. (Lesson 3.1)
63. 4 + y = 12 64. t- 2= 1 65. -14 = r + 5
66. —6 + x = —15 67. x — (—6) = 8 68. a — (—9) = —2
69. PHOTOGRAPHY You take 24 pictures. Six of the pictures cannot be
developed because of bad lighting. Let x represent the number of pictures
that can be developed successfully. Which of the following is a correct model
for the situation? Solve the correct equation. (Lesson 3.1)
A. x + 6 = 24 B. 6x = 24 C. x — 6 = 24 D. x + 24 = 6
GREATEST COMMON FACTOR Find the greatest common factor of the
pair of numbers. (Skills Review p. 761)
70.5,35 71.30,40 72.12,22
73.10,25 74.17,51 75.27,36
76. 14, 42 77. 9, 24 78. 21, 49
3.2 Solving Equations Using Multiplication and Division
Solving Multi-Step Equations
Goal
Use two or more steps to
solve a linear equation.
Key Words
• like terms
• distributive property
How hot is Earth's crust?
Temperatures within Earth’s crust
can get hot enough to melt rocks. In
Example 2 you will see how a multi-
step equation can be used to predict
the depth at which the temperature of
Earth’s crust is 114°C.
Solving a linear equation may require more than one step. Use the steps you
already know for transforming an equation. Simplify one or both sides of the
equation first, if needed. Then use inverse operations to isolate the variable.
Student MeCp
►Vocabulary Tip
The prefix multi-
means "more than
one". A multi-step
equation is solved by
transforming the
equation more than
one time.
k _ J
i Solve a Linear Equation
Solve 3x + 7 = — 8 .
Solution
To isolate the variable, undo the addition and then the multiplication.
3x + 7 — — 8
Write original equation.
3x+ 7- 7= -8-7
Subtract 7 from each side to undo the addition.
U>
X
II
1
Ln
(Subtraction Property of Equality)
Simplify both sides.
3x _ -15
Divide each side by 3 to undo the multiplication.
3 3
(Division Property of Equality)
x = —5
Simplify.
CHECK Check by substituting —5 for x in the original equation.
3x + 7 = — 8
Write original equation.
3(—5) + 71-8
Substitute -5 for x.
-15 + 7 1 -8
Multiply.
S
00
1
II
00
1
Solution is correct.
Solve a Linear Equation
Solve the equation. Check your solution in the original equation.
1- 6 x — 15 = 9 2. lx — 4 = —11 3. 2y + 5 = 1
Chapter 3 Solving Linear Equations
Student Ho dp
2 Use a Verbal Model
^More Examples
More examples
are ava j| a bie at
www.mcdougallittell.com
SCIENCE LINK The temperature
within Earth’s crust increases about
30°C for each kilometer beneath the
surface. If the temperature at Earth’s
surface is 24°C, at what depth would
you expect the temperature to be 114°C?
Solution
Verbal
Model
Labels
Algebraic
Model
Temperature
Temperature
at Earth’s
Rate of
Depth
inside
=
+
temperature •
below
Earth
surface
increase
surface
Temperature inside Earth = 114
(degrees Celsius)
Temperature at Earth’s surface = 24 (degrees Celsius)
Rate of temperature increase = 30 (degrees Celsius per kilometer)
Depth below surface = \d (kilometers)
114 = 24 + 30 d Write equation.
90 = 30 d Subtract 24 from each side.
3 — d Divide each side by 30.
ANSWER ► The temperature will be 114°C at a depth of 3 kilometers.
Use a Verbal Model
4. If the temperature at Earth’s surface is 24°C, at what depth would you expect
the temperature to be 174°C? Use the verbal model in Example 2 to solve.
BEEQIB 3 Combine Like Terms First
Solve lx — 3x — 8 = 24.
Solution 7x — 3x - 8 =
24
Write original equation.
II
OO
1
4?
24
Combine like terms 7x and -3x.
4x — 8 + 8 —
24 + 8
Add 8 to each side to undo the
subtraction.
Ax —
32
Simplify.
Ax
32
Divide each side by 4 to undo the
4
4
multiplication.
JC =
v_
8
Simplify.
3.3 Solving Multi-Step Equations
Student HeCp
► Study Tip
Remember to distribute
the negative sign to
each term inside the
parentheses, not to
just the first term. ******
**►
4 Use the Distributive Property
Solve the equation,
a. 8 x — 2(x + 7) = 16
Solution
a. Distribute a negative number.
8x — 2(x + 7)
8x - 2x - 14
6 x - 14 + 14
6x
6
x = 5
b. 5x + 3(x + 4) — 28
b. Distribute a positive number.
16
5x + 3(x + 4) =
28
16
5x + 3x + 12 =
28
16
8 v + 12 =
28
16 + 14
8 x + 12 - 12 =
28 - 12
30
8 x =
16
30
8v
16
6
8
8
x — 2
Use the Distributive Property and Combine Like Terms
Solve the equation. Check your solution in the original equation.
5- 6 (x + 2) = 15 6- 8 — 4(x + 1) = 8 7.3m + 2(m — 5) = 10
Student HeCp
► Study Tip
In Example 5 you can
clear the equation of
fractions by multiplying
2
by the reciprocal of g.
L _/
J 5 Multiply by a Reciprocal First
Solve 4 = -^(x + 3).
Solution 4 = j(x + 3)
“(4) = -(§)(* + 3)
6 = x + 3
6 — 3 = x + 3 — 3
3 = x
Write original equation.
Multiply each side by j, the reciprocal of
Simplify.
Subtract 3 from each side.
Simplify both sides.
Multiply by a Reciprocal First
Solve the equation. Check your solution in the original equation.
8. 6 = |(x + 7) 9. |(x - 2) = 8 10. -|(x + 1) = 9
Chapter 3 Solving Linear Equations
UJjNJ
KHl Exercises
Guided Practice
Vocabulary Check
Skill Check
Identify the like terms
1. 3x 2 + 5x + 3 + x
4. 4x + 2(x + 1)
Solve the equation.
7. 4x + 3 = 11
10 . 3r- r + 15 = 41
13. 5 (d - 7) = 90
16. |(x + 6) = 12
in the expression.
2 . 8 x — 4 + 5x 2 — 4x
5. 3 — m + 2 (m — 2)
8 . ly - 3 = 25
11 . 13 = 12? — 5 — 3 t
14. 3(8 + b) = 27
17. |(x- 1) = 6
3. 2t + t 2 + 6 1 2 — 6 1
6 . 8 — 3(x + 4) + 3x
9. 2x — 9 = -11
12 . —8 + 5a — 2 = 20
15. — 4(x + 6) = 12
18. |(x + 8 ) = 8
Practice and Applications
SOLVING EQUATIONS
19. 48 = 1 In + 26
22 . 3g - 1 = 8
25. 4a + 9a = 39
28. 22x - I2x = 60
Solve the equation.
20 . 2x + 7 = 15
23. 3)/ + 5 = 11
26. 5w + 2w = 11
29. 4c + (-7c) = 9
21.5 p — 16 = 54
24. 7* - 9 = 19
27. 8/7 — 3/7 — 4 = 21
30. 9t — 15 1= -18
Student HeCp
► Homework Help
Example 1: Exs. 19-24
Example 2: Exs. 44, 45
Example 3: Exs. 25-30
Example 4: Exs. 31-36,
47-52
Example 5: Exs. 37-39
SOLVING EQUATIONS WITH PARENTHESES Solve the equation.
31. 5(6 + j) = 45 32. 3(Jfc — 2) = 18 33. -2(4 - m) = 10
34. x + 4(x + 3) = 17 35. 8 _y — (8 + 6 y) = 20 36. x — 2(3x — 2) = —6
3.3 Solving Multi-Step Equations
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 44-46 is available at
www.mcdougallittell.com
Solution Step
1 + 3 = 6
f = 3
5x = 6
6
x = —
Explanation
Original Equation
43. LOGICAL REASONING Copy the
solution steps shown. Then write
an explanation for each step in the
right-hand column.
44. STUDENT THEATER Your school’s
drama club charges $4 per person for
admission to the play Our Town. The
club borrowed $400 from parents to
pay for costumes and props. After
paying back the parents, the drama club has $100. How many people attended
the play? Choose the equation that represents this situation and solve it.
A. 4x + 400 = 100
B. 4x + 100 = 400
C. 4x - 400 = 100
45. FARMING PROJECT You have a 90-pound calf you are raising for a 4-H
project. You expect the calf to gain 65 pounds per month. In how many
months will the animal weigh 1000 pounds?
Yi
46. FIREFIGHTING The formula d = — + 26 relates nozzle pressure n (in
pounds per square inch) and the maximum distance the water reaches d (in
feet) for a fire hose with a certain size nozzle. Solve for n to find how much
pressure is needed to reach a fire 50 feet away. ►Source: Fire Department Hydraulics
CHALLENGE Solve the equation.
47. 4(2y + 1) — 6y = 18 48. 22x + 2(3x + 5) = 66 49. 6x + 3(x + 4) = 15
50. 7 - (2 - g) = -4 51. x + (5x - 7) = -5 52. 5 a - (2a - 1) = -2
StSndBKCUz&d Test 53. MULTIPLE CHOICE Which is a solution to the equation 9x — 5x — 19 = 21?
Practice
® -10 CD— CDj CD 10
54. MULTIPLE CHOICE The bill (parts and labor) for the repair of a car is $458.
The cost of parts is $339. The cost of labor is $34 per hour. Which equation
could you use to find the number of hours of labor?
® 34(x + 339) = 458 (G) 34 + 339x = 458
CH) 34x + 339 = 458 CD 34 + x + 339 = 458
Chapter 3 Solving Linear Equations
Mixed Review
Maintaining Skills
Quiz 7
WRITING POWERS Write the expression in exponential form.
(Lesson 1.2)
55. a*a*a*a*a*a 56. x to the fifth power 57. 4*4*4
58. five squared 59. t cubed 60. 3x • 3x • 3x • 3x • 3x
EVALUATING EXPRESSIONS Evaluate the expression.
(Lessons 1.3, 2.4, 2.8)
61.5 + 8-3
64. -6 -h 3 — 4 • 5
62. 32 • 4 + 8
65. 2 — 8 +
63. 5 • (12 - 4) + 7
(3 - 6) 2 + 6
66
-5
COMPARING FRACTIONS AND DECIMALS Complete the statement using
<, >, or =. (Skills Review pp. 767, 770)
67. \ ? 0.35
71. -y ^9.5
68 . 1.5 ? |
72.0 ? y
69.0.30 ? |
73.2.7 ? -y
70. | $ 1.6
74. | § 0.75
Solve the equation. (Lessons 3.1, 3.2)
1 .x- 14 = 7 2. _y + 8 = —9 3.5 = m-(-12)
4. 10* =-10 5.47 = | 6 . 3 = |x
6 5
7. HISTORY TEST You take a history test that has 100 regular points and
8 bonus points. You get a score of 91, which includes 4 bonus points. Let x
represent the score you would have had without the bonus points. Which
equation represents this situation? Solve the equation. (Lesson 3.1)
A.x + 91 = 100 B. x + 4 = 91 C.x + 4 = 108
8 . TICKET PRICE You buy six tickets for a concert that you and your friends
want to attend. The total charge for all of the tickets is $72. Write and solve
an equation to find the price of one concert ticket. (Lesson 3.2)
Solve the equation. (Lesson 3.3)
9. 2 x — 5 = 13 10. 12 + 9x = 30 11. Sn - 10 - 12n = -18
12. 6(5y — 3) + 2 = 14 13. 7x - 8 (x + 3) = 1 14. |(x + 1) = 10
15. FOOD PREPARATION You are helping to make potato salad for a family
picnic. You can peel 2 potatoes per minute. You need 30 peeled potatoes.
How long will it take you to finish if you have already peeled 12 potatoes?
(Lesson 3.3)
3.3 Solving Multi-Step Equations
DEVELOPING CONCEPTS
For use with
Lesson 3.4
Goal
Use algebra tiles to solve
equations with variables
on both sides.
Question
Materials
• algebra tiles
How can you use algebra tiles to solve an equation with a variable on
both the left and the right side of the equation?
Explore
O Use algebra tiles to model the
equation 4x + 5 = 2x + 9.
M B B
+ =
+ +
+ +
© You want to have x-tiles on only one
side of the equation. Subtract two
x-tiles from each side. Write the new
equation. ? + 5 = ?
■ ■
■
■
■
IB
IB
+ 1
+ 1
S
+ ■
+
+ ■
B
BB
Bfl
BB
+ B
B
e To isolate the x-tiles, subtract
five 1-tiles from each side. Write
the new equation. 2x = ?
+
S'
+
BB
B
+ +
+1
—
+ +
B
B ■
+
_ ^
y
■r
Q You know the value of 2x. To find
the value of x, split the tiles on
each side of the equation in half to
get x = ? .
+
B +
BB
MB
1 + +
+ +
Think About It
MB
M)l+
= y
+ +
+IB
Use algebra tiles to solve the equation.
1. 4x + 4 = 3x + 7 2. 2x + 3 = 6 + x
3. 6x + 5 = 3x + 14 4. 5x + 2 = 10 + x
5. 8x + 3 = lx + 3 6. x + 9 = 1 + 3x
+ [+
+
S
+
/
+
+
7. The model at the left shows the solution of an
equation. Copy the model. Write the solution
step and an explanation of the step beside
each part of the model.
Chapter 3 Solving Linear Equations
Solving Equations with
Variables on Both Sides
Goal
Solve equations that
have variables on
both sides.
Key Words
• identity
• variable term
• coefficient
Can a cheetah keep up the pace?
The cheetah is the fastest animal on land for running short distances. In
Exercises 48 and 49 you will solve an equation to find out if a cheetah can
catch up to a running gazelle.
Some equations have variables on both sides. To solve these equations, you can
first collect the variable terms on one side of the equation. The examples will
show you that collecting the variable terms on the side with the greater variable
coefficient will result in a positive coefficient.
Student HeCp
B3Z!mZ219 1 Collect Variables on Left Side
Solve lx + 19 = — 2x + 55.
y —.
► Study Tip
Since variables
represent numbers,
you can transform an
equation by adding
and subtracting
variable terms. . *-*-•
h ^
Solution
Look at the coefficients of the x-terms. Since 7 is greater than — 2, collect the
x-terms on the left side to get a positive coefficient.
lx + 19 = — 2x + 55
Write original equation.
► lx + 19 + 2x = — 2x + 55 + 2x
Add 2x to each side.
9x+ 19 = 55
Combine like terms.
9x + 19 - 19 = 55 - 19
Subtract 19 from each side.
9x = 36
Simplify both sides.
9x _ 36
9 9
Divide each side by 9.
x = 4 Simplify.
ANSWER ► The solution is 4.
CHECK /
lx + 19 = — 2x + 55
7(4) + 19 1 -2(4) + 55
47 = 47 /
Write original equation.
Substitute 4 for each x.
Solution is correct.
v.
3.4 Solving Equations with Variables on Both Sides
Student HeCp
->
^ Look Back
For help with
identifying terms of an
expression, see p. 87.
k _ )
J 2 Collect Variables on Right Side
Solve 80 — 9y = 6y.
Solution Remember that 80 — 9y is the same as 80 + (—9 y). Since 6 is
greater than —9, collect the y-terms on the right side to get a positive coefficient.
80 — 9y = 6 y Write original equation.
80 — 9y + 9y — 6y + 9y Add 9 y to each side.
80 = 15y
80 = 15 y
15 ” 15
16
Combine like terms.
Divide each side by 15.
Simplify.
16 1
ANSWER ^ The solution is — or 5—. Check this in the original equation.
Collect Variables on One Side
Solve the equation. Check your solution in the original equation.
1- 34 — 3x = I4x 2. 5_y — 2 = y + 10 3- —6x + 4 = —8x
3 Combine Like Terms First
Solve 3x — 10 + 4x = 5x — 7.
Solution 3x - 10 + 4x = 5x - 7
7x — 10 = 5x — 1
lx — 10 — 5x = 5x — 7 — 5x
2x - 10 = -7
2x - 10 + 10 = -7 + 10
2x = 3
2x = 3
2 2
3
Write original equation.
Combine like terms.
Subtract 5xfrom each side.
Combine like terms.
Add 10 to each side.
Simplify both sides.
Divide each side by 2.
Simplify.
3 1
ANSWER The solution is — or 1— Check this in the original equation.
Combine Like Terms First
Solve the equation. Check your solution in the original equation.
4. 5x — 3x + 4 = 3x + 8 5- 6x + 3 = 8 + lx + 2x
Chapter 3 Solving Linear Equations
NUMBER OF SOLUTIONS So far you have seen linear equations that have only
one solution. Some linear equations have no solution. An identity is an equation
that is true for all values of the variable, so an identity has many solutions.
Student HeCp
► Morl Examples
More examples
1are available at
www.mcdougallittell.com
4 Identify Number of Solutions
Solve the equation if possible. Determine whether it has one solution ,
no solution , or is an identity.
a. 3(x + 2) = 3x + 6 b. 3(x + 2) = 3x + 4 c. 3(x + 2) = 2x + 4
Solution
a. 3(x + 2) = 3x + 6
3x + 6 = 3x + 6
3x + 6 — 3x = 3x + 6 — 3x
6 = 6
Write original equation.
Use distributive property.
Subtract 3xfrom each side.
Combine like terms.
ANSWER ► The equation 6 = 6 is always true, so all values of x are
solutions. The original equation is an identity.
b. 3(jc + 2) = 3x + 4
3x + 6 = 3x + 4
3x + 6 — 3x = 3x + 4 — 3x
6^4
Write original equation.
Use distributive property.
Subtract 3xfrom each side.
Combine like terms.
ANSWER ► The equation 6 = 4 is never true no matter what the value of x.
The original equation has no solution.
c_ 3(x + 2) = 2x + 4
3x + 6 = 2x + 4
3x + 6 — 2x = 2x + 4 — 2x
x + 6 = 4
x + 6 — 6 = 4 — 6
x = —2
Write original equation.
Use distributive property.
Subtract 2xfrom each side.
Combine like terms.
Subtract 6 from each side.
Simplify both sides.
ANSWER ► The solution is —2. The original equation has one solution.
Identify Number of Solutions
Solve the equation if possible. Determine whether the equation has one
solution , no solution , or is an identity.
6. 2(x + 4) = 2x + 8 7_ 2(x + 4) = x — 8
8. 2(x + 4) = 2x — 8 9- 2(x + 4) = x + 8
3.4 Solving Equations with Variables on Both Sides
Exercises
Guided Practice
Vocabulary Check 1. Complete: An equation that is true for all values of the variable is called
a(n) _J_.
2. Is the equation — 2(4 — x) = 2x — 8 an identity? Explain why or why not.
Identify the coefficient of each variable term.
3- 16 + 3y = 22 4. 3x + 12 = 8x — 8 5. 4x — 2x = 6
6- 5x — 4x + 3 = 9 — x 7. 5m + 4 = 8 — 7m 8- 2(x + 1) = 14
Skill Check Solve the equation if possible. Determine whether the equation has one
solution , no solution , or is an identity.
9. lx + 3 = 2x — 2 10. 5(x — 5) = 5x + 24 11. 12 — 5 a = —2a — 9
12. 3(4c + 7) = 12c 13. v — 2x + 3 = 3 — x 14. 6y — 3y + 6 = 5y — 4
15. FUNDRAISING You are making pies to sell at a fundraiser. It costs $3 to
make each pie, plus a one-time cost of $20 for a pastry blender and a rolling
pin. You plan to sell the pies for $5 each. Which equation could you use to
find the number of pies you need to sell to break even, or recover your costs?
A. 3x = 20 + 5x B. 3x + 20 = 5x
C. 3x — 20 = 5x D. 20 — 5x = 3x
16. Solve the correct equation in Exercise 15 to find the number of pies you need
to sell to break even.
Practice and Applications
WRITING Describe the first step you would use to solve the equation.
17. v + 2 = 3x — 4 18. 5t + 12 = 2 1
19. 2x — 1 = — 8x + 13 20. — 4x = — 9 + 5x
Student HeCp
► Homework Help
Example 1: Exs. 17-20,
21-26
Example 2: Exs. 17-20,
21-26
Example 3: Exs. 27-34
Example 4: Exs. 37-46
SOLVING EQUATIONS Solve the equation.
21 . 15 — 2 y = 3 y
23. 5x — 16 = 14 — 5x
25. llx — 21 = 17 — 8jc
27. 5x — 4x = —6x + 3
29. r — 2 + 3r=6 + 5r
31.2f- 3* + 8 = 3f- 8
33. — x + 6 — 5x = 14 — 2x
22 . 2p - 9 = 5p + 12
24. -3 g + 9 = 15g - 9
26. 4x + 27 = 3x + 34
28. lOy = 2y — 6y + 7
30. 4 + 6x — 9x = 3x
32. 13x + 8 + 8x = —9x — 22
34. 5x — 3x + 4 = 3x + 8
Chapter 3 Solving Linear Equations
ERROR ANALYSIS In Exercises 35 and 36, find and correct the error.
35.
36.
Student HeCp
IDENTIFYING NUMBER OF SOLUTIONS Solve the equation if possible.
Determine whether the equation has one solution , no solution , or is an
identity.
^Homework Help
^ Xtra
txira neip wun
W 7 problem solving
problem solving in
37. 8c - 4 = 20 - 4c
38. 24 - 6 r = 6(4 - r)
Exs. 37-42 is available at
www.mcdougallittell.com
39. —7 + 4m = 6m — 5
40. 6m — 5 = 1m + 7 — m
j
41. 3x — 7 = 2x + 8 + 4x
42. 6 + 3c = —c — 6
MENTAL MATH Without writing the steps of a solution, determine
whether the equation has one solution , no solution , or is an identity.
43. 8 + 6a = 6a — 1
44. 6a + 8 = 2a
45. 8 + 6a = 2a + 8
46. 8 + 6a = 6a + 8
47. His tory Link / Steamboats carried cotton
and passengers up and down the Mississippi
River in the mid-1800s. A steamboat could
travel 8 miles per hour downstream from
Natchez, Mississippi, to New Orleans,
Louisiana, and only 3 miles per hour
upstream from New Orleans to Natchez.
It was about 265 miles each way.
^ • Natchez, MS
If it took a steamboat 55 more hours to
go upstream than it did to go downstream,
how long did it take to complete the roundtrip?
Solve St = 3 (t + 55), where t is the time (in hours) it takes the steamboat to
travel downstream and (t + 55) is the time it takes to travel upstream.
48. CHEETAH AND GAZELLE A cheetah running 90 feet per second is 100 feet
behind a gazelle running 70 feet per second. How long will it take the
cheetah to catch up to the gazelle? Use the verbal model to write and solve a
linear equation.
Speed of cheetah • Time = 100 + Speed of gazelle • Time
49. WRITING A cheetah can run faster than a gazelle, but a cheetah can only run
at top speed for about 20 seconds. If a gazelle is too far away for a cheetah to
catch it within 20 seconds, the gazelle is probably safe. Would the gazelle in
Exercise 48 be safe if the cheetah starts running 500 feet behind it? Explain
your answer. HINT: Use 500 feet instead of 100 feet in the verbal model.
3.4 Solving Equations with Variables on Both Sides
CHALLENGE Solve the equation.
50. 2(2x + 3) = — 6(x + 9)
51.7 - (-40 = 4t- 14-21*
Standardized Test
Practice
Mixed Review
Maintaining Skills
52. -|x + 5 = jx - 3 53. 7 - |x = |x + 4
54. (MULTIPLE CHOICE Which equations are equivalent?
I. 3x — 4x + 18 = 5x II. 4 + 6x = 8x — 2 III. 2x — 8 = 7 — x
(a) I and II Cb) II and III CcT) All Cd) None
55. MULTIPLE CHOICE For which equation is j = 4 a solution?
CD -10 + 5; = -2 + 2 j <3D Ij - 3/ + 2 = 4/ - 2
CH) 6; - 4 = 4; + 4 GD 3/ + 7 = 2/ - 2
56. MULTIPLE CHOICE Solve 15x + 6 - x = 16x + 6 - 2x.
(A) v = 0 CS) — 6 Cc) No solution Cp) Identity
57. DRIVING DISTANCE It takes you 3 hours to drive to your friend’s house at
an average speed of 48 miles per hour. How far did you travel? (Lesson 1.1)
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lessons 1.1, 1.2)
x
58. 7 • y when y = 8 59. x — 5 when x = 13 60. — when x = 56
61. x 3 when x = 6 62. 4 1 2 when t = 3 63. (3x) 2 when x = 4
NUMERICAL EXPRESSIONS Evaluate the expression. (Lesson 1.3)
64. (10 + 6) -h 2 — 3 65. 8 + 4 -h (3 — 1) 66. 14 - 2 • 5 - 3
67. MENTAL MATH You want to buy a pair of sneakers that costs $49.99. The
state sales tax adds $2.99 to the total cost. If you have $53, do you have
enough money to buy the sneakers? (Lesson 1.4)
RULES OF ADDITION Find the sum. (Lesson 2.3)
68. 3 +(-4) 69.-6 + 2 70.-11 + (-8)
71. 5 + 16 +(-9) 72. 8 + (-7) + (-10) 73.-22 + (-5) + 4
SOLVING EQUATIONS Solve the equation. (Lesson 3.2)
74. 15x = 255 75. 236x = 0 76. |x = 9 77. |x = 60
DIVIDING DECIMALS Divide. (Skills Review p. 760)
78. 15 + 0.05 79. 4 4 - 0.002 80. 20 +
81.8.1 + 0.9 82.0.72 + 0.3 83.6.4 +
0.4
0.8
84. 46.2 + 0.02
85. 39.1 + 0.01
86 . 23.4 + 0.04
Chapter 3 Solving Linear Equations
More on Linear Equations
Goal
Solve more complicated
equations that have
variables on both sides.
Key Words
• inverse operations
• distributive property
Will it save money to join a health club?
In this lesson you will solve more
linear equations that have variables
on both sides. You will solve an
equation to compare the costs of
different payment plans at a health
club in Example 4.
You have learned several ways to transform an equation into an equivalent
equation. As you solve more complicated equations, you will continue to use
these same steps to isolate the variable.
STEPS FOR SOLVING LINEAR EQUATIONS
Q Simplify each side by distributing and/or combining like terms.
0 Collect variable terms on the side where the coefficient
is greater.
© Use inverse operations to isolate the variable.
© Check your solution in the original equation.
L _ J
i Solve a More Complicated Equation
Solve 4(1 — x) + 3x = —2(x + 1).
Student MeCp
► Study Tip
To isolate the variable
xin Example 1, you can
eliminate -2xfrom the
right side by adding 2x
to each side. ..
Solution
4(1 — x) + 3x = —2(x + 1)
4 — 4x + 3x = —2x — 2
4 — x — —2x — 2
.... ► 4 — x + 2x — —2x — 2 + 2x
4 + x = —2
4 + x — 4 = —2 — 4
x = —6
ANSWER ^ The solution is -6.
Check by substituting —6 for each x in the original equation.
Write original equation.
Use distributive property.
Combine like terms.
Add 2x to each side.
Combine like terms.
Subtract 4 from each side.
Simplify.
3.5 More on Linear Equations
SOLUTION STEPS Simplify an equation before you decide whether to collect
the variable terms on the right side or the left side. In Examples 2 and 3, use the
distributive property and combine like terms to make it easier to see which
coefficient is larger.
J 2 Solve a More Complicated Equation
Solve —3(4x + 1) + 6x = 4(2x — 6).
Student HeCp
^ -
► Study Tip
You can use mental
math to add 6xto
each side of the
equation. ..
Solution
— 3(4x + 1) + 6x = 4(2x — 6)
— 12x — 3 + 6x = 8x — 24
—6x — 3 = 8x — 24
-.► —3 = 14x — 24
21 = 14x
21
14
3
2
x
x
Write original equation.
Use distributive property.
Combine like terms.
Add 6x to each side.
Add 24 to each side.
Divide each side by 14.
Simplify.
ANSWER ^ The
3 1
solution is — or 1— Check this in the original equation.
3 Solve a More Complicated Equation
Solve ^(12x + 16) = 10 - 3(x - 2).
Solution
|(12x + 16) = 10 - 3(x - 2)
Write original equation.
i|^ + -^=10-3x + 6
Use distributive property.
3x + 4 = 16 — 3x
Simplify.
6x + 4 = 16
Add 3x to each side.
(N
r-H
II
so
Subtract 4 from each side.
x = 2
Divide each side by 6.
ANSWER ► The solution is 2. Check this in the original equation.
Solve a More Complicated Equation
Solve the equation.
1 - 6(x + 3) + 3x = 3(x — 2) 2 . 4x + (2 — x) = —3(x + 2)
3. — 2(4x + 2) = —2(x + 3) + 9
4. }(3j - 12) = 6 - 2(y - 1)
Chapter 3 Solving Linear Equations
Student HeCp
p More Examples
More examples
l/ are available at
www.mcdougallittell.com
4 Compare Payment Plans
HEALTH CLUB COSTS A health club has two payment plans. You can become
a member by paying a $10 new member fee and use the gym for $5 a visit. Or,
you can use the gym as a nonmember for $7 a visit. Compare the costs of the
two payment plans.
Solution Find the number of visits for which the plans would cost the same.
Verbal
Model
New member
fee
+
Member’s
fee per visit
Number
•
of visits
v._
V
_ S
Members cost
Nonmember’s
fee per visit
•
Number
of visits
v V
/
Nonmember's cost
Labels New member fee = 10 (dollars)
Member’s fee per visit = 5 (dollars)
Nonmember’s fee per visit = 7 (dollars)
Number of visits = x
Algebraic 10 + 5 • x = 7 • x
Model
10 = 2x
x = 5
A table can help you interpret the result.
Write linear equation.
Subtract 5xfrom each side.
Divide each side by 2.
Number of visits
i
2
3
4
5
6
7
Member’s cost
$15
$20
$25
$30
($35)
$40
$45
Nonmember’s cost
$7
$14
$21
$23
($35)
$42
$4<?
-V-' *-V-'
Nonmember’s cost is less Member’s cost is less
ANSWER ► If you visit the health club 5 times, the cost would be the same as a
member or a nonmember. If you visit more than 5 times, it would
cost less as a member. If you visit fewer than 5 times, it would cost
less as a nonmember.
Compare Payment Plans
5. A video store charges $8 to rent a video game for five days. Membership to
the video store is free. A video game club charges only $3 to rent a game for
five days, but membership in the club is $50 per year. Compare the costs of
the two rental plans.
3.5 More on Linear Equations
^ Exercises
Guided Practice
Vocabulary Check State the inverse operation needed to solve the equation.
1.x + 5 = 13 2.x- 4 =-9 3. lx = 28 4.36 = ^
6
Decide whether the equation is true or false. Use the distributive property
to explain your answer.
5. 3(2 + 5) = 3(2) + 5 6. (2 + 5)3 = 2(3) + 5(3)
7. 8(6 - 4) = 8(6) - 8(4) 8. (6 - 4)8 = 6 - 4(8)
9. -2(4 + 3) = -8 + 6 10. -2(4 - 3) = -8 + 6
Skill Check
Solve the equation. Check your
11. 2(x - 1) = 3(jc + 1)
13. 6(8 + 3a) = —2 (a - 4)
15. —4 (m + 6) + 2m = 3 (m + 2)
17. |(16x - 8) = 9 - 5(x - 2)
O
solution in the original equation.
12 . 3(x + 2) = 4(5 + x)
14. 8(4 — r) + r = —6(3 + r )
16. 7(c - 7) + 4c = — 2(c + 5)
18. |(25 - 5k) = 21 - 3 (k - 4)
Practice and Applications
SOLVING EQUATIONS Solve the equation.
19. 3{x + 6) = 5(x — 4)
21 . 5(x + 2) = x + 6(x — 3)
23. 24 y - 2(6 -y) = 6(3y + 2)
25. 4(m + 3) — 2/77 = 3(m — 3)
27. 4 + 5(3 — jc) = 4(8 + 2x)
29. 10(2x + 4) = — (—8 — 9x) + 3x
20 . 7(6 -y)= -3(y - 2)
22 . 8(x + 5) — l(x + 8)
24. 7(fe + 2) - 46 = 2 (b + 10)
26. 2(a + 4) = 2 (a — 4) + 4a
28. 5(—x + 2) = —3(7x + 2) + 8x
30. 9(f - 4) - It = 5(t - 2)
r Student HeCp
► Homework Help
Example 1: Exs. 19-30
Example 2: Exs. 19-30
Example 3: Exs. 31-36
Example 4: Exs. 40-43
V _
SOLVING EQUATIONS Solve the equation by distributing the fraction
first.
31. 3{x + 2) = ^(12x + 4) — 5x
33. ^(8n — 2) = —(—8 + 9 n) — 5 n
35. |(24 - 200 + 9 1 = 2(5 1 + 1)
32. |(10x + 25) = -10 - 4(x +3)
34. 2(8 - 4x) = |(33 - 18x) + 3
36. |(9 n - 6) = 4(n + 1)
Chapter B Solving Linear Equations
37, LOGICAL REASONING Write the steps you would use to solve the equation
3(x — 4) + 2x = 6 — x. Beside each step, write an explanation of the
step. Then show how to check your answer.
ERROR ANALYSIS In Exercises 38 and 39, find and correct the error.
40, COMPUTER TIME A local computer center charges nonmembers $5 per
session to use the media center. Members are charged a one-time fee of $20
and $3 per session. Use the verbal model to write an equation that can help
you decide whether to become a member. Solve the equation and explain
your solution.
Member
+
Member
Number
Nonmember
Number
one-time fee
session fee
of sessions
session fee
of sessions
41. PAINT YOUR OWN POTTERY You want to paint a piece of pottery. The
total price is the cost of the piece plus an hourly painting rate. Studio A sells
a vase for $12 and lets you paint for $7 an hour. Studio B sells a similar vase
for $15 and lets you paint for $4 an hour. Which equation would you use to
compare the total price at each studio?
A. lx — 12 = 4x — 15 B. 12 + lx — 15 + 4x
42. COMPARE COSTS Use the information in Exercise 41. If it takes you 2 hours
to paint a vase, would Studio A or Studio B charge less to paint a vase?
43. ROCK CLIMBING A rock-climbing gym charges nonmembers $16 per day
to use the gym and $8 per day for equipment rental. Members pay a yearly
fee of $450 for unlimited climbing and $6 per day for equipment rental.
Which equation represents this situation? Solve the equation to find how
many times you must use the gym to justify becoming a member.
A. (16 + 8)x = 450 — 6x B. 24x = 450 6x
C. (16 + 8)v = 450 + 6x D. I6x + 8 = 450 + 6x
ROCK CLIMBING Indoor
rock-climbing gyms have
climbing walls and routes to
simulate real outdoor climbs.
More about rock
climbing is available at
www.mcdougallittell.com
CHALLENGE Solve the equation.
44. —3(7 — 3 n) + 2n = 5(2 n — 4) 45. 4x + 3(v — 2) = —5(x — 4) — x
46. -7 + 8(5 - 3 q) = 3(7 - 9 q) 47. y + 2(y - 6) = -(2 y - 14) + 49
48. j(3x - 12) = 6 - 2(x - 1) 49. 2(6 - 2x) = ~9x - \{~4x + 6)
3.5 More on Linear Equations
Student HeCp
► Homework Help
Help with problem
~<r® ^ solving in Exs. 50
and 51 is available at
www.mcdougallittell.com
Standardized Test
Practice
Mixed Review
Maintaining Skills
Science Link y Use the following information for Exercises 50 and 51.
The diagram shows the orbits
of Jupiter’s four largest moons:
Io, Europa, Ganymede, and
Callisto. The orbits are circular.
Io’s orbit is x kilometers (km)
from Jupiter. The distance
between Io and Europa is
300,000 km. The distance
between Europa and Ganymede
is 400,000 km. The distance
between Ganymede and Callisto
is 800,000 km. The distance
from Jupiter to Callisto is
3 3
4—x, or 4— times the distance from Jupiter to Io
800,000 km
400,000 km
300,000 km
50, Find the distance x between Jupiter and Io, using the equation
x + 300,000 + 400,000 + 800,000 — 4—x.
51- Use the solution to Exercise 50 to find the distance of each moon’s orbit
from Jupiter.
Moon s Orbit
Io
Europa
Ganymede
Callisto
Distance from Jupiter (km)
?
?
?
?
52. MULTIPLE CHOICE Which inverse operation can be used to solve the
equation 6 + x = 15?
(A) Add 6 to each side.
CcT) Multiply each side by 6.
CD Subtract 6 from each side.
(D) Divide each side by 6.
53. MULTIPLE CHOICE Solve -(7x + 5) = 3x - 5.
CD -5
(H) io
GD 15
MULTIPLYING REAL NUMBERS Find the product. (Lesson 2.5)
54. 6(-6) 55. —3(—12) 56. -8(-5) 57. 11(—7)
COMBINING LIKE TERMS Simplify the expression by combining like
terms if possible. If not possible, write already simplified. (Lesson 2.7)
58. h + 7 — 6h 59. 3 w 2 + 2w — 3w 60. ab + 4a — b
61. 35 + 5t — 2s + 61 62. x — y + 2xy
63. —8 m — m 2 + 2m
SUBTRACTING DECIMALS Subtract. (Skills Review p. 759)
64.11.9 - 1.2 65.15.75 - 4.25 66.3.6 - 0.5
67. 12.44 - 6.02
68 . 22.87 - 2.99
69. 56.32 - 33.83
Chapter 3 Solving Linear Equations
Solving Decimal Equations
Goal
Find exact and
approximate solutions of
equations that contain
decimals.
Key Words
• rounding error
What's the price of a slice?
Exact answers are not always practical.
Sometimes rounded answers make
more sense. In Example 3 you will
round for a practical answer for each
person’s share in the cost of a pizza.
The giant pizza slice shown here was
made in San Francisco, California, in
1989. Its shape was very close to a
triangle with a base of 4 meters and a
height of 6.5 meters.
Round for the Final Answer
Solve — 38x — 39 = 118. Round to the nearest hundredth.
Solution
— 38x — 39 = 118 Write original equation.
— 38x = 157 Add 39 to each side.
157
x = —— Divide each side by -38.
— Jo
x~ —4.131578947 Use a calculator to get an
approximate solution.
x ~ —4.13 Round to nearest hundredth.
ANSWER ^ The solution is approximately —4.13.
CHECK /
— 38x — 39 = 118 Write original equation.
— 38(— 4.13) — 39 X 118 Substitute -4.13 for x.
117.94 -118%/ Rounded answer is reasonable.
When you substitute a rounded answer into the original equation, the two sides
of the equation may not be exactly equal, but they should be approximately
equal. Use the symbol ~ to show that quantities are approximately equal.
Round for the Final Answer
Solve the equation. Round to the nearest hundredth.
1. 24x + 43 = 66 2. -42x + 28 = 87 3. 22x - 39x = 19
3.6 Solving Decimal Equations
Student HeCp
► More Examples
More exam Pl es
are available at
www.mcdougallittell.com
2 Solve an Equation that Contains Decimals
Solve 3.5x — 37.9 = 0.2x. Round to the nearest tenth.
Solution
3.5x - 37.9 = 0.2x
3.3x - 37.9 = 0
3.3x = 37.9
3.3x 37.9
3.3
3.3
11.48484848
Write original equation.
Subtract 0.2xfrom each side.
Add 37.9 to each side.
Divide each side by 3.3.
Use a calculator to get an
approximate solution.
Round to nearest tenth.
* - 11.5
ANSWER ► The solution is approximately 11.5.
CHECK /
3.5x — 37.9 = 0.2x Write original equation.
3.5(11.5) — 37.9 2= 0.2(11.5) Substitute 11.5 for each x.
2.35 ~ 2.3 / Rounded answer is reasonable.
Solve an Equation that Contains Decimals
Solve the equation. Round to the nearest tenth.
4. 2.4x - 0.9 = 12.4 5. 1.13y - 25.34 = 0.26y
6. 14.7 + 2.3x = 4.06 7. 3.25 n - 4.71 = 0.52 n
ROUNDING ERROR Using a rounded solution in a real-life situation can lead to
a rounding error, as in Example 3.
3 Round for a Practical Answer
Three people want to share equally in the cost of a pizza. The pizza costs
$12.89. What is each person’s share?
Solution
Find each person’s share by solving 3x = 12.89.
3x — 12.89 Write original equation.
x — 4.29666. . . Use a calculator to divide each side by 3.
Exact answer is a repeating decimal.
x ~ 4.30 Round to nearest cent.
ANSWER ^ Each person’s share is $4.30.
Three times the rounded answer is one cent too much due to rounding error.
Chapter 3 Solving Linear Equations
PERCENTS When you solve a problem involving percents, remember to write
the percent in decimal form.
Student HeCp
^
► Skills Review
To review writing a
percent as a decimal,
see p. 768.
^ _ /
5% of $23.45 = 0.05(23.45) Rewrite 5% as 0.05.
= 1.1725 Multiply.
~ $1.17 Round to nearest cent.
■i'/.IJIJU 4 Use a Verbal Model
You buy a sweatshirt for a total cost of $20. The total cost includes the price of
the sweatshirt and a 5% sales tax. What is the price of the sweatshirt?
Solution
Price
+
Sales tax
•
Price
= Total cost
Model
Labels
Price = x
(dollars)
Sales tax = 0.05
(no units)
Total cost = 20
(dollars)
Algebraic
x + 0.05 • x = 20
Write linear equation.
Model
1.05jc = 20
Combine like terms.
20
X = 1.05
Divide each side by 1.05.
x « 19.04761905
Use a calculator to get an
approximate solution.
x- 19.05
Round to nearest cent.
ANSWER ^ The price of the sweatshirt is $19.05.
CHECK /
x + 0.05 • x = 20
Write linear equation.
19.05 + (0.05X19.05) 1 20
Substitute 19.05 for each x.
19.05 + 0.95 1 20
Multiply and round to nearest cent.
20 = 20 /
Solution is correct.
The total cost of $20 includes the $19.05 price of the sweatshirt and the 5%
sales tax of $.95.
Use a Verbal Model
8 . You spend a total of $25 on a gift. The total cost includes the price of the gift
and a 7% sales tax. What is the price of the gift without the tax? Use the
verbal model to solve the problem.
Price
+ Sales tax •
Price
= Total cost
3.6 Solving Decimal Equations
Exercises
Guided Practice
Vocabulary Check 1 , Give an example of rounding error.
2 . The solution of I3x = 6 rounded to the nearest hundredth is 0.46.
Which of the following is a better way to list the solution? Explain.
A. x = 0.46 B. x ~ 0.46
Tell what each symbol means.
3- = 4. ~ 5. 1 6. ^
Skill Check Round to the nearest tenth.
7. 23.4459 8. 108.2135
11.56.068 12.0.555
9.-13.8953 1 0.62.9788
13.8.839 14.-75.1234
Solve the equation. Round the result to the nearest hundredth. Check the
rounded solution.
15. 2.2x = 15 16. 14 - 9x = 37
17. 3(31 - 14) = -4 18. 2.69 - 3.64x = 23.78x
19. BUYING DINNER You spend a total of $13.80 at a restaurant. This includes
the price of dinner and a 15% tip. What is the price of dinner without the tip?
Use a verbal model to solve the problem.
Practice and Applications
SOLVING AND CHECKING Solve the equation. Round the result to the
nearest hundredth. Check the rounded solution.
20.
13x — 7 = 27
21.
38 = -14 + 9 a
22.
17x- 33 = 114
23.
-lx + 32 = -21
24.
—lx + 17 = -6
25.
18 — 3_y = 5
26.
99 = 21l + 56
27.
—35 m + 75 = 48
28.
CO
II
00
1
t—H
1
29.
42 = 23x - 9
Student ftcCp
^Homework Help
Example 1: Exs. 20-29
Example 2: Exs. 30-35
Example 3: Ex. 36
Example 4: Exs. 37, 38
X
SOLVING EQUATIONS Solve the equation. Round the result to the
nearest hundredth.
30. 9.47x = 7.45x -8.81 31. 39.21x + 2.65 = 42.03x
32. 12.67 + 42.35x = 5.34x 33. 4.65x - 4.79 = -6.84x
34. 7.87 - 9.65x = 8.52x - 3.21 35. 8.79x - 6.54 = 6.48 + 13.75x -
Chapter 3 Solving Linear Equations
Link to
economics
COCOA PRODUCTION
All chocolate products are
made from the beans of the
cacao tree. The beans are
known as cocoa beans in
English-speaking countries.
36. COCOA CONSUMPTION The 267.9 million people in the United States
consumed 639.4 million kilograms of cocoa produced in the 1996-1997
growing year. Which choice better represents the amount of cocoa
consumed per person that year? Explain your reasoning.
► Source: International Cocoa Organization
A. 2.38671146 kilograms B. about 2.4 kilograms
FUNDRAISING To raise money, your student council is selling magazine
subscriptions. The student council will receive a one-time bonus of $150
from the magazine publisher plus 38% of the subscription money. The
following verbal model represents the situation.
Money
Ronus
+
Subscription
•
Subscription
raised
J —9 V/ll Li IJ
percentage
money
37. Write a linear equation from the verbal model.
HINT: Remember to write the percent in decimal form: 38% = 0.38.
38. How much subscription money is needed for the council to raise a total of
$300? Round your answer to the nearest dollar.
Changing Decimals to integers
Solve 4.5 — 7.2x = 3.4x — 49.5. Round to the nearest tenth.
Solution
You can multiply an equation with decimal coefficients by a power of ten to
get an equivalent equation with integer coefficients. Multiply each side of this
equation by 10 to rewrite the equation without decimals.
4.5 - 1.7.x = 3.4x - 49.5
Write original equation.
10(4.5 - 1.2k) = 10(3.4x - 49.5)
Multiply each side by 10.
45 - 72x = 34x - 495
Use distributive property.
45 = 106x - 495
Add 72x to each side.
540 = 106x
Add 495 to each side.
540
106 X
Divide each side by 106.
5.094339623 ~x
Use a calculator to get an
approximate solution.
5.1 ~ x
Round to the nearest tenth.
Student HeCp
► Homework Help
Extra help with
^ problem solving in
Exs. 39-42 is available at
www.mcdougallittell.com
ANSWER [► The solution is approximately 5.1. Check this in the original equation.
Solve the equation. Round to the nearest tenth.
39. 2.5x + 0.7 = 4.6 - 1.3x 40. I.Ijc + 3.2 = 0.2x - 1.4
41. 3.35x + 2.29 = 8.61 42. 0.625y - 0.184 = 2.506y
3.6 Solving Decimal Equations
Standardized Test
Practice
Mixed Review
FIELD TRIP In Exercises 43-45, use the following information.
School buses that have 71 seats will be used to transport
162 students and 30 adults.
43, Write an equation to find the number of buses needed.
44, Solve the equation in Exercise 43. Is the exact answer practical? Explain
your reasoning.
45- Would you round the answer to Exercise 44 up or down? Why?
46. IVIULTIPLE CHOICE What power of ten would you multiply the equation
5.692x — 1.346 = 8.45lx by to change it to an equivalent equation with
integer coefficients?
(a) io 1 d) io 2 c© io 3 d) io 4
47. IVIULTIPLE CHOICE What is the solution of the equation
7.2x + 5.6 = —8.4 — 3.7x rounded to the nearest hundredth?
CD -1.28 CD -1.284 CD -1.29 GD 1.29
48. MULTIPLE CHOICE The cross-country track team ran 8.7 kilometers in
42.5 minutes during their workout. Which equation could you use to find r,
the team’s average running speed (in kilometers per minute)?
(A) 8.7r = 42.5 CD 42.5r = 8.7
CD 42.5 + r = 8.7 CD 8.7 + r = 42.5
49. MULTIPLE CHOICE Solve the equation you chose in Exercise 48 to find the
team’s average running speed (in kilometers per minute).
CD 0.2 CD 4.1 CD 32.0 CD 32.2
ACCOUNT ACTIVITY In Exercises
50-52, use the table. It shows all
the activity in a checking account
during June. Deposits are positive
and withdrawals are negative.
(Lesson 1.7)
Day
Activity
June 6
-$225.00
June 10
+ $310.25
June 17
+ $152.33
June 25
-$72.45
June 30
-$400.00
50. How did the amount of money in the account change from the beginning of
June through June 10?
51. Find the total amount withdrawn in June.
52. What was the total change in the account balance over the course of the
month?
53. INPUT-OUTPUT TABLES Make an input-output table for the function
A = 8 + 5/. Use 2, 3, 4, 5, and 6 as values for t. (Lesson 1.8)
Chapter 3 Solving Linear Equations
Maintaining Skills
Quiz 2
FINDING OPPOSITES Find the opposite of the number. (Lesson 2.2)
54. 8
58. 7.5
55. -3
59. 5.6
56. 0.2
60. -4.9
57.
61. -16
ADDING FRACTIONS Add. Write the answer as a fraction or as a mixed
number in simplest form. (Skills Review p. 764)
2 3
62. lj + 2j
“■ 5 Ta + * 7 ?
68. 3^ + 5-jy
7 1
63. 477 + 9^
66 - + <
69. 4 + 4
64 3— + 2—
J 12 Z 12
__ n 12 . ,_13
67 ' 9 T6 + 15 l6
70 - 4 + 7 i
Solve the equation. Tell whether it has one solution , no solution , or is an
identity. (Lesson 3.4)
1
r-
II
1
r-
<N
t"
2. 18 + 5n = 8n
3. -40c + 4)= -2 (2x + 8)
4. |(64r + 32) = kl6r - 8)
O Z
Solve the equation. (Lessons 3.4, 3.5)
5. 2y + 5 = -y — 4
6. 13m = 15m + 14
7. 8x — 3 — 5x = 2x + 7
8. 9 — 4x = 6x + 2 — 3x
9. 5 + 4(x - 1) = 3(2 + x)
10. -3(4 — r) + 4r = 2(4 + r)
11. 8n + 4(—5 — In) = — 2(n + 1)
12. v — 5(x + 2) = x + 3(3 — 2x)
13. \(2k - 4) = 3 (k + 2) - 3 k
14. |-(6x — 3) = 6(2 + x) — 5x
15. BIKE SAFETY You live near a mountain bike trail. You can rent a mountain
bike and a safety helmet for $10 an hour. If you have your own helmet, the
bike rental is $7 an hour. You can buy a helmet for $28. How many hours do
you need to use the trail to justify buying your own helmet? (Lesson 3.5)
Solve the equation. Round to the nearest hundredth. (Lesson 3.6)
16. lx + 19 = 11 17. -13c + 51c = -26
18. 3.6y + 7.5 = 8.2y 19. 18y - 8 = 4y - 3
20. 2.24x - 33.52 = 8.91* 21. 3.2x - 4.9 = 8.4x + 6.7
22. BASEBALL CARDS You have 39 baseball cards that you want to give to
5 of your friends. You want each friend to get the same number of cards.
How many baseball cards should you give to each friend? (Lesson 3.6)
3.6 Solving Decimal Equations
For use with
Lesson 3.6
USING A GRAPHING CALCULATOR
j \ / I I i r-t i i—I
*V JVJUII
One way to solve multi-step equations is to use a graphing calculator to generate
a table of values. The table can show a value of the unknown variable for which
the two sides of the equation are approximately equal.
Student HeCp
►Software Help
See steps for
using a computer
spreadsheet as an
alternative approach at
www.mcdougallittell.com
Samplt
Use a table on a graphing calculator to solve 4.29x + 3.89(8 — x) = 2.65x.
Round your answer to the nearest tenth.
Solution
0 Use the Table Setup function on your graphing
calculator to set up a table. Choose values of x
beginning at 0 and increasing by 1.
0 Press • Enter the left-hand side of the
equation as Y, and the right-hand side of the
equation as Y r Enter jQ| for each
multiplication. It prints as *.
0 View your table. The first column of the table
should show values of x. Scroll down until you
find values in the Yj and Y 2 columns that are
approximately equal. The values are closest to
being equal when x = 14, so the solution must
be greater than 13 and less than 15.
X
Y1
Y 2
12
35.92
31.8
13
36.32
34.45
m
36.72
37.1
15
37.12
39.75
16
37.52
42.4
Q To find the solution to the nearest tenth, change
the Table Setup so that x starts at 13.1 and
increases by 0.1. You can see that the values in
the Yj and Y 2 columns are closest to being equal
when x = 13.8. The solution to the nearest tenth
is 13.8.
X
Y1
Y 2
13.6
36.56
36.04
13.7
36.6
36.305
Efclfa
36.64
36.57
13.9
36.68
36.835
14
36.72
37.1
TryThas*
Use a graphing calculator to solve the equation to the nearest tenth.
1. 19.65x + 2.2(x - 6.05) = 255.65 2 . 16.2(3.1 - jc) - 31.55* = -19.5
3. 3.56* + 2.43 = 6.17* - 11.40 4. 3.5(* - 5.6) + 0.03* = 4.2* - 25.5
Chapter 3 Solving Linear Equations
Formulas
Goal
Solve a formula for one
of its variables.
Key Words
• formula
How fast did Pathfinder travel to Mars?
The Mars Pathfinder Mission used
a solar-powered spacecraft to carry
a robotic explorer, Sojourner rover,
to Mars. Sojourner, shown in the
photograph, was the first wheeled
vehicle operated on Mars.
In Example 5 you will solve a
formula for one of its variables to
estimate Pathfinder’s average
speed on its flight to Mars.
A formula is an algebraic equation that relates two or more quantities. You can
solve a formula to describe one quantity in terms of the others as shown in the
examples that follow.
Student HeCp
>
► Look Back
To review steps for
solving linear
equations, see p. 157.
\ _ /
i Solve a Temperature Conversion Formula
The Celsius and Fahrenheit temperature scales are related by the formula
C = g(F — 32), where Crepresents degrees Celsius and F represents degrees
Fahrenheit. Solve the temperature formula for degrees Fahrenheit F.
Solution
To solve for the variable F, transform the original formula to isolate F.
Use the steps you already know for solving a linear equation.
c = |(F - 32)
9 9 5
- • C = - • |(F - 32)
9
|C = F ~ 32
|c + 32 = F ~ 32 + 32
|c + 32 = F
Write original formula.
9 5
Multiply each side by the reciprocal of
Simplify.
Add 32 to each side.
Simplify.
ANSWER ► The new formula is F — ^C + 32.
3.7 Formulas
H
Student HeCp
p More Examples
More examples
are ava j| a b| e a t
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2 Solve an Area Formula
The formula for the area
of a triangle is A = ~bh.
Find a formula for the
base b in terms of area A
and height h.
Solution
A = ^bh
2A = bh
2A
ir = b
Write original formula.
Multiply each side by 2.
Divide each side by h.
3 Solve and Use an Area Formula
The formula for the area of a rectangle is A = £w.
a. Find a formula for length £ in terms of area A and width w.
b. Use the new formula to find
the length of a rectangle that
has an area of 35 square feet
and a width of 5 feet.
Solution
a. Solve for length l.
A = £w Write original formula.
— = £ Divide each side by w.
w
b. Substitute the given values into the new formula.
ANSWER ^ The length of the rectangle is 7 feet.
k_
Solve and Use an Area Formula
1. In the formula for the area of
a triangle, solve for height h.
2. Use the new formula to find the
height of a triangle that has an
area of 25 square inches and a
base of 10 inches.
Chapter 3 Solving Linear Equations
4 Solve and Use a Density Formula
The density of a substance is found by dividing its mass by its volume.
a. Solve the density formula d = — for mass m.
b. Use the new formula to find the mass of a lead sample that has a density
of 11.3 grams per cubic centimeter and a volume of 0.9 cubic centimeters.
Round to the nearest tenth.
Solution
Link to
Careers
AEROSPACE ENGINEER
Donna Shirley was the
manager of the Mars
Exploration Program from
1994-1998 and the original
leader of the team that built
the Sojourner rover.
More about
" Pathfinder is at
www.mcdougallittell.com
jfi
a. d = — Write original formula.
dv = m Multiply each side by v.
b. Substitute the given values into the new formula.
m = dv= 11.3 • 0.9 = 10.17 - 10.2
ANSWER ► The mass of the lead sample is approximately 10.2 grams.
Solve and Use a Distance Formula
SPACE TRAVEL Mars Pathfinder was launched on December 4, 1996. During
its 212-day flight to Mars, it traveled about 310 million miles.
a. Solve the distance formula d = rt for rate r.
b- Estimate Pathfinder's average speed in miles per hour. Round your answer
to the nearest whole number.
Solution
a o d — rt Write original formula.
~ — r Divide each side by t.
b. You are given the flight time in days , but you want to find the average
speed in miles per hour. Because there are 24 hours in each day, there are
212(24) hours in 212 days. Time in hours, t = 212(24) = 5088 hours.
Substitute the given values into the new formula.
d = 310,000,000
t 5088
60,928
ANSWER ► Pathfinder's average speed was about 60,928 miles per hour.
s_
Solve and Use a Distance Formula
3. Solve the distance formula d = rt for t.
4. Use the result to find the time (in days) that it takes to travel 40 million miles
at an average speed of 50,000 miles per day.
3.7 Formulas
Exercises
Guided Practice
Vocabulary Check Complete the sentence.
1. A formula is an algebraic ? that relates two or more real-life quantities.
2 . You can ? a formula to express one quantity in terms of the others.
Skill Check
Solve the equation for the indicated variable.
3 . r — s = t; r 4. ax + b = c; b
6 . 2j + 5 = k;j
7. x = ^(y + 4);y
5. 3y = x; y
8 . 6(s — 1) = t\ s
In Exercises 9 and 10, use the formula for the area of a rectangle,
A = iw.
9- Find a formula for w in terms of A and £.
10. Use the new formula in
Exercise 9 to find the width
of a rectangle that has an area
of 104 square inches and a _
length of 13 inches.
Practice and Applications
CONVERTING TEMPERATURE In Exercises 11 and 12, use the
g
temperature conversion formula F=|C+32.
11. Solve the formula for degrees Celsius C. Show all your steps.
12. Normal body temperature is given as 98.6°F. Use the formula you wrote in
Exercise 11 to find this temperature in degrees Celsius.
Student HeCp
| ►Homework Help
Example 1: Exs. 11,12
Example 2: Exs. 13-17
Example 3: Exs. 13-17
Example 4: Exs. 18,19
Example 5: Exs. 20, 21
V _ J
SOLVING AN AREA FORMULA Solve the formula for the indicated
variable. Show all your steps. Then evaluate the new formula by
substituting the given values.
13. Area of a rectangle: A = i w
Solve for w.
i
w
Find the value of w
when A = 36 andi = 9.
14. Area of a triangle: A = ^bh
Solve for h.
I — b ~\
Find the value of h
when A = 24 and b = 8.
Chapter 3 Solving Linear Equations
SOLVING AN AREA FORMULA Solve the formula for the indicated
variable. Show all your steps. Then evaluate the new formula by
substituting the given values.
15. Solve A = £w fori. 16. Solve A = j^bh for b.
w
1
Find the value of £
when A = 112 and w — 1.
Find the value of b
when A — 22 and h — 4.
17. 4Puzz^ The formula for
the perimeter of a rectangle is
P = 2£ + 2w. Find the area of a
rectangle that has perimeter P of
18 centimeters and length £ of
6 centimeters. HINT: You can
begin by solving the perimeter
formula for w.
w
1
Science Lie In Exercises 18 and 19, use the formula for density,
d = —, where m = mass and v = volume.
v
18. Find a formula for v in terms of d and m.
BOTTLE-NOSED WHALES
usually stay under water for
15 to 20 minutes at a time, but
they can stay under water for
over an hour.
19. Use the formula you wrote in Exercise 18 to find the volume of a piece of
cork that has a density of 0.24 grams per cubic centimeter and a mass of
4.0 grams. Round to the nearest hundredth.
20. BOTTLE-NOSED WHALES A bottle-nosed whale can dive at a rate
of 440 feet per minute. You want to find how long it will take for a
bottle-nosed whale to dive 2475 feet at this rate. Which equation represents
this situation?
A. t = 2475 - 440
B. t =
2475
440
21. Solve the correct equation in Exercise 20. Round your answer to the nearest
whole minute.
SCUBA DIVING In Exercises 22 and 23, use the following information.
A scuba diver starts at sea level. The pressure on the diver at a depth of d feet
is given by the formula below, where P represents the total pressure in pounds
per square foot.
P = 64J+ 2112.
22 . Find a formula for depth d in terms of pressure P.
23. If the current pressure on a diver is 4032 pounds per square foot, what is the
diver’s current depth?
3.7 Formulas
Standardized Test
Practice
Mixed Review
Maintaining Skills
24. LOGICAL REASONING Given a
triangle whose sides have lengths Solution Step
a, b, and c, the formula for its P = a ■+■ b +- c
perimeter P is P = a + b + c. P-b = a+-c
The steps at the right show a p_^_ c==a
formula for finding a side length
of a triangle given the perimeter
and the lengths of the two other sides.
Write an explanation for each step.
Explanation
Original Equation
?
?
25. MULTIPLE CHOICE You plan to drive to the mountains to go hiking. You
estimate that you will travel on a highway for 205 miles at an average speed
of 55 miles per hour. How much time will you need for this part of the trip?
Round your answer to the nearest whole hour.
(A) 1 hour Cb) 2 hours Cep 3 hours Cp) 4 hours
26. MULTIPLE CHOICE What is the equivalent of 25°C in degrees
Fahrenheit? Use the formula F = + 32.
CD -4°F ® 13°F CED 46°F CD V7°F
CHECKING SOLUTIONS OF INEQUALITIES Check to see if the given value
of the variable is or is not a solution of the inequality. (Lesson 1.4)
27. v - 8 < 5; x = 12 28. 4 + k > 32; k = 30 29. 9a > 54; a = 5
30. t + 17 < 46; t = 21 31. I2x < 70; x = 6 32. y — 33 > 51; y = 84
33. 6x < 35; x — 6 34. 14 - y > 12; y = 4 35. 42 + x < 65; x = 23
EVENT ATTENDANCE Use the
bar graph for Exercises 36-38.
Each bar shows the percent
of teenagers that attended a
selected event during a
12-month period. (Lesson 1.7)
36. Which event was attended by the
most teens?
37. What percent of teens attended
rock concerts?
38. On average, how many teens out of 100 attended art museums?
SIMPLIFYING FRACTIONS Write the fraction in simplest form.
(Skills Review p. 163)
21
39 ' 49
50
40 ‘ 85
41 ^
72
42.
48
64
43 ^
32
44 ^
48
28
45 ' 32
46.
9
27
Where Teens Go
Pro sports
Art museum
44%
31%
28%
26%
0% 10% 20% 30% 40% 50%
► Source: YOUTHv iews
Chapter 3 Solving Linear Equations
Ratios and Rates
Goal
Use ratios and rates to
solve real-life problems.
Key Words
• ratio
• rate
• unit rate
• unit analysis
How far can you drive on a full tank of gas?
Rates are useful for estimating the
distance a truck can travel on a tank
of gasoline. In Example 5 you will
use a truck’s average mileage to
estimate how many miles it can
travel on 18 gallons of gasoline.
The ratio of a to b is y. If a and b are measured in different units, then y is called
b b
the rate of a per b . Rates are often expressed as unit rates. A unit rate is a rate
per one given unit, such as 60 miles per 1 gallon.
Student HeCp
1 ^ -
►Writing Algebra
A ratio, such as |, can
be written as 5 to 3 or
5:3.
^ j
l| Find a Ratio
The tennis team won 10 of its 16 matches. Find the ratio of wins to losses.
matches won 10 matches 5
Solution
Ratio =
matches lost 6 matches
ANSWER ^ The win-loss ratio is —, which is read as “five to three.”
2 Find a Unit Rate
You run a 10 kilometer race in 50 minutes. What is your average speed in
kilometers per minute?
Solution Rate = IP km = 1 2 3 = 0.2 km/min
50 mm 5 mm
ANSWER ^ Your average speed is 0.2 kilometers per minute.
1. Your school football team won 8 out of 15 games, with no tie games.
What was the team’s ratio of wins to losses?
Find the unit rate.
2 . A plane flies 1200 miles in 4 hours.
3. You earn $45 for mowing 3 lawns.
3.8 Ratios and Rates
H
Student HeCp
► Study Tip
A ratio compares two
quantities measured in
the same unit. The
ratio itself has no
units. A rate compares
two quantities that
have different units,
v _ )
3 Find a Rate
You kept a record of the number of miles you drove in your truck and the
amount of gasoline used for 2 months.
Number of miles
290
242
196
237
184
Number of gallons
12.1
9.8
8.2
9.5
7.8
What was the average mileage for a gallon of gasoline? Round your result in
miles per gallon (mi/gal) to the nearest tenth.
Solution Average mileage is a rate that compares miles driven to the
amount of gasoline used. To find the average mileage, divide the number of
miles driven by the number of gallons of gasoline used.
Rate =
290 + 242 + 196 + 237 + 184 1149 mi
12.1 + 9.8 + 8.2 + 9.5 + 7.8 47.4 gal
ANSWER ► The average mileage was about 24.2 miles per gallon.
24.2 mi/gal
UNIT ANALYSIS Writing the units when comparing each quantity of a rate is
called unit analysis. You can multiply and divide units just like you multiply and
divide numbers. When solving a rate problem, you can use unit analysis to help
determine the units for the rate.
|3Z!mZ 2I9 4 Use Unit Analysis
Use unit analysis to convert the units,
a. 3 hours to minutes b. 72 inches to feet
Solution
a. Use the fact that 60 minutes = 1 hour. So, equals 1.
0 ^ 60 minutes 1
3 houfs • —=180 minutes
ANSWER ► 3 hours equals 180 minutes.
b. Use the fact that 1 foot = 12 inches. So, inches e ^ ua ^ s ^
72 metres • * = 6 feet
^ 12 inches
ANSWER ^ 72 inches equals 6 feet.
Use Unit Analysis
4. Use unit analysis to convert 8 pounds to ounces. (1 pound = 16 ounces.)
5, Use unit analysis to convert 84 days to weeks.
Chapter 3 Solving Linear Equations
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p More Examples
M°r e examples
are available at
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GZHIZ9 5 Use a Rate
Use the average mileage you found in Example 3 to estimate the number of
miles you can drive on a full 18 gallon tank of gasoline. Round your answer to
the nearest mile.
Solution
In Example 3 you found the average mileage to be about 24.2 miles per gallon.
Multiply this rate by 18 gallons to estimate the distance you can drive.
mi
distance = ( 24.2 J(18 gal)
Substitute rate and gallons.
= (24.2 mi)(18) Use unit analysis.
= 435.6 mi Multiply.
ANSWER ► You can drive about 436 miles on an 18 gallon tank.
Use a Rate
6 . A car uses fuel at a rate of 19 miles per gallon. Estimate how many miles the
car can travel on 13 gallons of fuel.
Link to
Currency
iflOa oc m t\c
II \
PESOS The peso is the basic
unit of money in Mexico.
Currency exchange rates vary
according to economic
conditions.
6 Apply Unit Analysis
PESOS You are visiting Mexico and want to exchange $150 for pesos. The
rate of currency exchange is 9.242 Mexican pesos per United States dollar on
the day you exchange the money. How many pesos will you receive? Round to
the nearest whole number.
Solution
You can use unit analysis to write an equation to convert dollars into pesos.
Use the fact that 9.242 pesos = 1 dollar. So, P esos _ j
r 1 dollar
P = 150 dollars Write equation.
P = 150 (9.242 pesos) Use unit analysis.
P = 1386.3 pesos Multiply.
ANSWER ► You will receive 1386 pesos.
Apply Unit Analysis
I. You are visiting Canada and you want to exchange $140 in United States
dollars for Canadian dollars. The rate of currency exchange is 1.466
Canadian dollars per United States dollar. How many Canadian dollars will
you get? Round to the nearest whole number.
3.8 Ratios and Rates
■Eia Exercises
Guided Practice
Vocabulary Check Complete the sentence.
1. If a and b are two quantities measured in the same unit, then the ? of
a to bis —
b
2 . A rate compares two quantities measured in ? units.
3. A unit rate is a rate per ? given unit.
4. You can use ? to change from one unit of measure to another.
Skill Check Write the ratio in simplest form.
5. || 6. || 7. 14 to 21 8.77 to 55
Find the unit rate. Round your answer to the nearest hundredth.
9- Swim 2 miles in 40 minutes 10- Pay $1.50 for 24 tea bags
11 . MILEAGE The average mileage for your old truck is 10.5 miles per gallon.
Estimate the number of miles you can travel on a full 22 gallon tank of
diesel fuel.
Practice and Applications
SIMPLIFYING RATIOS Write the ratio in simplest form.
12- 5 to 10 13- 30 to 120 14- 8 to 136 15- 60 to 100
16-
6
8
17.
66
18
18.
11
20
19.
28
35
20. FOOTBALL During a football game, a quarterback throws 30 passes and
completes 15 of them. What is the ratio of passes completed to passes
thrown?
Student HeCp
^ .
► Homework Help
Example 1: Exs. 20, 21
Example 2: Exs. 22-27
Example 3: Exs. 28, 38
Example 4: Exs. 29-37,
39, 40
Example 5: Ex. 41
Example 6: Exs. 42, 43
^ j
21. DENTISTRY Humans produce a set of 20 teeth during early jaw
development. A second set of 32 permanent teeth replaces the first set of
teeth as the jaw matures. What is the ratio of first teeth to permanent teeth?
UNIT RATE Find the unit rate.
22. Earn $126 for working 18 hours 23. Hike 45 miles in 3 days
24. $3 for 5 containers of yogurt 25. $2 for 5 cans of dog food
26. 440 grams of cereal in 8 servings 27. 20 ounces in 2.5 servings
Chapter 3 Solving Linear Equations
28. BOOK CLUB You belong to a book
club at the library. You keep a list
of how many books you read each
month. Find the average number of
books you read per month from
September through December.
Month
Books
Sept.
1
Oct.
3
Nov.
4
Dec.
3
DETERMINING UNITS Write the appropriate unit.
29. 5 ° miles • 2 hours = 100 ? 30. 108 inches • . ' fo( f = 9 ?
1 hour 12 inches
31. UNIT ANALYSIS Choose the expression that completes the following
equation: 720 seconds • ? = 12 minutes.
A.
1 minute
60 seconds
60 seconds
1 minute
CONVERTING UNITS In Exercises
result to the nearest tenth.
32. 60 eggs to dozens of eggs
34. 168 days to weeks
36. 100 yards to feet
(1 yard = 3 feet)
32-37, convert the units. Round the
33. 2 years to months
35. 1270 minutes to hours
37. 2000 meters to kilometers
(1 kilometer = 1000 meters)
38. AVERAGE SPEED You ride a stationary bike at the gym. After your last five
visits you wrote down how long you rode the bike and how many miles you
pedaled. What was your average speed in miles per minute?
Number of miles
9
10
12
15
18
Number of minutes
30
30
35
45
45
39. UNIT ANALYSIS Use unit analysis to write your answer to Exercise 38 in
miles per hour.
BALD EAGLES In Exercises 40 and 41, use the following information
from page 129. A bald eagle can fly at a rate of 30 miles per hour.
40. Use unit analysis to find a bald eagle’s flying rate in miles per minute.
41. Use the result of Exercise 40 to find how many minutes it would take a bald
eagle to fly 6 miles.
Student HeCp
► Homework Help
Extra help with
p ro b| e m solving in
Exs. 42-43 is available at
www.mcdougallittell.com
EXCHANGE RATE In Exercises 42 and 43, use 9.242 pesos per dollar as
the rate of currency exchange. You are visiting Mexico and have taken
$325 United States dollars to spend on your trip. Round to the nearest
whole number.
42. If you exchange the entire amount, how many pesos will you receive?
43. You have 840 pesos left after your trip. How many dollars will you get back?
3.8 Ratios and Rates
Standardized Test
Practice
Mixed Review
Maintaining Skids
44. MULTIPLE CHOICE There are a total of 28 marbles in a bag. Six of the
marbles are red and the rest are blue. What is the ratio of red marbles to blue
marbles?
2 6
® 11 ® 28
45. IV1ULTIPLE CHOICE If you drive a car m miles in 2 hours, which expression
will give the average speed of the car?
©m + 2 C© 2m (H) CD f
46. (MULTIPLE CHOICE You travel 154 miles on half a tank of fuel. Your car
gets 22 miles per gallon. How many gallons of fuel can your tank hold?
(A) 7 CD 14 CD 22 (© 132
47. (MULTIPLE CHOICE You want to exchange $60 Canadian dollars into United
States dollars. The exchange rate is 1.466 Canadian dollars per United States
dollar on the day you exchange the money. How many United States dollars
will you get?
CD $41 ® $46 (H) $131 CD $221
48. POPULATION PROJECTIONS The table shows the projected number (in
millions) of people 85 years and older in the United States for different
years. Make a line graph of the data. (Lesson 1.7)
Year
2000
2010
2020
2030
2040
2050
Number of people 85 years
and older (in millions)
4.1
5.0
5.0
5.8
8.3
9.6
DATA UPDATE of U.S. Bureau of the Census data at www.mcdougallittell.com
COMPARING INTEGERS Graph the numbers on a number line. Then
write two inequalities that compare the numbers. (Lesson 2.1)
49. 4,-3 50.-5,-2 51. -6, 3
SOLVING AND CHECKING Solve the equation. Round the result to the
nearest hundredth. Check the rounded solution. (Lesson 3.6)
52. —la - 9 = 6 53. 10 - 3x = 4x 54. 5x + 14 = -x
55. SOCCER FIELD What is the width of a rectangular soccer field that has an
area of 9000 square feet and a length of 120 feet? (Lesson 3.7)
LEAST COMMON DENOMINATOR Find the least common denominator of
the pair of fractions. (Skills Review p. 762)
56.
60.
3 2
4’ 5
_L A
16’ 20
57
61
2 _3_
9’ 18
14 3J_
54’ 81
58.
62.
5 _ 8 _
6’ 30
A 11
64’ 24
59.
63.
A A
49’ 70
Chapter 3 Solving Linear Equations
Percents
Goal
Solve percent problems.
Key Words
• percent
• base number
What is the discount percent?
There are three basic types
of percent problems. In
Example 4 you will solve
one type to find the discount
percent on a sale item.
A percent is a ratio that compares a number to 100. You can write a percent
as a fraction, as a decimal, or as a number followed by a percent symbol %.
40
For example, you can write forty percent as yyy, 0.40, or 40%.
You can use a verbal model to help you write a percent equation.
Verbal
Model
Labels
Algebraic
Model
Number being = ent . base
compared to base p number
Number compared to base = a (same units as b)
P
Percent — p% — - (no units)
Base number = b (assigned units)
• b
The base number is the number that is being compared to in any percent equation.
1 Number Compared to Base is Unknown
What is 30% of 70 feet?
Verbal a —
Model
★
Labels Number compared to base = a (feet)
30
Percent = 30% = yyy = 0.30 (no units)
Base number = 70 (feet)
Algebraic a = (0.30)(70) = 21
Model
ANSWER ► 21 feet is 30% of 70 feet.
p percent] • \ b]
3.9 Percents
Student HeCp
► Study Tip
When you solve a
percent equation, first
convert the percent to
a decimal or a fraction.
^ _ )
2 Base Number is Unknown
Fourteen dollars is 25% of what amount of money?
Verbal
\a\ = \p percent
• b
Model
Labels
Number compared to base = 14
(dollars)
Percent = 25% :
25 _ 1
100
(no units)
Base number =
b
(dollars)
Algebraic
Model
II
|H*
•
4(14) = 4 (f)
56 = b
ANSWER ► $14 is 25% of $56.
3 Percent is Unknown
One hundred thirty-five is what percent of 27?
Verbal
Model
[a] = p percent • [ftj
Labels
Number compared to base = 135
(no units)
P
Percent = p% = —
(no units)
Base number = 27
(no units)
P
Algebraic
135 = —(27)
Model
135 = P
27 100
500 = p
ANSWER ^ 135 is 500% of 27.
Solve a Percent Equation
1. What is 15% of 100 meters? 2 . 12 is 60% of what number?
3. 8 is what percent of 20? 4. 20 is what percent of 8?
■
Chapter 3 Solving Linear Equations
DISCOUNT When an item is on sale, the difference between the regular price
and the sale price is called the discount. To find the discount percent, use the
regular price as the base number in the percent equation.
Student HeCp
^ More Examples
More examples
are ava j| a i)| e at
www.mcdougallittell.com
Model and Use Percents
DISCOUNT PERCENT You are shopping for a portable CD player. You find
one that has a regular price of $90. The store is selling the CD player at a sale
price of $72. What is the discount percent?
Verbal
Model
Labels
Discount
p percent • Regular price
Discount = Regular price — Sale price
= 90 - 72 = 18
P
Percent = p% =-
Regular price = 90
(dollars)
(no units)
(dollars)
Algebraic
Model
18 = —(90)
18 =
90
0.20
P
100
P
' 100
20 = p
ANSWER ^ The discount percent is 20%.
Model and Use Percents
5. A radio has a regular price of $80. You receive a 10% discount when you
purchase it. Find the amount of the discount. Then find the sale price of the
radio with the 10% discount.
Three Types of Percent Problems
QUESTION
GIVEN
NEED TO FIND
EXAMPLE
What is p percent of b?
b and p
Number compared to base, a
Example 1
a is p percent of what?
a and p
Base number, b
Example 2
a is what percent of b?
a and b
Percent, p
Example 3
3.9 Percents
Exercises
Guided Practice
Vocabulary Check In Exercises 1 and 2, consider the statement "10% of 160 is 16."
1. Write an equation that represents the statement.
2 . What is the base number in the equation you wrote in Exercise 1?
Write an equation for each question. Do not solve the equation.
3- 15% of what number is 12? 4. 99 is what percent of 212?
5- What is 6% of 27? 6-13 is 45% of what number?
Skill Check Solve the percent problem.
7. 35 is what percent of 20?
9. 18 is 25% of what number?
8- 12% of 5 is what number?
10- 24 is 120% of what number?
SALES TAX The price of a book without tax is $10. The sales tax rate on
the price of the book is 6%.
P
11 - Model the situation with an equation of the form a =
12- Solve the equation to find the amount of the tax.
Practice and Applications
UNDERSTANDING PERCENT EQUATIONS Match the percent problem
with the equation that represents it.
13- a = (0.39)(50) A- 39 is 50% of what number?
14- 39 = p(50) B- 39% of 50 is what number?
15- 39 = 0.50 b C- $39 is what percent of $50?
SOLVE FOR a Solve the percent problem.
16- How much money is 35% of $750? 17. What number is 25% of 80?
18. What distance is 24% of 710 miles? 19. 14% of 220 feet is what length?
Student HeCp
^ -‘X
► Homework Help
Example 1: Exs. 16-21
Example 2: Exs. 22-27
Example 3: Exs. 28-33
Example 4: Ex. 35
^ J
20. How much is 8% of 800 tons? 21 . What number is 200% of 5?
SOLVE FOR b Solve the percent problem.
22. 52 is 12.5% of what number? 23. 42 feet is 50% of what length?
24. 45% of what distance is 135 miles? 25. 2% of what amount is $20?
26. 30 grams is 20% of what weight? 27. 90 is 45% of what number?
Chapter 3 Solving Linear Equations
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 28-33 is available at
www.mcdougallittell.com
SOLVE FOR p Solve the percent problem.
28, 3 inches is what percent of 40 inches? 29. $240 is what percent of $50?
30. 55 years is what percent of 20 years? 31. 18 is what percent of 60?
32. 9 people is what percent of 60 people? 33. 80 is what percent of 400?
34. 1 ^^ What percent _ 60_
of the region is shaded blue?
20
What percent is shaded yellow?
40
20
r
20
20
_L
60
35. THE BETTER BUY Store A has a coat on sale for 30% off the regular price
of $60. The same coat is on sale at Store B for 20% off the regular price of
$60. You also have a Store B coupon for 10% off the sale price. Will you
save money by going to Store B? Explain why or why not.
36. Hist ory Link x The table below shows the number of electoral votes each
candidate in the Election of 1860 received. What percent of the total number
of electoral votes did each candidate receive?
LinkJ&±
Careers
COLLEGE RECRUITERS
give tours of the campus,
arrange orientation seminars,
and visit high schools.
More about college
recruiters is available
at www.mcdougallittell.com
Party
Candidate
Electoral votes
Republican
Abraham Lincoln
180
Southern Democratic
J.C. Breckinridge
72
Constitutional Union
John Bell
39
Northern Democratic
Stephen Douglas
12
CHOOSING A COLLEGE In Exercises 37-39, use the graph. It shows the
responses of 3500 seniors from high schools around the United States.
Reason for Choosing a College
Other 280
Size 175
Academic-
Reputation 630
Cost 945
Availability
of Major 735
Location 735
► Source: Careers and Colleges
37. What percent of the seniors said location was the reason for their choice?
38. What percent of the seniors said academic reputation was the reason for their
choice?
39. What percent of the seniors said cost most influences their choice?
3.9 Percents
LOGICAL REASONING In Exercises 40 and 41, use a = ^b.
Standardized Test
Practice
Mixed Review
Maintaining Skills
Quiz 3
■
40. Complete the sentence: When the percent p is a number greater than 100, the
value of a is ? than the value of the base number b.
41 . Write a percent equation for the statement “a is 300 percent of hr Then
choose one set of values for a , b , and p that make the equation true.
42. (MULTIPLE CHOICE Choose the equation you would use to find 25% of 120.
(A) 0.25x = 20 CDjt = ^f| C©x = ^ CD x = (0.25X120)
43. MULTIPLE CHOICE What amount would you leave for a 20% tip on a $35
restaurant bill?
CD $-70 (D $2.80 CD $7 CD $28
WRITING AND SOLVING EQUATIONS Write the sentence as an equation.
Let x represent the number. Use mental math to solve the equation. Then
check your solution. (Lesson 1.5)
44. The sum of a number and 18 is 45.
45. The product of a number and 21 is 105.
FINDING ABSOLUTE VALUE Evaluate the expression. (Lesson 2.2)
46. | 9 | 47. |-32 | 48. — 1 5 | 49. — | — 16 |
ORDERING NUMBERS Write the numbers in order from least to greatest.
(Skills Review p. 770)
50. 1301, 1103, 1031, 1013, 1130 51. 217, 2017, 270, 2170, 2701
52. 23.5, 23.45, 23.4, 23.53, 23.25 53. 5.09, 5.9, 5.1, 5.19, 5.91
Solve the formula for the indicated variable. (Lesson 3.7)
1. Solve d — rt for t. 2. Solve A = \bh for h. 3. Solve d — — for v.
2 v
Use unit analysis to complete the equation. (Lesson 3.8)
4. 7 weeks • ? = 49 days 5. 108 inches • ? = 9 feet
_ 20 students 15 classrooms _ 24 hours . _ t
6. —-• —;—;—;— = ? 7. ——;-• 10 days = ?
1 classroom 1 school 1 day
DISCOUNT PERCENT In Exercises 8 and 9, the regular price of a shirt is
$23. You buy it on sale for $17.25. (Lesson 3.9 )
8. What is the amount of the discount?
9. Write and solve a percent equation to find the discount percent.
Chapter 3 Solving Linear Equations
fika|irir
O Chapter Summary
^ and Review
• equivalent equations, p. 132
• rounding error, p. 164
• unit rate, p. 177
\
• inverse operations, p. 133
• formula, p. 171
• unit analysis, p. 178
• linear equation, p. 134
• ratio, p. 177
• percent, p. 183
• properties of equality, p. 140
• rate, p. 177
• base number, p. 183
• identity, p. 153
Solving Equations Using Addition and Subtraction
Examples on
pp. 133-134
Use inverse operations of addition and subtraction to isolate the variable.
Simplify first if necessary.
USE ADDITION
y —4 = -6
Write original
equation.
USE SUBTRACTION
x- (-2) = 12
x + 2 = 12
y — 4 + 4= —6 + 4 Add 4 to each side.
y = — 2 Simplify both sides.
Write original equation.
Simplify.
x + 2 — 2 — 12 — 2 Subtract 2 from each side.
x = 10
Simplify both sides.
Solve the equation. Check your solution in the original equation.
1- y ~ 15 = -4 2. 7 + x = -3 3. f - (-10) = 2
Solving Equations Using Multiplication and Division
Examples on
pp. 138-140
Use inverse operations of multiplication and division to isolate the variable.
USE MULTIPLICATION
1
Write original equation.
8( — m I = 8( — 5) Multiply each side by the
' ' reciprocal, 8.
m = — 40 Simplify.
USE DIVISION
—In — 28 Write original equation.
-In = _28_
-7 -7
n = —4
Divide each side by -7.
Simplify.
Solve the equation. Check your solution in the original equation.
4. 81 = 3 1 5. — 6x = 54
6 - J =-16
Chapter Summary and Review
Chapter Summary and Review continued
Solving Multi-Step Equations
You may need more than one step to solve an equation.
-2 p - (-5) - 2p = 1 3
Write original equation.
—2 p + 5 — 2p = 13
Use subtraction rule to simplify.
-4 p + 5 = 13
Combine like terms -2 p and -2 p.
—4p + 5 — 5 = 13 — 5
Subtract 5 from each side to undo the addition.
00
II
1
Simplify both sides.
II
l 00
Divide each side by -4 to undo the multiplication.
p= -2
Simplify.
Solve the equation.
7. 26 + 9x = -1
8. -32 = 4c - 12 9. 9r - 2 — 6r = 1
10. -2(4 - x) - 7 = 5
11. n + 3(1 + 2n) = 17 12. |(x + 8) = 9
Solving Equations with Variables on Both Sides
Examples on
pp. 151-153
Linear equations can have one solution , no solution , or many
solutions. To solve, collect the variable terms on one side of the equation.
Equation with one solution:
Equation with no solution:
Equation with many solutions:
15 d + 20 = Id - 4
1
&
1
U\
II
1
U\
1
&
2n — 5n +11 = 2 — 3/2 + 9
8d = -24
—6x — 5 + 6x = —15 — 6x + 6x
— 3/i +11 = 11— 3/7
d= -3
-5 ^ -15
11 = 11
The solution is —3.
The original equation
has one solution.
—5 = —15 is never true no
matter what the value of x. The
original equation has no solution.
11 = 11 is always true, so
all values of n are solutions.
The original equation is
an identity.
Solve the equation if possible. Determine whether the equation has
one solution, no solution, or is an identity.
13. 24 — 3x = 9x 14. 15x — 23 = 15x 15. 2m — 9 = 6 — m
16. 26 — Ad = 4(9 — d) 17. 12 + \\h = -18 - Ah 18. 2x + 18 + 4x = -2x + 10
Chapter 3 Solving Linear Equations
Chapter Summary and Review continued^
More on Linear Equations
Examples on
pp. 157-159
You can use a verbal model to write and solve linear equations.
FUNDRAISER You are making sandwiches to sell at a fundraiser. It costs $.90
to make each sandwich plus a one-time cost of $24 for packaging. You plan to
sell each sandwich for $2.50. Write and solve an equation to find how many
sandwiches you need to sell to break even.
Verbal
Cost to make
Number of
_L
Cost of
Price of
Number of
Model
sandwich
sandwiches
packaging
sandwich
sandwiches
Labels Cost to make sandwich = 0.90 (dollars)
Cost of packaging = 24 (dollars)
Number of sandwiches = x
Price of sandwich = 2.50 (dollars)
▼
Algebraic 0.90 • x + 24 = 2.50 • x
Model
24 = 1.6x
15 = x
ANSWER ► You need to sell 15 sandwiches to break even, or recover your costs.
19. TOMATOES One tomato plant is 12 inches tall and grows 1 inch per week.
Another tomato plant is 6 inches tall and grows 2 inches per week. Write and
solve an equation to find when the plants will be the same height.
Solving Decimal Equations
Examples on
pp. 163-165
3.45m =
6.38m =
m ~
m ~
For some equations, only an approximate solution is required.
— 2.93m — 2.95 Write original equation.
— 2.95 Add 2.93m to each side.
—0.462382445 Divide each side by 6.38.
Use a calculator to get an approximate solution.
—0.46 Round to nearest hundredth.
Solve the equation. Round the result to the nearest hundredth.
20 . 3x — 4 = 3 21 . 5x — 9 = 18* - 23 22 . 13.7 1 - 4.7 = 9.9 + 8.1*
Chapter Summary and Review
Chapter Summary and Review continued
3.7 Formulas
Examples on
pp. 171-173
You can solve a formula for any one of its variables.
The formula for the area of a rectangle is A = Hw. Find a formula for width w in
terms of area A and length i.
A
A
lw
= w
Write original formula.
To isolate w, divide each side by L
Solve the formula for the indicated variable.
23- Solve for l : V = JLwh. 24- Solve for m: d = —. 25- Solve for b: P — a + b + c.
v
3.2-3A Ratios, Rates, and Percents
Examples on
pp. 177-179, 183-185
Ratios, rates, and percents can be used to compare
real-life quantities.
a. A football team has a record of 7 wins and 3 losses. What percent of the
games did the team win?
The team won 7 out of 10 games. Use [«]=/? percent • [&].
7 = —(10)
7
P_
10
70 = p
ANSWER ► The team won 70% of its games.
b. The football team has a total of 900 rushing yards this season. Find the team’s
average rushing yards per game.
900 rushing yards 90 rushing yards
Rate =- 77 ^-=- 7 -
10 games 1 game
ANSWER ► The team’s average is 90 rushing yards per game.
CAR MILEAGE At 60 miles per hour, a car travels 340 miles on 20 gallons
of gasoline.
26- What is the average mileage per gallon of gasoline?
27- How many miles could this car travel on 5 gallons of gasoline at the same speed?
28- What percent of the 20 gallons is 5 gallons?
Chapter 3 Solving Linear Equations
Solve the equation. Check your solution in the original equation.
1.x + 3 = 8 2. 19 = 0-4 3. —2y = —18 4. 22 = 3p - 5
5 . r — (—7) = 14 6 .|=— 6 7.|(9 + w) = -10 8 . -|x - 2 = -8
Solve the equation if possible. Determine whether the equation has
one solution , no solution , or is an identity.
9. 14 — 5 1 = 3 1 10. 6x — 9 = 10* + 7 11. — 3(*
12. |^(9* + 2) = 15* 13. 24y — (5y + 6 ) = 21y + 3 14. —5r -
Solve the equation. Round the result to the nearest hundredth.
15. 26 + 9 p = 58 p 16. -34 = 8 * - 15 17. 15jc — 18 = 37
18.13.2* + 4.3 = 2.1k 19. 42.6* - 29.4 = -3.5* 20. 3.82 + 1.25* = 5.91
Solve the formula for the indicated variable.
21 .A = £w 22.jC + 32 = F
Solve for £. Solve for C.
Convert the units.
24. 98 days to weeks 25. 37 hours to minutes 26. 15 yards to feet
Solve the percent problem.
27. What number is 30% of 650? 28. 15% of what amount is $36?
29.4 is what percent of 20? 30. How much is 45% of 200 pounds?
31 . SHOVELING SNOW You shovel snow to earn extra money and charge
$12 per driveway. You earn $72 in one day. Let * represent the number of
driveways you shoveled. Which of the following equations is an algebraic
model for the situation?
A. 72x = 12 B. *Lc = 72 C. 12x = 72 D. ~x = 12
32. SUMMER JOB At your summer job you earn $8 per day, plus $3 for each
errand you run. Write and solve an equation to find how many errands you
need to run to earn $26 in one day.
33. EXCHANGE RATE You are visiting Canada and want to exchange $175 in
United States dollars for Canadian dollars. The rate of currency exchange is
1.466 Canadian dollars per United States dollar. How many Canadian dollars
will you get? Round to the nearest whole number.
23. A = jbh
Solve for h.
— 2) = 6 — 3*
- 6 + 4r = —r+2
Chapter Test
Chapter Standardized Test
Tip
Start to work as soon as the testing time begins.
Keep working and stay focused on the test.
1 _ Which number is a solution of
4 — x = —5?
CD -9
CD -l
CD 9
2. Which step can you use to solve the
3
equation ^x = 12?
3
(A) Divide each side by
Cb ) Divide each side by |\
3
Cc) Multiply each side by
(D) Multiply each side by
3- The perimeter of the rectangle is
40 centimeters. Find the value of x.
x
3x
(D 4
CD 5
CD 8
CD 10
If 9x — 4(3x —
2) = 4, then x
(D -4
CD-f
cd|
CD 2
5. Solve the equation |-(27x + 18) = 12.
CD -f
CDf
CD 2
CDf
Chapter 3 Solving Linear Equations
6 . How many solutions does the equation
—2y + 3(4 — y) = 12 — 5y have?
(A) none CD one
CD two Q5) more than two
7. If 0.75? = 12, then t = ?.
CD 3 CD 9
C© 16 CD 36
8 . What is the value of y if
13.6y - 14.8 = 4.1y - 6.3?
CA) -2.2 CD -1.2
CD 0.5 CD 0.9
9. Use the temperature conversion formula
F = + 32 to convert 10° Celsius to
degrees Fahrenheit.
® — 12°F CD 38°F
CD 42°F CD 50°F
CD None of these
10. You can stuff 108 envelopes in 45 minutes.
At this rate, how many envelopes can you
stuff in 2 hours?
CA) 50 CD 144
CD 216 CD 288
11. What is 26% of 250 meters?
Ca) 9.6 meters CD 65 meters
CD 185 meters CD 961.5 meters
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
J 1 Make a Line Graph
Make a line graph of the following average monthly temperatures:
January, 27°F; February, 34°F; March, 41°F; and April, 53°F.
Solution Draw the vertical scale
from 0°F to 60°F. Mark the months
on the horizontal axis. Label the axes.
Draw a point on the grid for each data
point. Connect the points with lines
as shown at the right.
Try These
1. The table gives the wind speed and the wind chill temperature when the
outside temperature is 40°F. Make a line graph of the data.
Wind speed (mi/h)
25
30
35
40
45
Wind chill temperature (°F)
15
13
11
10
9
2 Evaluate a Function
A campsite charges $85 for two people to rent a cabin and $10 for each
additional person. The total cost is given by C = 85 + 10 n, where n is the
number of additional people. Make an input-output table for the cost when
there is a total of 2, 4, or 6 people.
Student HeCp
► Extra Examples
More examples
anc j p rac tj ce
exercises are available at
www.mcdougallittell.com
Solution
INPUT
FUNCTION
OUTPUT
2 people
n — 0
C = 85 + 10(0)
C = $85
4 people
n — 2
C = 85 + 10(2)
C = $105
6 people
n — 4
C = 85 + 10(4)
C = $125
Input n
0
2
4
Output C
$85
$105
$125
Try These
2 . The green fee for up to four people to golf 18 holes is $100. The cost of
renting clubs is $9 per person. The total cost is given by C = 100 + 9r,
where r is the number of people who rent clubs. Make an input-output
table for 0, 2, and 4 people renting clubs.
Maintaining Skills
Evaluate the expression for the given value of the variable.
(Lessons 1.1,1.2,1.3)
1. 20 — 4 y when y = 3 2. ^ + 12 when x = 8 3- x 2 — 8 when x — 1
4. (6 + x) + 3x when x = 6 5. (30 3 when t — 2
6 . 8x 2 when x = 4
Evaluate the expression. (Lessons 1.3, 2.3,2.4)
7. 9 4- 3 + 2
8. -5 + 3 • 8 - 6
10.20 - (-3) - 8
11 . 2 • 35 + (-13)
12 . [(6 • 4) + 5] - 7
Check to see if the given value of the variable is or is not a solution of
the equation or inequality. (Lesson 1.4)
13.4 + 2x = 12;x = 2 14. 6x - 5 = 13;x = 3 15. 3y + 7 = 31; y = 8
16.x - 4 < 6;x = 9
17. 5m + 3 > 8; m = 1
18. 9 <22 - 4x;x = 3
In Exercises 19-22, write the phrase or sentence as a variable expression,
equation, or inequality. (Lesson 1.5)
19. A number x cubed minus eight
20. Four less than twice a number x is equal to ten.
21. The product of negative three and a number x is less than twelve.
22 . A number x plus fifteen is greater than or equal to thirty.
23. Draw a line graph to represent the data given by the input-output table.
(Lesson 1.8)
Input x
2
4
6
8
10
12
Output y
1
5
9
13
17
21
Complete the statement using < or >. (Lesson 2.1)
28. -2.1 ? 1.2 29.-109 P -101 30. 2 0-3
31. -6 ? 9
Simplify the expression. (Lessons 2.5, 2.6,2.7)
32. —4(x)(6) 33. 5(—y) 3
34. 8(—3)(—jc)(— x) 35. -(4 - 2 1)
36. ~2(x + 3) - 1 37. (6x - 9)|
38. 3 + 6(x — 4) 39. 5(9x + 5) - 2x
Chapter 3 Solving Linear Equations
FLYING SQUIRRELS In Exercises 40 and 41, use the following information.
A flying squirrel drops from a tree with a downward velocity of -6 feet
per second. (Lesson 2.5)
40. Write an algebraic model for the displacement of the squirrel (in feet) after
t seconds.
41 . Find the squirrel’s change in position after 5 seconds. Is your answer a
positive or negative number? Explain.
In Exercises 42 and 43, use the area model shown below. (Lesson 2.6)
42. Find two expressions for the area of the large rectangle.
43. Write an algebraic statement that shows that the two
expressions are equal.
15
Solve the equation. (Lessons 3.1-3.5)
+
x
44.x + 11 = 19
45.x - (-7) = -2
46. 9b = 135
47. 35 = 3c - 19
4S.f-9=-l
49. 4(2x - 9)
= 6(10x — 6)
50. 3 (q - 12) = 5q + 2
51.|(2x + 5) = 6
52. 9(2 p + 1)
1
II
CO
1
53. FUNDRAISER Your school band is planning to attend a competition. The
total cost for the fifty band members to attend is $750. Each band member
will pay $3 toward this cost and the rest of the money will be raised by
selling wrapping paper. For each roll of wrapping paper sold, the band makes
$2. Write and solve an equation to find how many rolls the band members
need to sell to cover the cost. (Lesson 3.3)
Solve the equation. Round the result to the nearest hundredth. (Lesson 3.6)
54. 8x - 5 = 24 55. 70 = 9 — 3x 56. -3.46y = -5.78
57.4.17ft + 3.29 = 2.74 n 58. 2.4(0.3 + jc) = 8.7 59. 23.5a + 12.5 = 9.3a - 4.8
In Exercises 60 and 61, use the formula for the area of a triangle,
A = £ bh . (Lesson 3.7)
60. Find a formula for h in terms of A and b.
61. Use the new formula to find the height of a triangle
that has an area of 120 square centimeters and a base
of 24 centimeters.
62. CAR TRIP You start a trip at 9:00 A.M. and the car’s odometer reading is
66,300 miles. When you stop driving at 3:00 P.M., the odometer reading is
66,660 miles. What was your average speed? (Lesson 3.8)
63. SALE PRICE You buy a sweater that is on sale for 30% off the regular price
of $65. How much did you pay for the sweater? (Lesson 3.9)
Cumulative Practice
Materials
• graphing calculator or
computer
Planning
OBJECTIVE Compare the income and expenses of a car wash to
determine profit.
Investigating the Data
The booster club is planning a car wash to raise funds for the football
team. Use the information below to answer Exercises 1-8.
Booster Cl ub Car Wash
Income
• Wash only
• Wash and vacuum
Expenses
• 20 sponges
• 3 bottles of detergent
• 3 bottles of window cleaner
• 20 rolls of paper towels
• 6 poster boards
$4 per car
$6 per car
$1.47 each
$2.44 for a 24-ounce bottle
$1.44 for a 32-ounce bottle
$:44 for an 30-sheet roll
$.74 each
1. The club members plan to bring buckets, hoses, and towels from home.
Name another important item the group will need to bring from home.
2 . Find the total car wash expenses.
3. Copy and complete the table. You may want to use a spreadsheet.
Number of cars washed
20
40
60
80
100
Income earned
?
?
?
?
?
4. Copy and complete the table. You may want to use a spreadsheet.
Number of cars washed
and vacuumed
20
40
60
80
100
Income earned
?
?
?
?
?
5. How much income will be earned if 60 cars are washed and 40 cars are
washed and vacuumed?
6_ The profit earned from the car wash is the income minus the expenses. The
profit earned is a function of the number of customers. How much profit will
the club make if 60 cars are washed and 40 cars are washed and vacuumed?
Chapter 3 Solving Linear Equations
7. Complete the equation for the total income I earned when x cars are washed
and y cars are washed and vacuumed. I = ? x + ? y
8 . Complete the equation for the total profit P earned when x cars are washed
and y cars are washed and vacuumed. P = ?x+ ? y - ?
Presenting Your results
Write a report about the car wash fundraiser.
• Include a discussion of income, expenses,
and profit.
• Include your answers to Exercises 1-8.
• Find how many cars the group
would need to wash to break even
(when income equals expenses).
• Find how many cars the group would
need to wash and vacuum to break even.
• Suppose the booster club wants to earn at least $200 profit. Find three
different combinations of car washes x, and car washes and vacuums y,
so that the club would meet its goal.
• Find what local car washes charge for similar services.
Extending the Project
Think of a different fundraising event that a club could use to earn money.
1. Describe the fundraiser.
2 . Do some research to find out what kind of supplies or equipment you would
need to get started and the cost of these items.
3- Survey some of your friends and neighbors to find what price they would be
willing to pay for the product or service.
4. Decide what you would charge customers for your merchandise or service.
5. Write an equation for the total income I earned as a function of the number
of customers n.
6. Write an equation for the total profit P earned as a function of the number of
customers n.
7. Determine the number of customers n needed to break even.
8 . Choose a profit goal. How many customers do you need to reach that goal?
Project
jTi
How steep are the hills of
San Francisco?
* /, ■■ ■; * • 'i ■ ** i S b jIi ■
"> v v V‘V' */■'• <•>-.■« %
* - . . »* 1 . a. - - ♦ V, ■ * * ji ,■
" ‘--v. 5c*AWfpnj,x^:w®<: wv<>-: . : i ,
• o*;.*, ... , • , ’ « • • * . . ■ p
1
\
APPLICATION: Cable Cars
In the lS70s, Andrew Hallidie designed the first
cable car system in the United States to make it easier
to climb the steep hills of San Francisco.
To design a transportation system, he needed a
mathematical way to describe and measure the
steepness of a hill.
Think & Discuss
1. Name another real-life situation where steepness is
important. When is steepness helpful? When is
steepness a problem?
2. How would you describe the steepness of the
sections of the street below?
350 ft
Clay Street
(cross section of side view)
Learn More About It
You will calculate the steepness of the sections of this
street in Exercise 55 on p. 247.
application link More about cable cars is available at
www.mcdougallittell.com
m ■m"~»2'V“ "*
4 Study Guide
PREVIEW
What’s the chapter about ?
• Graphing linear equations
• Finding the slope of a line
• Determining if a graph represents a function
Key Words
- ^
• ordered pair, p. 203
• y-intercept, p. 222
• slope-intercept form,
• linear equation, p. 210
• slope, p. 229
p. 243
• x- intercept, p. 222
• direct variation, p. 236
• function notation,
p. 254
_ >
PREPARE
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to p. 49)
1 _ What is the domain of the following input-output table?
Input
-2
0
2
4
Output
3
5
7
9
CD 3, 5, 7, 9 CD -2, 0,2, 4
CD 3, 5, 7, 9, —2, 0, 2, 4 CD all real numbers
Skill Check (refer to pp. 4,10, 86)
x - y
2 . Evaluate —-— when v = — 3 and y = — 1.
CD -2 CD-I
3. Evaluate (2x) 2 when x = 5.
(A) 10 CD 20
(© i
CD 2
(© 25
CD 100
STUDY TIP
Make Vocabulary
Cards
Including a sketch will
help you remember
a definition.
y-intercept
JhejHnterc ept is the y-coordin ate c f the
where a line crosses they^axis _
point
Chapter 4 Graphing Linear Equations and Functions
The Coordinate Plane
Goal
Plot points in a
coordinate plane.
Key Words
• coordinate plane
• origin
• x-axis, y-axis
• ordered pair
• x-coordinate
• /-coordinate
• quadrant
• scatter plot
Student HeCp
► Study Tip
Note in general (x, /).
x-coordinate-1 t
/-coordinate-1
How are wing length and wing beat related?
In Exercises 33-35 you will use a
coordinate plane to picture the
relationship between the length
of a bird’s wing and the bird’s
wing-beat rate.
A coordinate plane is formed by two real
4
number lines that intersect at a right angle at
vertical _
the origin. The horizontal axis is the x-axis
or y- axis
2
horizontal
and the vertical axis is the/-axis.
1
or x-axis
1
\
Each point in a coordinate plane corresponds
-4 -2
0
1 2 3 4 x
to an ordered pair of real numbers. The first
/
w
number in an ordered pair is the x-coordinate
origin
—2
(3, -2)*
and the second number is the/-coordinate.
(0,0)
— 3
-4
In the graph at the right, the ordered pair
(3, —2) has an x-coordinate of 3 and a
/-coordinate of —2.
i) Identify Coordinates
Write the ordered pairs that correspond to points A , B , C, and D.
Solution
In the coordinate plane at the right, Point A
is 3 units to the right and 2 units down from
the origin. So, its x-coordinate is 3 and its
/-coordinate is —2. The ordered pair is (3, —2).
Point B has coordinates (—2, — 1).
Point C has coordinates (0, 2).
Point D has coordinates (—3, 4).
Identify Coordinates
1. Write the ordered pairs that correspond
to points A, B , C, and D.
3
O'
B
c
1
A
5
~
1
-1
]
[
X
D.
k
4.1 The Coordinate Plane
Student MeCp
► More Examples
More examples
~<!^S v are available at
www.mcdougallittell.com
2 Plot Points in a Coordinate Plane
Plot the point in a coordinate plane,
a. (3, 4) M-2,-3)
Solution
a. To plot the point (3, 4), start at
the origin. Move 3 units to the
right and 4 units up.
(3,4)
3
n
1
4
1
t
3 3
5 x
b. To plot the point (—2, —3), start
at the origin. Move 2 units to
the left and 3 units down.
\y
2_
5
3
-
1
-1
X
-3
(-
2,-
■3)
—3
The x-axis and the y-axis divide the coordinate
plane into four regions called quadrants. Each
point in a coordinate plane is located in one of the
four quadrants or on one of the axes.
You can tell which quadrant a point is in by
looking at the signs of its coordinates. In the
graph at the right, the point (4, 3) is in Quadrant I.
The point (0, —4) is on the y-axis and is not inside
any of the four quadrants.
4
3
Quadrant II 2
( ,+) x
—2 O
Quadrant III —2
w
(4,3)
•
Quadrant I
(+, +)
1 2 3 4 x
Quadrant IV
(+,-)
40,-4)
3 Identify Quadrants
Name the quadrant the point is in.
a. (-2, 3) b. (4, —2)
Solution
a- (—2, 3) is in Quadrant II because its x-coordinate is negative and its
y-coordinate is positive.
b- (4, —2) is in Quadrant IV because its x-coordinate is positive and its
y-coordinate is negative.
Plot Points and Identify Quadrants
Plot the point in a coordinate plane.
2. (-2,5) 3. (3,7) 4. (-1,-3) 5. (-2,
Name the quadrant the point is in.
6. (-5, -3) 7. (2,0) 8. (4, -1) 9. (-3,
0 )
6 )
Chapter 4 Graphing Linear Equations and Functions
USING A SCATTER PLOT Many real-life situations can be described in terms of
pairs of numbers. Medical charts record both the height and weight of a patient,
while weather reports may include both temperature and windspeed. One way to
analyze the relationships between two quantities is to graph the pairs of data on a
coordinate plane. Such a graph is called a scatter plot.
C23mZ 319 4 Make a Scatter Plot
SNOWMOBILE SALES The amount (in millions of dollars) spent in the United
States on snowmobiles is shown in the table. Make a scatter plot and explain
what it indicates. ►Source: National Sporting Goods Association
Year
1990
1991
1992
1993
1994
1995
1996
Spending
322
362
391
515
715
924
970
Solution
Because you want to see how spending changes over time, put time t on the
horizontal axis and spending s on the vertical axis. Let t be the number of years
since 1990. The scatter plot is shown below.
ANSWER ► From the scatter plot, you can see that the amount spent on
snowmobiles tends to increase as time increases.
Make a Scatter Plot
10, The age a (in years) of seven cars and the price p (in hundreds of dollars)
paid for the cars are recorded in the following table. Make a scatter plot and
explain what it indicates.
Age
4
5
3
5
6
4
7
Price
69
61
75
52
42
71
30
4.1 The Coordinate Plane
PJ Exercises
Guided Practice
Vocabulary Check In Exercises 1-3, complete the sentence.
1. Each point in a coordinate plane corresponds to an ? of real numbers.
2. In the ordered pair (2, 5), the ^-coordinate is ? .
3. The x-axis and the y-axis divide the coordinate plane into four ? .
Skill Check Plot and label the ordered pairs in a coordinate plane.
4. A(4, -1),£(5,0) 5.A(—2, -3),fl(-3, -2)
IDENTIFYING QUADRANTS Complete the statement with always,
sometimes , or never.
6. A point plotted in Quadrant IV ? has a positive y-value.
7. A point plotted in Quadrant IV ? has a positive x-value.
8. A point plotted on the x-axis ? has ^-coordinate 0.
9. A point with a positive x-coordinate is ? in Quadrant I, Quadrant IV, or
on the x-axis.
Practice and Applications
IDENTIFYING ORDERED PAIRS Write the ordered pairs that correspond
to the points labeled A, B, C, and D in the coordinate plane.
■ Student fteCp
^—■n
I p Homework Help
Example 1: Exs. 10-12
Example 2: Exs. 13-18
Example 3: Exs. 19-26
Example 4: Exs. 27-35
PLOTTING POINTS Plot and label the ordered pairs in a coordinate plane.
13. A(0, 3), B(-2, - 1), C(2, 0) 14. A(5, 2), 5(4, 3), C(-2, -4)
15. A(4, 1), 5(0, -3), C(3, 3) 16. A(0, 0), 5(2, -2), C(-2, 0)
17. A(—4, 1), 5(—1, 5), C(0, -4)
18. A(3, -5), 5(5, 3), C(—3, -1)
IDENTIFYING QUADRANTS Without plotting the point, tell whether it is
in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV.
19. (5,-3) 20. (-2,7) 21.(6,17) 22. (14,-5)
23. (-4, -2) 24. (3, 9) 25. (-5, -2) 26. (-5, 6)
Chapter 4 Graphing Linear Equations and Functions
CAR COMPARISONS In Exercises 27-32, use the scatter plots below to
compare the weight of a car to its length and to its gas mileage.
27. In the Weight vs. Length graph,
what are the units on the
horizontal axis? What are the
units on the vertical axis?
28. In the Weight vs. Length graph,
estimate the coordinates of the
point for a car that weighs
about 4000 pounds.
29. Which of the following is true?
A. Length tends to decrease as
weight increases.
B. Length is constant as
weight increases.
C. Length tends to increase as
weight increases.
D. Length is not related to weight.
Weight vs. Length
n.
L
210
£ 180
J 150
0
•
m
.*
1
ft 1
ft
m
V
0 2000
3000
Weight (lb)
4000 1/1/
REGULAR C
m
PREMIUM
m
Weight vs. Gas Mileage
G
28
24
20
(/>
eg
CD
1
ft
m
•
•1
*
1
2000 3000 4000 1/1/
Weight (lb)
30. In the Weight vs. Gas Mileage graph, in the
ordered pair (2010, 29), what is the value of W7
What is the value of G?
Link to
Careers
31. SNTERPRETING DATA In the Weight vs. Gas Mileage graph, how does a
car’s gas mileage tend to change as the weight of the car increases?
32. CRITICAL THINKING How would you expect the length of a car to affect its
gas mileage? Explain your reasoning.
B ioiogy Unify In Exercises 33-35, the table shows the wing length (in
millimeters) and the wing-beat rate (in beats per second) for five birds.
Bird
Flamingo
Shellduck
Velvet Scoter
Fulmar
Great Egret
Wing length
400
375
281
321
437
Wing-beat rate
2.4
3.0
4.3
3.6
2.1
BIOLOGISTS study animals
in natural or controlled
surroundings. Biologists who
study birds are called
ornithologists.
More about biologists
is available at
www.mcdougallittell.com
33. Make a scatter plot that shows the wing-beat rates and wing lengths for the
five birds. Use the horizontal axis to represent the wing-beat rate.
34. What is the slowest wing-beat rate shown on the scatter plot? What is the
fastest? Where are these located on your scatter plot?
35. INTERPRETING DATA Describe the relationship between the wing length
and the wing-beat rate for the five birds.
4.1 The Coordinate Plane
Standardized Test
Practice
Mixed Review
Maintaining Skills
36 . iVIULTIPLE CHOICE Which ordered pair has an x-coordinate of —7?
(K>(3,-7) CD (-7, 3) CD (7, 3) CD (3, 7)
37 . IVIULTIPLE CHOICE The point (—9, —8) is in which quadrant?
CD Quadrant I CD Quadrant II
CH) Quadrant III CD Quadrant IV
38 . MULTIPLE CHOICE Which ordered pair is in Quadrant IV?
CD (7, 12) CD (-4, 3) CD (-5,-2) CD (8,-7)
39 . MULTIPLE CHOICE The vertical axis is also called the ? .
CD x-axis CD y -axis
CFO coordinate plane GD origin
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lessons 1.1, 2.4, and 2.5)
40. 3x + 9 when x = 2 41 . 13 — (y + 2) when y = 4
42. 4.2 1 + 17.9 when t — 3 43. — x — y when x — —2 and y = — 1
USING EXPONENTS Evaluate the expression. (Lessons 1.2, 1.3)
44. x 2 — 3 when x — 4 45. 12 + y 3 when y = 3
46. x 5 + 10 when x = 1.5 47. - + f when a = 2 and b — 3
a — b
ABSOLUTE VALUE Evaluate the expression. (Lesson 2.2)
48 . |-2.6 | 49 . 1 1.07 |
51 .
^2
3
SOLVING EQUATIONS Solve the equation. (Lesson 3.3)
52 . 3x - 6 = 0 53 . 6x + 5 = 35 54 . x + 1 = -3
55 . a — 3 = —2 56 . |x-l = -l 57 . |r + 3 = 4
SUBTRACTING FRACTIONS Subtract. Write the answer as a fraction or as
a mixed number in simplest form. (Skills Review p. 765)
4 1
58 . 7 | - 4j
«’■ 9 t? - 4
64 . 17f - 10§
2 1
59. 3j ~ if
9 2
62 . 8 ^- “ 5 ^-
17 2
65 . 12 ^- - 7 Yl
60 . 8f - l\
63 . 6 ^
66 . 18 ^
Chapter 4 Graphing Linear Equations and Functions
DEVELOPING CONCEPTS
For use with
Lesson 4.2
Goal
Discover the relationship
between ordered pairs that
are solutions to a linear
equation.
Materials
• ruler
• graph paper
• pencil
Question
What can you observe about the graph of the ordered
pairs that are solutions to a linear equation?
A linear equation in x and y is an equation that can be written in the form
Ax + By = C where A and B are not both zero. A solution of a linear equation is
an ordered pair (x, y) that makes the equation true. For example, (0, 3) is a
solution of the equation x + Ay = 12 because 0 + 4(3) = 12.
Explore
6^' *"
0 Show that (8, 1) is also a
solution to the equation
x + Ay =12. Plot the two
solutions, (0, 3) and (8, 1),
on a coordinate graph.
Draw a line through them.
0 Determine whether the following ordered pairs are also solutions of the
equation x + Ay = 12.
£>(-4,4) £(-1,2)
F( 2, 1) G(4, 2)
© Plot the points in Step 2.
Q Make a conjecture about the
graph of the ordered pairs
that are solutions to the
equation x + Ay = 12.
Think About It
In each exercise you are given a linear equation and two solutions.
Plot the solutions and draw the line that connects them. Plot the points
represented by the ordered pairs given and use the graph to guess
whether the other ordered pairs are solutions to the equation. Test your
results by substituting in the equation.
1- 2x + y = 3; solutions: 7(0, 3) and J(A, —5)
ordered pairs: K( 2, 4), L( 1, 4), and Af(l, 0)
2 . 3x — 2 y = 12; solutions: P(4, 0) and <2(2, —3)
ordered pairs: R( 6, 3), 5(2, — 1), and T{ 2, 4)
3 . Write a generalization about the solutions of a linear equation.
Developing Concepts
Graphing Linear Equations
Goal
Graph a linear equation
using a table of values.
Key Words
• linear equation
• solution of an equation
• function form
• graph of an equation
How long will it take an athlete to bum 800 calories?
Many relationships between two
real-life quantities are linear. In
Exercise 49 you will see that the
time an athlete exercises has a
linear relationship to the number
of calories burned.
As you saw in Developing Concepts 4.2, page 209, a linear equation in x and y
is an equation that can be written in the form
Ax + By = C
where A and B are not both zero. A solution of an equation in two variables is
an ordered pair (x, y) that makes the equation true.
B2EEEH 1 Check Solutions of Linear Equations
Determine whether the ordered pair is a solution of x + 2y = 5.
a. (1,2) b. (7, —3)
Solution
a. x + 2y = 5 Write original equation.
1 + 2(2) ]= 5 Substitute 1 for x and 2 for y.
5 = 5 Simplify. True statement.
ANSWER ► (1, 2) is a solution of the equation x + 2y = 5.
b. x + 2y = 5
7 + 2( — 3) 1 5
1 A 5
Write original equation.
Substitute 7 for x and -3 for y.
Simplify. Not a true statement.
ANSWER ► (7, —3) is not a solution of the equation x + 2y = 5.
Check Solutions of Linear Equations
1. Determine whether the ordered pair is a solution of 2x + y = 1.
a. (-3, 7) b. (3, —7) c. (|, o) d. (f,-6)
Chapter 4 Graphing Linear Equations and Functions
FUNCTION FORM A two-variable equation is written in function form if one of
its variables is isolated on one side of the equation. For example, y = 3x + 4 is in
function form while 2x + 3y = 6 is not in function form.
Student tteCp
► Study Tip
You can find solutions
of equations in two
variables by choosing
a value for one
variable and using it to
find the value of the
other variable.
K J
B32J22EB 2 Find Solutions of Linear Equations
Find three ordered pairs that are solutions of —2x + y = — 3.
0 Rewrite the equation in function form to make it easier to substitute values
into the equation.
— 2x + y = —3 Write original equation.
y = 2x — 3 Add 2x to each side.
© Choose any value for x and substitute it into the equation to find the
corresponding y- value. The easiest x- value is 0.
y = 2(0) — 3 Substitute 0 for x.
y = — 3 Simplify. The solution is (0, -3).
© Select a few more values
of x and make a table to
record the solutions.
X
0
1
2
3
-1
-2
y
-3
-1
1
3
-5
-7
ANSWER ^ (0, —3), (1, —1), and (2, 1) are three solutions of —2x + y = —3.
GRAPHS OF LINEAR EQUATIONS The graph of an equation in x and y is the
set of all points (x, y) that are solutions of the equation. The graph of a linear
equation can be shown to be a straight line.
Student MeCp
► Study Tip
Try to choose values
of xthat include
negative values, zero,
and positive values to
see how the graph
behaves to the left and
right of the y-axis.
L _ J
®23322BB 3 Graph a Linear Equation
Use a table of values to graph y — 3x — 2.
O Rewrite the equation in function form. This equation is already written in
function form: y = 3x — 2.
© Choose a few values of x and make a table
of values.
X
-2
-1
0
1
2
V
-8
-5
-2
1
4
With this table of values you have found
five solutions.
(-2, -8), (-1, -5), (0, -2), (1, 1), (2, 4)
© Plot the points and draw a line through them.
ANSWER ► The graph of y = 3x — 2 is shown at
the right.
4.2 Graphing Linear Equations
Student HcCp
►Study Tip
When choosing values
of x, try to choose
values that will
produce an integer.
^ _ )
4 Graph a Linear Equation
Use a table of values to graph 4y — 2x = 8.
Solution
o Rewrite the equation in function form by solving for y.
Ay — 2x = 8 Write original equation.
Ay = 2x + 8 Add 2x to each side.
y = —x + — Divide each side by 4.
y = ^x + 2 Simplify.
Choose a few values of x and make a table of values.
X
-A
-2
0
2
4
V
0
1
2
3
4
e Plot the points and draw a line
through them.
ANSWER ► The graph of 4y — 2x = 8 is
shown at the right.
I Graphing a Linear Equation
step Q Rewrite the equation in function form, if necessary.
step @ Choose a few values of x and make a table of values.
step 0 Plot the points from the table of values. A line through
these points is the graph of the equation.
Because the graph of a linear equation is a straight line, the graph can be drawn
using any two of the points on the line.
Graph Linear Equations
Rewrite the equation in function form.
2. 2x — y = 7 3- 6x + 3y = 18 4. 4y — 3x = —28
Find three ordered pairs that are solutions of the equation. Then graph
the equation.
5. y = —2x +1 6- x — y = 7 7. 4x + y = — 3
Chapter 4 Graphing Linear Equations and Functions
Exercises
Guided Practice
Vocabulary Check 1 . Complete: An ordered pair that makes an equation in two variables true is
called a ? .
2. Complete: A linear equation in x and y can be written in ? form.
Skill Check Determine whether the ordered pair is a solution of the equation.
3- x — y = —7, (—3, 4) 4. x + y = 10, (2, —12)
5. 4x — y = 23, (5, —3) 6- 5x + 3 y = —8, (2, —4)
Rewrite the equation in function form.
7. x + y = —2 8- x + 3y = 9 9. 4x + 2 + 2y = 10
Find three ordered pairs that are solutions of the equation.
10. y = 4x-l 11. y = 5x + 7 12 y = + 3
Use a table of values to graph the equation.
13. y — x — 4 14. y — x + 5 15. x + y = 6
Practice and Applications
CHECKING SOLUTIONS Determine whether the ordered pair is a solution
of the equation.
16. y = 2x + 1 , (5, 11) 17. y = 5 — 3x, (2,0)
18. 2y — 4x = 8, (—2, 8) 19. 5x — 8 y = 15, (3, 0)
20. 6y - 3x = -9, (1, -1) 21. -2x ~9y = l, (-1, -1)
Rewrite the equation in function form.
23. 2x + 3j = 6 24. x + 4y = 4
! Student HeCp
► Homework Help
Example 1: Exs. 16-21
Example 2: Exs. 22-39
Example 3: Exs. 40-48
Example 4: Exs. 40-48
FUNCTION FORM
22. — 3x + y = 12
25. 5x + 5 y = 19
28. 2x + 5 y = —15
FINDING SOLUTIONS
the equation.
31. y = 3x — 5
34. x + 2y = 8
37. 5x + 2 y= 10
26. 5y — 2x = 15
29. 3x + 2 y = —3
32. y = 7 — 4x
35. 2x + 3y = 9
38. y — 3x = 9
27. -x - y = 5
30. 4x — y = 18
33. j = —2x — 6
36. 3x - 5j = 15
39. - 5x - 3y = 12
Find three ordered pairs that are solutions of
4.2 Graphing Linear Equations
Link to
Sports
GRAPHING EQUATIONS Use a table of values to graph the equation.
40. y = 3x + 3 41. y = 4x + 2 42. y = 3x - 4
43. y — 5x = —2 44. x + y = 1 45. 2x + y = 3
46. y - 4x = -1 47. x + 4y = 48 48. 5x + 5y = 25
TRAINING FOR A TRIATHLON In Exercises 49-51, Mary Gordon is training
for a triathlon. Like most triathletes she regularly trains in two of the
three events every day. On Saturday she expects to burn about 800
calories during her workout by running and swimming.
Running: 7.1 calories per minute
Swimming: 10.1 calories per minute
Bicycling: 6.2 calories per minute
49. Copy and complete the model below. Let x represent the number of minutes
she spends running, and let y represent the number of minutes she spends
swimming.
TRIATHLON A triathlon is a
Verbal
Calories burned
•0+0
n #
Swimming
Total calories
race that has three parts:
Model
while running
j
time
burned
bicycling.
More about triathlons
is available at
www.mcdougallittell.com
Labels
Calories burned while running = |TJ (calories/minute)
Running time = |^xj (minutes)
[~?j = 10.1 (calories/minute)
[T] = |yj (minutes)
Total calories burned = 800 (calories)
Algebraic |Tj • x + |Tj • y
800
Write a linear model.
Model
50. Make a table of values and sketch the graph of the equation from
Exercise 49.
51 . If Mary Gordon spends 45 minutes running, about how many minutes will
she have to spend swimming to burn 800 calories?
Science Link / In Exercises 52 and 53, use the table showing the boiling
point of water (in degrees Fahrenheit) for various altitudes (in feet).
Altitude
0
500
1000
1500
2000
2500
Boiling Point
212.0
211.1
210.2
209.3
208.5
207.6
52. Make a graph that shows the boiling point of water and the altitude. Use the
horizontal axis to represent the altitude.
53. INTERPRETING DATA Describe the relationship between the altitude and
the boiling point of water.
Chapter 4 Graphing Linear Equations and Functions
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 54-55 is available at
www.mcdougallittell.com
INTERNET ACCESS In Exercises 54-56, use the following information. An
Internet service provider estimates that the number of households h (in millions)
with Internet access can be modeled by the equation h = 6.76t + 14.9 where t
represents the number of years since 1996.
54. Make a table of values. Use 0 < t < 6 for 1996-2002.
55. Graph the equation using the table of values from Exercise 54.
56. CRITICAL THINKING What does the graph mean in the context of the
real-life situation?
Standardized Test
Practice
57. MULTIPLE CHOICE Which ordered pair is a solution of — 3x + y = —5?
(A) (8,-16) CD(8, -29) (©(8,-64) 0(8,19)
58. MULTIPLE CHOICE Rewrite the equation — 2x + 5 y = 10 in function form.
CE) y = 2x + 2 CD y = 2x + 5
(]±) y — \x + 2 GD y — + 5
59.
MULTIPLE CHOICE Which equation does the graph represent?
(A) x + y = -2 ^
CD 6x + 3 y = 0
CD 2x — y = 3 -►
CD — v + 2y = 6 j
Mixed Review EVALUATING EXPRESSIONS Find the sum. (Lesson 2.3)
60. 5+ 2 +(-3) 61. -6 + (-14) + 8
62. -18 + (-10) + (-1) 63. -j + 6 + j
SIMPLIFYING EXPRESSIONS
64. 2a — 5b — la + 2b
66 . n 2 + 3/7? — 9m — 3 n 2
68 . 2c 2 - Ac + 8c 2 - 4c 3
Simplify the expression. (Lesson 2.7)
65. —6x + 2y - 8x + 4y
67. —4r — 5^ 3 + 2r — 7r
69. —3k 3 — 5k + h + 5k
SOLVING EQUATIONS Solve the equation. (Lesson 3.2)
70. -2z=-26 71. =-10 72. 6c =-96 73. ~ = -9
Maintaining Skills DECIMALS AND PERCENTS Write the decimal as a percent.
(Skills Review p. 768)
74.0.15 75.0.63 76.0.5 77.0.02
78.0.005 79.1.27 80.3 81.8.6
4.2 Graphing Linear Equations
Graphing Horizontal and
Vertical Lines
Goal
Graph horizontal and fj Q Volcano's height O function of time?
vertical lines. _
Key Words
• horizontal line
• vertical line
• coordinate plane
• x-coordinate
• /-coordinate
• constant function
• domain
• range
In Example 4 you will explore how the height of Mount St. Helens has changed.
All linear equations in x and y can be written in the form Ax + By = C. When
A = 0, the equation reduces to By = C and the graph of the equation is a
horizontal line. When B = 0, the equation reduces to Ax = C and the graph of
the equation is a vertical line.
Student HeCp
1 ^ — -\
► Study Tip
The equations y = 2
and Ox + 1/ = 2 are
equivalent. For any
value of x, the ordered
pair (x, 2) is a solution
of y = 2.
\ _
i Graph the Equation y = b
Graph the equation y = 2.
Solution
The equation does not have rasa variable.
The y-coordinate is always 2, regardless of
the value of x. For instance, here are some
points that are solutions of the equation:
(-3,2), (0, 2), and (3,2)
ANSWER ► The graph of the equation y = 2 is a
horizontal line 2 units above the x-axis.
i
(-
-3, 2)
3
(0,2)
(3,2)
1
y =
= 2*
-
1
-1
L
3
X
Graph the equation.
1.y=-3 2. y = 4
3- y =
1
2
Chapter 4 Graphing Linear Equations and Functions
Student MeCp
^ More Examples
4^1 More examples
' are available at
www.mcdougallittell.com
2 Graph the Equation x = a
Graph the equation x = — 3.
Solution
The x-coordinate is always — 3, regardless
of the value of y. For instance, here are some
points that are solutions of the equation:
(-3, -2), (-3, 0), and (—3, 3)
ANSWER ► The graph of the equation
x = —3 is a vertical line 3 units
to the left of the y-axis.
J
(-
3 , 3 :
i
1 x= '
-3
1
i-
3,0
1
-1
]
L x
T<-
3, -
-2)
Jj
Graph the Equation x = a
Graph the equation.
4. x = 2 5- x = — 1 6. x = 3^
3 Write an Equation of a Line
Write the equation of the line in the graph.
1
3
-
1
-1
1 X
t
3
y
1
-
1 |
r 1
3 x
Solution
a. The graph is a vertical line. The x-coordinate is always —2.
ANSWER ► The equation of the line is x = —2.
b. The graph is a horizontal line. The y-coordinate is always 4.
ANSWER ► The equation of the line is y = 4.
Write the equation of the line in the graph.
1
i .y
-1
-1
]
L
5 X
-3
L .y
1
1
-1
[
X
4.3 Graphing Horizontal and Vertical Lines
CONSTANT FUNCTION A function of the form y = b, where b is a number, is
called a constant function. Its range is the single number b and its graph is a
horizontal line.
Student HeCp
^
p Look Back
For help with domain
and range, see p. 49.
k _/
4 Write a Constant Function
The graph below shows the height of Mount St. Helens from 1860 to May
1980. Write an equation to represent the height of Mount St. Helens for this
period. What is the domain of the function? What is the range?
Height of Mount St. Helens
10,000
O)
'53
9,000
20 40 60 80 100
Years since 1860
120
Solution
From the graph, you can see that between 1860 and 1980, the height H was
about 9700 feet. Therefore, the equation for the height during this time is
H = 9700. The domain is all values of t between 0 and 120. The range is
the single number 9700.
Write a Constant Function
9. On May 18, 1980, Mount St. Helens erupted. The eruption blasted away
most of the peak. The height of Mount St. Helens after the 1980 eruption was
8,364 feet. Write an equation that represents the height of Mount St. Helens
after 1980. What is the domain of the function? What is the range?
Equations of Horizontal and Vertical Lines
In the coordinate plane,
the graph of y = b is a
horizontal line.
iy
X
II
aa
X
In the coordinate plane,
the graph of x = a is a
vertical line.
Chapter 4 Graphing Linear Equations and Functions
Exercises
Guided Practice
Vocabulary Check 1 . Is the x-axis a horizontal or a vertical line?
2 . Is the y-axis a horizontal or a vertical line?
3. Complete: A function of the form y = b is called a ? function.
Skill Check Graph the equation.
4. y=l 5.x =-10 6.y=-5 7. x = 7
Write the equation of the line in the graph.
1
1
-1
J
X
1
-1
J
5 X
LOGICAL REASONING Complete the statement with always , sometimes ,
or never.
10. The graph of an equation of the form y = b is ? a horizontal line.
11 . A line that passes through the point (2, —3) is ? a vertical line.
12. The graph of an equation of the form x = a is ? a horizontal line.
13 . The range of the function y = 4 is ? equal to 4.
Practice and Applications
CHECKING SOLUTIONS Determine whether the given ordered pair is a
solution of the equation.
14. y = —2, (—2, —2) 15. y = 3,(3, -3)
16. j = 0, (0, 1) 17. x = 5, (—5, —5)
FINDING SOLUTIONS Find three ordered pairs that are solutions of
the equation.
18.x = 9 19.x = | 20. y = 10
Student HeCp
► Homework Help
Example 1: Exs. 18-29
Example 2: Exs. 18-29
Example 3: Exs. 30-32
Example 4: Exs. 33, 34
21.y=-5 22.x =-10 23. y = 7
GRAPHING EQUATIONS Graph the equation.
24. y = —1 25. > ? = 8 26. x = 4
27. x = -9 28. x = | 29. x =
4.3 Graphing Horizontal and Vertical Lines
Link
Science
MAMMOTH CAVE is the
longest recorded cave system
in the world, with more than
348 miles explored and
mapped. About 130 forms of
life can be found in the
Mammoth Cave system.
i
WRITING EQUATIONS Write the equation of the line in the graph.
32.
33. HEART RATE You decide to exercise using a treadmill. You warm-up with a
5 minute walk then do a 10 minute fast run. The graphs below show your
heart rate during your warm-up and during the fast run.
a. Write an equation that gives your heart rate during your warm-up. What is
the domain of the function? What is the range?
b. Write an equation that gives your heart rate during the fast run. What is
the domain of the function? What is the range?
During Fast Run
y i
_ 160
V
t rate
r minu
o
i- m
cc o.
“2 120
CO
V
1 100
0 2 4 6 8 10 *
Time (minutes)
During Warm-Up
_ 160
V
+*
3
O) 3
S'i 140
*-
t: v
cc a.
®C2 120
CO
V
e 100
0^
I
3 ’
12 3 4!
Time (minutes)
5 *
34. Science Link x You are visiting Kentucky on your summer vacation. You
go to Mammoth Cave National Park, the second oldest tourist attraction in
the United States. One interesting fact about Mammoth Cave is that it has a
constant temperature of 54° year round. The temperature outside the cave on
the day you visited was 80°. The graphs below show the temperatures
outside Mammoth Cave and inside Mammoth Cave.
a. Write an equation to represent the temperature outside the cave. What is
the domain of the function? What is the range of the function?
b. Write an equation to represent the temperature inside Mammoth Cave
What is the domain of the function? What is the range of the function?
j j
e 80
o
i
J J
or 80
o
1 70
**
1 70
CG
| 60
E
i®
■“ 50
“i
CO
1 60
E
i£
K 50
0^
(
h
L
)
20 40 60 80
Time (minutes)
100
X
)
20 40 60 80
Time (minutes)
100
X
22 <
Chapter 4 Graphing Linear Equations and Functions
Standardized Test
Practice
Mixed Review
Maintaining Skills
Quiz 1
35. MULTIPLE CHOICE Which point does not lie on the graph of y = 3?
(A) (0,3) (©(-3,3) (©(3,-3) (5) (|,3)
36. MULTIPLE CHOICE The ordered pair (3, 5) is a solution of ? .
(f~) y = 5 (© x = 5 (R) y = —3 (T) x = —5
EVALUATING EXPRESSIONS Evaluate the expression. (Lesson 1.3)
37. 17 - 6 + 4 - 8 38. 6 + 9 h- 3 + 3 39. 4 • 5 - 2 • 6
40. 9 • 6 + 3 • 18
41.22 - 8 -s- 2 • 3
42. 0.75 -h 2.5 • 2 + 1
SOLVING EQUATIONS Solve the equation. (Lesson 3.1)
43. r - (-4) = 9 44. -8 - (-c) = 10 45. 15 - (~b) = 30
LEAST COMMON DENOMINATOR Find the least common denominator
(LCD) of each pair of fractions. Then rewrite each pair with their LCD.
(Skills Review p. 762)
2 7
3’ 8
47 — —
7 , 3
48 — —
2’ 7
^54
49 ' 7 ’ 21
8 7
9’ 12
12 5
51 ‘ 13’ 26
7 2
52 —-—
18’ 15
53 ———
20’ 15
Plot and label the ordered pairs in a coordinate plane. (Lesson 4.1)
1. A(—4, 1), 5(0, 2), C(—3, 0) 2 . A(— 1, -5), 5(0, -7), C(l, 6)
3. A(— 1, —6), 5(1, 3), C(—1, 1) 4. A(2, -6), 5(5, 0), C(0, -4)
Without plotting the point, name the quadrant the point is in.
(Lesson 4.1)
5. (6, 8) 6. (-4,-15) 7. (5,-9) 8. (-3,3)
Rewrite the equation in function form. (Lesson 4.2)
9. 2x + y = 0 10. 5x — 2y = 20 11. — 4x — 8_y = 32
Find three ordered pairs that are solutions of the equation. Then graph
the equation. (Lesson 4.2)
12. y = 2x - 6 13. y = 4x + 1
15. y = —3(x — 4) 16. lOx + y = 5
Graph the equation. (Lesson 4.3)
18. x = —5 19. _y = 2
14. y = 2(—3x + 1)
17. 6 = 8x — 3_y
20 . x = 4
4.3 Graphing Horizontal and Vertical Lines
Graphing Lines Using
Intercepts
Goal
Find the intercepts of
the graph of a linear
equation and then use
them to make a quick
graph of the equation.
Key Words
• x-intercept
• /-intercept
• x-axis
• /-axis
How much should you charge for tickets?
In Exercises 48-51 you will use
the graph of a linear equation to
determine how much you should
charge for tickets to raise money
for animal care in a zoo.
An x-intercept is the x-coordinate of a point where a graph crosses the x-axis.
Ay-intercept is the y-coordinate of a point where a graph crosses the y-axis.
The /-intercept is the—
value of / when x = 0.
Here, the /-intercept is 2.
The x-intercept is the
value of x when / = 0.
Here, the x-intercept is 3.
Because two lines that are not parallel intersect in exactly one point:
• The vertical line given by x = a, a A 0, has one x-intercept and no y-intercept.
• The horizontal line given by y = b, b A 0, has one y-intercept and no
x-intercept.
• A line that is neither horizontal nor vertical has exactly one x-intercept and
one y-intercept.
i) Find an x-fntercept
Find the x-intercept of the graph of the equation 2x + 3y = 6.
Solution
To find an x-intercept, substitute 0 for y and solve for x.
2x + 3y = 6 Write original equation.
2x + 3(0) = 6 Substitute 0 for /.
x = 3 Solve for x.
ANSWER ► The x-intercept is 3. The line crosses the x-axis at the point (3, 0).
2 Find a y-intercept
Find the ^-intercept of the graph of the equation 2x + 3y = 6.
Solution
To find a ^-intercept, substitute 0 for x and solve for y.
2x + 3y = 6 Write original equation.
2(0) + 3y = 6 Substitute 0 for x.
y — 2 Solve for y.
ANSWER ► The ^-intercept is 2. The line crosses the y -axis at the point (0, 2).
Find Intercepts
1. Find the x-intercept of the graph of the equation 3x — 4 y = 12.
2 . Find the ^-intercept of the graph of the equation 3x — 4 y = 12.
Student ftedp
\
► Study Tip
The Quick Graph
process works
because only two
points are needed
to determine a line.
V _ j
2EEEH3
Making a Quick Graph
step Q Find the intercepts.
step © Draw a coordinate plane that includes the
intercepts.
step © Plot the intercepts and draw a line through them.
3 Make a Quick Graph
Graph the equation 3x + 2y = 12.
Solution
Q Find the intercepts.
3x + 2y= 12
3x + 2(0) = 12
x = 4
3x + 2y = 12
3(0) + 2y = 12
y = 6
Write original equation.
Substitute 0 for y.
The x-intercept is 4.
Write original equation.
Substitute 0 for x.
The y-intercept is 6.
© Draw a coordinate plane that includes
the points (4, 0) and (0, 6).
© Plot the points (4, 0) and (0, 6) and
draw a line through them.
4.4 Graphing Lines Using intercepts
When you make a quick graph, find the intercepts before you draw the coordinate
plane. This will help you find an appropriate scale on each axis.
( Student MeCp
p More Examples
More examples
are ava j| a bie at
www.mcdougallittell.com
j
a Choose Appropriate Scales
Graph the equation y = 4x + 40.
Solution
© Find the intercepts.
y — 4x + 40
0 = 4x + 40
-40 = 4x
— 10 = x
Write original equation.
Substitute 0 for y.
Subtract 40 from each side.
Divide each side by 4.
ANSWER ► The x-intercept is —10. The line crosses the x-axis at
the point (—10, 0).
y = 4x + 40 Write original equation.
y = 4(0) + 40 Substitute 0 for x.
y — 40 Simplify.
ANSWER ► The y-intercept is 40. The line crosses the y-axis at
the point (0, 40).
© Draw a coordinate plane that includes the
points ( —10, 0) and (0, 40). With these
values, it is reasonable to use tick marks
at 10-unit intervals.
You may want to draw axes with at least
two tick marks to the left of —10 and to
the right of 0 on the x-axis and two tick
marks below 0 and above 40 on
the y-axis.
© Plot the points (—10, 0) and (0, 40)
and draw a line through them.
Make a Quick Graph
Find the intercepts of the graph of the equation.
3- 3x — 6y = 18 4. 4x — 5y = 20 5. y = —2x + 50
Graph the equation using intercepts.
6. 2x + 5y = 10 7. x — 6y = 6 8- 12x — 4y = 96
Chapter 4 Graphing Linear Equations and Functions
!■ | Exercises
Guided Practice
Vocabulary Check 1 . Complete: In the ordered pair (3, 0) the ? is the x-intercept.
2. Complete: In the ordered pair (0, 5) the ? is the ^-intercept.
Skill Check Find the x-intercept of the graph of the equation.
3 . 5x + 4y = 30 4 . y = 2x + 20 5 . —lx — 3y = 21
Find the /-intercept of the graph of the equation.
6. 6x + 3y = 51 7. —2x — 8j = 16 8. lOx — y = —5
Find the x-intercept and
Graph the equation.
9. y = x + 2
12 . 3 y = —6x + 3
the /-intercept of the
10. y — 2x = 3
13. 5_y = 5x + 15
graph of the equation.
11. 2x - v = 4
14. x — y = 1
Practice and Applications
USING GRAPHS TO FIND INTERCEPTS Use the graph to find the
x-intercept and the /-intercept of the line.
1
—1
L --_
1 X
— z
i
/
/
/
1
/
3 /
-
1 )
, J
L x
FINDING X-INTERCEPTS
18. x - 2y = 4
21 . 5x + 6 y = 95
24. -x - 5y = 12
Find the x-intercept of the line.
19. x + 4y = —2 20. 2x — 3y = 6
22. — 6x - 4y = 42 23. 9x - 4 y = 54
25. 2x + 6y = -24 26. -13x - y = 39
I Student HeCp
► Homework Help
Example 1: Exs. 15-26
Example 2: Exs. 15-17,
27-32
Example 3: Exs. 33-47
Example 4: Exs. 45-47
L j
FINDING y-INTERCEPTS Find the /-intercept of the line.
27. y — 4x — 2 28. y — —3x + 7 29. y — 13x + 26
30. y = 6x — 24 31. 3x — 4y = 16 32. 2x — 17y = —51
USING INTERCEPTS Graph the line that has the given intercepts.
33. x-intercept: —2 34. x-intercept: 4 35. x-intercept: —7
^-intercept: 5 ^-intercept: 6 ^-intercept: —3
4.4 Graphing Lines Using intercepts
MATCHING GRAPHS Match the equation with its graph.
36. y = 4x — 2 37. y = 4x + 2 38. y = 4x + 3
GRAPHING LINES Find the x-intercepts and the /-intercepts of the line.
Graph the equation. Label the points where the line crosses the axes.
39. y = x + 3 40. y = x + 9 41. y = -4 + 2x
42. y = 2 — x
45. 36x + Ay = 44
43. y = —3x + 9
46. y = lOx + 50
44. y = 4x — 6
47. —9x + y = 36
Link to
Business
ZOO EXPENSES
The American Zoo and
Aquarium Institute estimates
that it costs a zoo about
$22,000 per year to house,
feed, and care for a lion and
about $18,500 for a polar bear.
More about zoo
4® * expenses available at
www.mcdougallittell.com
E2EE3 Zoo Fundraising
ZOO FUNDRAISING You are organizing a breakfast tour to raise funds for
animal care. Your goal this quarter is to sell $1500 worth of tickets. Assuming
200 adults and 100 students will attend, how much should you charge for an
adult ticket and a student ticket? Write a verbal model and an algebraic model
to represent the situation.
Solution
Verbal
Model
Labels
Number
of adults
Adult ticket
price
+
Number of
students
Student
ticket price
Total
sales
Number of adults = 200
Adult ticket price = x
Number of students = 100
Student ticket price = y
Total sales = 1500
Algebraic 200 x + 100 y = 1500
Model
2x + y = 15
(people)
(dollars per person)
(people)
(dollars per person)
(dollars)
Write a linear model.
Divide each side by 100 to simplify.
48. Graph the linear function 2x + y = 15.
49. What is the x-intercept? What does it represent in this situation?
50. What is the y-intercept? What does it represent in this situation?
51. CRITICAL THINKING If students cannot afford to pay more than $3 for a
ticket, what can you say about the price of an adult ticket?
Chapter 4 Graphing Linear Equations and Functions
Standardized Test
Practice
Mixed Review
Maintaining Skills
RAILROAD EMPLOYEES In Exercises 52-54, use the following
information. The number of people who worked for the railroads in the
United States each year from 1989 to 1995 can be modeled by the equation
y = —6.6x + 229, where x represents the number of years since 1989 and
y represents the number of railroad employees (in thousands).
DATA UPDATE of the U.S. Bureau of the Census at www.mcdougallittell.com
52. Find the y-intercept of the line.
What does it represent?
53. About how many people worked
for the railroads in 1995?
54. CRITICAL THINKING Do you
think the line in the graph will
continue to be a good model for
the next 50 years? Explain.
Railroad Employees
Years since 1989
55. MULTIPLE CHOICE Find the x-intercept of the graph of the equation
3x + y — —9.
(a) -3 CD 3 CD 9 CD -9
56. MULTIPLE CHOICE Find the y-intercept of the graph of the equation
2x — 3 y = 12.
CD -4 CD-I CD 4 CD 3
EVALUATING DIFFERENCES Find the difference. (Lesson 2.4)
58. 17 - (-6) 59. 18 | - 13 60. 7 - | -8 |
62. -4 - (-5) 63. -8 - 9 64. 13.8 - 6.9
EVALUATING QUOTIENTS Find the quotient. (Lesson 2.8)
65. 54 - 9 66. -72 - 8 67. 26 - ( -13) 68. -1 - 8
® 9 - 12 (-?) 70 ' 3 + ? 71 -|4 72.-20^(f
73. SCHOOL BAKE SALE You have one hour to make cookies for your school
bake sale. You spend 24 minutes mixing the dough. It then takes 12 minutes
to bake each tray of cookies. If you bake one tray at a time, which model can
you use to find how many trays you can bake during the hour? (Lesson 3.5)
A. x(24 + 12) = 60 B. 12x + 24 = 60
ROUNDING Round to the nearest cent. (Skills Review p. 774 )
74. $.298 75. $1,649 76. $.484 77. $8,357
78. $7,134 79. $3,152 80. $.005 81. $5,109
57. 5 - 9
4.4 Graphing Lines Using Intercepts
DEVELOPING CONCEPTS
For use with
Lesson 4.5
Goal
Use slope to describe the
steepness of a ramp.
Question
How can you use numbers to describe the steepness of a ramp?
Materials
• 5 books
• 2 rulers
• paper
You can use the ratio of the vertical
rise to the horizontal run to describe
the steepness, or slope , of a ramp.
. _ vertical rise _ 2
S °P e horizontal run 5
Explore
O Stack three books. Use a ruler as a ramp. Measure the rise and
the run as shown in the top photo. Record the rise, the run, and
the slope in a table like the one below.
Vertical rise
(incites)
Horizontal run
(incites)
Slope
© Keeping the rise the same, move the position of the ruler to
change the length of the run three times. Each time, measure
and record the results.
© Place a piece of paper under the edge of the stack of books.
Mark the point that is 6 inches from the base of the stack.
Place the end of the ramp on the mark as shown in the
middle photo. Record the results.
© Change the rise by adding or removing books as shown in the
bottom photo. Using a run of 6 inches each time, create three
more ramps with different rises. Each time, measure and record
the results.
Think About It
1. What happens to the slope when the rise stays the same and the run changes?
2 . What happens to the slope when the rise changes and the run stays the same?
3- Describe the relationship between the rise and the run when the slope is 1.
Chapter 4 Graphing Linear Equations and Functions
The Slope of a Line
Goal
Find the slope of a line.
Key Words
• rise
• run
• slope
How steep is a roller coaster?
You can describe steepness by a ratio
called slope. To find the slope, divide
the rise by the run. In Exercise 39 you
will find the slope of a roller coaster.
1 The Slope Ratio
Find the slope of a hill that has a vertical rise of 40 feet and
a horizontal run of 200 feet. Let m represent slope.
vertical
rise = 40 ft
horizontal run = 200 ft
Solution
_ vertical rise _ 40 _ J_
m horizontal run 200 5
ANSWER ► The slope of the hill is ~.
The slope of a line is the
ratio of the vertical rise to the
horizontal run between any
two points on the line. In the
diagram, notice how you can
subtract coordinates to find the
rise and the run.
i rise
slope =-
r run
4-2
8-3
2
5
4.5 The Slope of a Line
THE SLOPE OF A LINE
SLOPE When you use the formula for slope, you can label either point as (x v y x )
and the other as (x 2 , y 2 ). After labeling the points, you must subtract the
coordinates in the same order in both the numerator and the denominator.
■m3 2 Positive Slope
Find the slope of the line that passes through the points (1, 0) and (3, 4).
Solution
Let (x v jj) = (1, 0) and (x 2 , y 2 ) = (3, 4).
y 2 ~ Ji ■*-Subtract y-values.
171 — -
-Use the same order
to subtract x-values.
4-0 „ , .
= ^ j Substitute values.
4
= — Simplify.
= 2 Slope is positive.
ANSWER ► The slope of the line is 2.
5
/(3,4)
3
4
1
-1
:
J
X
— i
(1,0)
J
The line rises from left to right.
The slope is positive.
a Positive Slope
Find the slope of the line that passes through the two points. Draw a
sketch of the line to help you.
1. (x p Jj) = (3, 5) and (x 2 , y 2 ) = (1,4)
2 . (x v = (2, 0) and (x 2 , y 2 ) = (4, 3)
3. (x v jj) = (2, 7) and (x 2 , y 2 ) = (1, 3)
Chapter 4 Graphing Linear Equations and Functions
Student HeCp
N
► Study Tip
You can choose any
two points on a line
to find the slope. For
example, you can use
the points (0, 3) and
(3, 2) in Example 3 and
get the same slope.
You will see this proof
in Geometry.
V_
■3ZHIZI9 3 Negative Slope
Find the slope of the line that passes through the points (0, 3) and (6, 1).
Solution
Let (x p yj = (0, 3) and (x v y 2 ) = (6, 1).
y 2 — y x - Subtract y-values.
X 2 ~ X 1 * - Use Same or< ^ er
to subtract x-values.
1-3 , .
7 - 77 Substitute values.
6 — 0
1 + (-3) To subtract, add
6 — 0 the opposite.
-2 _ 1 Simplify to find the
6 — 3 negative slope.
ANSWER ► The slope of the line is
3*
The line falls from left to right.
The slope is negative.
Find a Negative Slope
Find the slope of the line that passes through the two points. Draw a
sketch of the line to help you.
4. (x r jj) = (2, 4) and (x 2 , y 2 ) = (-1, 5)
5. (x p jj) = (0, 9) and (x 2 , y 2 ) = (4, 7)
6 . (x r jj) = (-2, 1) and (x 2 , y 2 ) = (1, -3)
E2!mZ 219 4 Zero Slope
Find the slope of the line that passes through the points (1, 2) and (5, 2).
Solution
Let (x v y x ) = (1,2) and (x 2 , y 2 ) = (5, 2).
*— Subtract y-values.
**-—■ Use the same order
to subtract x-values.
Substitute values.
Simplify to find the slope
is zero.
The line is horizontal.
slope of the line is zero. The slope is zero.
^-•Vi
m — -
X 2 X 1
= 2-2
5 - 1
=!=°
ANSWER t The
4.5 The Slope of a Line
Student Hedp
► More Examples
More examples
~<!^S v are available at
www.mcdougallittell.com
5 Undefined Slope
Find the slope of the line that passes through the points (5, — 1) and (5,3).
Solution
Let (x v y x ) = (5, -1) and (x 2 , y 2 ) = (5, 3).
I
— Subtract y-values.
m —
x 2 -x l ^—
Use the same order
to subtract x-values.
3 -(-1)
Substitute values.
5-5
3 + 1
Subtracting a negative
5-5
is the same as adding
a positive.
W
Division by zero is
= z
undefined.
O'
3
(5,3)
1
L
-1
L
3
X
(5,-
-d'
—3
<
The line is vertical.
The slope is undefined.
ANSWER ► Because division by zero is undefined, the expression
4
— has no meaning. The slope of the line is undefined.
Find the Slope of a Line
For each line, determine whether the slope is positive , negative , zero, or
undefined. If the slope is defined, find the slope.
8 .
EMifiifiraa?
Slopes of Lines
A line with positive
slope rises from left
to right.
X
A line with negative
slope falls from left
to right.
A line with zero
slope is horizontal.
0 7
X
A line with
undefined slope
is vertical.
Chapter 4 Graphing Linear Equations and Functions
Exercises
Guided Practice
Vocabulary Check Use the photo of a ramp.
1. What is the rise of the ramp?
2 . What is the run of the ramp?
3. What is the slope of the
ramp?
Skill Check Plot the points and draw the line that passes through them. Without
finding the slope, determine whether the slope is positive, negative, zero,
or undefined.
4. (1,5) and (5, 5) 5. (-2,-2) and (0, 1) 6. (4, 2) and (4,-1)
7. (-3, 1) and (1,-3) 8. (2, 1) and (5, 3) 9. (-4,-3) and (0,-3)
Find the slope of the line.
Practice and Applications
THE SLOPE RATIO Plot the points and draw a line that passes through
them. Use the rise and run to find the slope.
13. (2, 3) and (0, 6) 14. (1, 4) and (3, 2) 15. (3, 1) and (-3, -2)
16. (2, 2) and (6, -1) 17. (-2, 1) and (2, 4) 18. (1, -3) and (4, 0)
Student HeCp
^ -\
► Homework Help
Example 1: Exs. 13-18,
29-34
Example 2: Exs. 21-28
Example 3: Exs. 19,
23-28
Example 4: Exs. 20,
29-34
Example 5: Exs. 29-34
x _J
GRAPHICAL REASONING Find the slope of the line.
22. CRITICAL THINKING Is the slope always positive if the coordinates of two
points on the line are positive? Justify your answer.
4.5 The Slope of a Line
FINDING SLOPE Find the slope of the line that passes through
the points.
23. (4, 3) and (8, 5) 24. (-2, 4) and (1, 6) 25. (3, 8) and (7, 7)
26. (3, -4) and (9,4) 27. (-3, 5) and (-5, 8) 28. (-6, -7) and (-4, -4)
Linkt^
History
jib sail
U.S.S. CONSTITUTION was
nicknamed "Old Ironsides"
by the crew in 1812 after the
defeat of the 38-gun British
frigate Guerriere.
ZERO OR UNDEFINED SLOPE Determine whether the slope is zero,
undefined\ or neither.
29. (0, 4) and (-5, 7) 30. (1, 2) and (1, 6) 31. (6, 2) and (9, 2)
32. (5, -8) and (3, -8) 33. (8, 7) and (14, 1) 34. (3, 10) and (3, 5)
35. History Link } The photo shows the
U.S.S. Constitution. Built in the late
1700s, it is the oldest warship afloat.
Find the slope of the edge of the
Constitution 's jib sail.
36. LADDER The top of a ladder is 12 feet from the ground. The base of the
ladder is 5 feet to the left of the wall. What is the slope of the ladder? Make
a sketch to help you.
37. INDUCTIVE REASONING Choose
three different pairs of points on the
line. Find the slope of the line
using each pair. What do you notice?
What conclusion can you draw?
38. INDUCTIVE REASONING Based on your conclusion from Exercise 37,
complete the following sentence: No matter what pair of points you choose
on a line, the ? is constant.
ROLLER COASTER In Exercises 39 and 40, use the following information.
You are supervising the construction of a roller coaster for young children. For
the first 20 feet of horizontal distance, the track must rise off the ground at a
constant rate. After your crew has constructed 5 feet of horizontal distance, the
track is 1 foot off the ground.
5 ft
20 ft
39. Plot points for the heights of the track in 5-foot intervals. Draw a line
through the points. Find the slope of the line. What does it represent?
40. After 20 feet of horizontal distance is constructed, you are at the highest
point of your roller coaster. How high off the ground is the track?
Chapter 4 Graphing Linear Equations and Functions
Standardized Test
Practice
Mixed Review
Maintaining Skills
Road Grade
Road signs sometimes describe the
slope of a road in terms of its grade,
The grade of a road is given as a
positive percent. Find the grade of
the road shown in the sketch.
Solution
Find the slope:
vertical rise _ 4
horizontal run 50'
4 4
Write as a fraction whose denominator is 100: =
g
Write — as a percent: 8%.
8
100 '
ANSWER ► The grade of the road is 8%.
41. Find the grade of a road that rises ly feet for every horizontal distance of
25 feet.
42. Find the grade of a road that rises 70 feet for every horizontal distance of
1000 feet.
43. MULTIPLE CHOICE What is the slope of the line through the points
(4, 3) and (11, 5)?
CD-f ©f CS>-\
44. MULTIPLE CHOICE Which word describes the slope of a vertical line?
CD zero Cg) positive (Tp undefined CD negative
SOLVING EQUATIONS Solve the equation. (Lesson 3.1)
45. x + 7 = 12 46.x- 3 = 11 47.x -(-2) = 6
REWRITING EQUATIONS Rewrite the equation so that y is a function
of x. (Lesson 3.7)
48. 5 y = lOx — 5 49. -^y = -|x + 3 50. —4x + y = 11
51. — 8x + 2y = 10 52. -3x + 6y = 12 53. x + |y = -1
OPERATIONS WITH DECIMALS Determine whether the equation is true
or false. (Skills Review p. 759)
54.1.3 - 2.7 = 1.4 55. y|- 1 = 0 56. ^ + 1 j = 10 = 0
57. 14.4 + 0.14 = 2.88 58. (7.8)(1.5) + 4.6 = 16.3 59. 12 + 0 • 7.18 = 12
4.5 The Slope of o Line
Direct Variation
Goal
Write and graph
equations that represent
direct variation.
Key Words
• direct variation
• constant of variation
• origin
How much do 36 gold bars weigh?
If you know the weight of one standard mint
gold bar, then you can determine the weight
of 2, 3, or more gold bars. In Example 3 you
will see that total weight is directly
proportional to the number of bars.
When two quantities y and x have a constant ratio k , they are said to have
direct variation. The constant k is called the constant of variation.
y
If — = k, then y = kx.
y
Model for Direct Variation: y = kx or — = k, where k A 0.
y x
Student MeCp
► Reading Algebra
The model for direct
variation y= /rxis
read as "/varies
directly with x."
K _ j
i Write a Direct Variation Model
The variables x and y vary directly. One pair of values is x = 5 and y = 20.
a. Write an equation that relates x and y.
b. Find the value of y when x = 12.
Solution
a. Because x and y vary directly, the equation is in the form of y = kx.
y = kx Write model for direct variation.
20 = k( 5) Substitute 5 for x and 20 for y.
4 — k Divide each side by 5.
ANSWER ► An equation that relates x and y is y = 4x.
b- y = 4(12) Substitute 12 for x.
y = 48 Simplify.
ANSWER ^ Whenx = 12, y = 48.
Write a Direct Variation Model
The variables x and y vary directly. Use the given values to write a direct
variation model that relates x and y.
1- x = 2, y = 6 2_ x = 3, y = 21 3. x = 8, y = 96
Chapter 4 Graphing Linear Equations and Functions
GRAPHING DIRECT VARIATION MODELS Because x = 0 and y = 0 is a
solution ofy = kx , the graph of a direct variation equation is always a line
through the origin.
Student HeCp
► Mor? Examples
More examples
are ava j| a b| e a t
www.mcdougallittell.com
2 Graph a Direct Variation Model
Graph the equation y — 2x.
Solution
O Plot a point at the origin.
© Find a second point by choosing any value for x and substituting it into
the equation to find the corresponding y- value. Use the value 1 for x.
y = 2x Write original equation.
y = 2(1) Substitute 1 for x.
y — 2 Simplify. The y-value is 2.
ANSWER ► The second point is (1, 2).
© Plot the second point and draw a line
through the origin and the second point.
ANSWER ► The graph of y = 2x is shown at
the right.
Graph a Direct Variation Model
Lint »«*_
History
FORT KNOX, the United
States Bullion Depository,
has stored many valuable
items. During World War II,
the English crown jewels and
the Magna Carta were stored
there.
More about Fort
Knox is available at
www.mcdougallittell.com
Graph the equation.
4. y = x 5. y = — 2x 6, y = 3x
M mz n m 3 Use a Direct Variation Model
FORT KNOX The gold stored in Fort Knox is in the form of standard
mint bars called bullion , of almost pure gold. Given that 5 gold bars weigh
137.5 pounds, find the weight of 36 gold bars.
Solution
Begin by writing a model that relates the weight W to the number n of gold bars.
W — kn Write model for direct variation.
137.5 = k( 5) Substitute 137.5 for W and 5 for n.
27.5 = k Divide each side by 5.
A direct variation model for the weight of a gold bar is W = 21.5n.
Use the model to find the weight of 36 gold bars.
W = 27.5(36) Substitute 36 for n.
W = 990 Simplify.
ANSWER ► Thirty six gold bars weigh 990 pounds.
4.6 Direct Variation
Student HeGp
► Study Tip
Sometimes real-life
data can be
approximated by a
direct variation model,
even though the data
do not fit this model
exactly.
L. _ j
Use a Direct Variation Model
ANIMAL STUDIES The tail length and body length (in feet) of 8 alligators are
shown in the table below. The ages range from 2 years to over 50 years. Write a
direct variation model that relates the tail length T to the body length B.
► Source: St. Augustine Alligator Farm
Body length B
- Tail length T -
Tail T
1.41
2.04
2.77
2.77
3.99
4.67
4.69
5.68
Body B
1.50
2.41
3.08
3.23
4.28
5.04
5.02
6.38
Solution
Begin by finding the ratio of tail length to body length for each alligator.
Tail T
1.41
2.04
2.77
2.77
3.99
4.67
4.69
5.68
Body B
1.50
2.41
3.08
3.23
4.28
5.04
5.02
6.38
Ratio
0.94
0.85
0.90
0.86
0.93
0.93
0.93
0.89
ANSWER ^ Since the values for the ratio are all close to 0.90 it is reasonable to
choose k = 0.90. A direct variation model is T = 0.90 B.
L
Use a Direct Variation Model
7. Use the direct variation model you found in Example 4 above to estimate the
body length of an alligator whose tail length is 4.5 feet.
21212052
Properties of Graphs of Direct Variation Models
The graph of y = kx is
a line through the origin.
The slope of the graph
of y = kx is k.
N
\ ;
X
k is negative.
k is positive.
Chapter 4 Graphing Linear Equations and Functions
Exercises
Guided Practice
Vocabulary Check 1 . Explain what it means for x and y to vary directly.
2. What point is on the graph of every direct variation equation?
Skill Check Find the constant of variation.
3- y varies directly with x, and y — 3 when x — 21.
4. y varies directly with x, and y = 8 when x = 32.
5- r varies directly with s, and r = 5 when s = 35.
The variables x and y vary directly. Use the given values to write an
equation that relates x and y.
6- x = 1, y = 2 7. x = 5, y = 25 8. x = 3, y = 36
Graph the equation.
9. y — x 10. y = — 3x 11. y = 5x
Practice and Applications
DIRECT VARIATION MODEL Find the constant of variation.
12. y varies directly with x, and y = 54 when x = 6.
13. y varies directly with x, and y — 12 when x — 6.
14. h varies directly with m, and h = 112 when m = 12.
15. W varies directly with m, and W = 150 when m — 6.
FINDING EQUATIONS In Exercises 16-24, the variables x and y vary
directly. Use the given values to write an equation that relates x and y.
16. x = 4, y = 12 17. x = 7, y = 35 18. x = 12, y = 48
19.x = 15, y = 90 20.x = 22, y = 11 21.x = 9, y = -3
22. x = — 1, y = — 1 23. x = —4, y = 40 24. x = 8, y = —56
RECOGNIZING DIRECT VARIATION In Exercises 25 and 26, state whether
the two quantities model direct variation.
25. BICYCLING You ride your bike at an average speed of 14 miles per hour.
The number of miles m you ride during h hours is modeled by m = 14 h.
26. Geom etry Link / The circumference C of a circle and its diameter d are
related by the equation C = i\d.
! Student HeCp
► Homework Help
Example 1: Exs. 12-26
Example 2: Exs. 27-30
Example 3: Exs. 34, 35
Example 4: Exs. 36, 37
4.6 Direct Variation
VIOLIN FAMILY The string
section of a symphony
orchestra has more than half
the musicians, and consists
of from 20 to 32 violins, 8 to 10
violas, 8 to 10 cellos, and
6 to 10 string basses.
Standardized Test
Practice
GRAPHING EQUATIONS Graph the equation.
27. y = 4x 28. y = — 3x 29. y = — x 30. y =
RECOGNIZING DIRECT VARIATION GRAPHS In Exercises 31-33, state
whether the graph is a direct variation graph. Explain.
1
O' '
-
1
-1
5
X
— 3
33.
34. Science Weight varies directly with gravity. With his equipment,
Buzz Aldrin weighed 360 pounds on Earth but only 60 pounds on the moon.
If Valentina V. Tereshkova had landed on the moon with her equipment and
weighed 54 pounds, how much would she have weighed on Earth with her
equipment?
35. TYPING SPEED The number of words typed varies directly with the time
spent typing. If a typist can type 275 words in 5 minutes, how long will it
take the typist to type a 935-word essay?
VIOLIN FAMILY In Exercises 36 and 37, use the following information.
The violin family includes the bass, the cello, the
viola, and the violin. The size of each instrument
determines its range. The shortest produces the
highest notes, while the longest produces the
deepest (lowest) notes.
Total
length
Violin family
Bass
Cello
Viola
Violin
Total length, t (inches)
72
47
26
23
Body length, b (inches)
44
30
?
14
Body
length, b
36. Write a direct variation model that relates the body length of a member of the
violin family to its total length. HINT: Round each ratio to the nearest tenth.
Then write a direct variation model.
37. Use your direct variation model from Exercise 36 to estimate the body length
of a viola.
38. MULTIPLE CHOICE Find the constant of variation of the direct variation
model 3x = y.
(A) 3 CD | ©1 © -3
39. MULTIPLE CHOICE The variables x and y vary directly. When x = 4,
y = 24. Which equation correctly relates x and y?
© x = 4y © y = 4x © x = 6y © y = 6x
Chapter 4 Graphing Linear Equations and Functions
Mixed Review
Maintaining Skills
Quiz 2
SOLVING EQUATIONS Solve the equation. (Lesson 3.3)
40. lx + 30 = —5 41. Ay = 26 — 9 y 42. 2 (w — 2) = 2
43. 9x + 65 = -Ax 44. 55 - 5y = 9y + 27 45. 7a - 3 = 4(a ~ 3)
FUNCTIONS In Exercises 46 and 47, solve the equation for y. (Lesson 3.7)
46. 15 = 7(x — y) + 3x 47. 3x + 12 = 5(x + y)
48. HOURLY WAGE You get paid $152.25 for working 21 hours. Find your
hourly rate of pay. (Lesson 3.8)
CHECKING SOLUTIONS Determine whether the ordered pair is a solution
of the equation. (Lesson 4.2)
49 . jc - y = 10, (5, -5) 50. 3x - 6 y= -2, (-4, -2)
51. 5a; + 6 y = — 1, (1, —1) 52. —4x — 3y = — 8, (—4, 2)
53. 3x + 4 y = 36, (4, 6) 54. 5x - 3y = 47, (2, 9)
EVALUATING EXPRESSIONS Find the number with the given prime
factorization. (Skills Review p. 761)
55. 2 • 3 • 11 56. 3 • 5 • 7 57. 2 3 • 7
58. 5 3 * 7 * 11 59. 2 • 3 • 5 • 7 • 17 60. 2 6 • 3 • 5 6
Find the x-intercept and the /-intercept of the line. Graph the equation.
Label the points where the line crosses the axes. (Lesson 4.4)
1 . y = 3x + 6 2. y — 8x = — 16 3. x — y = 10
4. 2x — y = 5 5. 4x + 2y = 20 6. x — 2y = 8
Find the slope of the line passing through the points. (Lesson 4.5)
7. (0, 0), (5, 2) 8. (1, -3), (-4, -5) 9. (3, 3), (-6, -4)
10. (-3, 2), (-5, -2) 11. (0, -4), (5, -4) 12. (1, -2), (-7, 6)
The variables x and y vary directly. Use the given values to write an
equation that relates xand y. (Lesson 4.6)
13. x = 3, y = 9 14. x = 5, y = 40 15. x = 15, y = 60
In Exercises 16-18, graph the equation. (Lesson 4.6)
16. y = 5x 17. y = — 6x 18. y = lOx
19. The number of bolts b a machine can make varies directly with the time t it
operates. The machine can make 4200 bolts in 2 hours. How many bolts can
it make in 5 hours?
4.6 Direct Variation
DEVELOPING CONCEPTS
For use with
Lesson 4.7
Goal
Determine the effect that
the slope and y-intercept
have on the graph of
y=mx+b.
Materials
• graph paper
• pencil
Question
How do the slope and /-intercept affect the graph of /= mx+ b ?
Explore
*0^===-*-^ I, r~ ^
© Graph each equation on the same coordinate plane. Describe any
patterns you see.
a. y = 2x
b. y = 2x + 2
c- y = 2x — 2
© For each equation in Step 1, give the slope of the line and write the
coordinates of the point where the graph crosses the y-axis.
© Graph each equation on the same coordinate plane. Describe any
patterns you see.
a- y = x
b. y = 2x
c. y = 3x
© For each equation in Step 3, give the slope of the line and write the
coordinates of the point where the graph crosses the y-axis.
Think About It
1. Based on your results in Steps 1 and 2, predict what the graph of
y = 2x + 5 will look like. Predict the y-intercept. Explain your prediction.
2 . Test your prediction by graphing the equation y = 2x + 5.
3_ Based on your results in Steps 3 and 4, predict what the graph of y = 5x will
look like. Predict the slope. Explain your prediction.
4. Test your prediction by graphing the equation y = 5x.
5- Based on your observations, what information do you think the numbers
m and b give you about a graph? Use graphs to support your answer.
Chapter 4 Graphing Linear Equations and Functions
Graphing Lines Using
Slope-Intercept Form
Goal
Graph a linear equation
in slope-intercept form.
How can you estimate production costs?
Key Words
• slope
• /-intercept
• slope-intercept form
• parallel lines
In Example 3 you will use the
graph of a linear model to estimate
the production costs for a small
hat-making business.
In Lesson 4.5 you learned to find the slope of a line given two points on the line.
There is also a method for finding the slope given an equation of a line.
SLOPE-INTERCEPT FORM OF THE EQUATION OF A LINE
The linear equation y = mx + b is written in slope-intercept form,
where m is the slope and b is the /-intercept.
slope /-intercept
* *
y = mx + b
If y = 2x + 3, then (0, 3) and (1,5) are on the line
and the slope is 2. More generally, if y = mx + b,
then (0, b ) and (1, m + b) are on the line and the
slope is
(m + b) — b
1 - 0
= m .
| Find the Slope and y-lntercept
Find the slope and y-intercept of 2x — y = —3.
Solution Rewrite the equation in slope-intercept form.
2x — y = —3 Write original equation.
— y = — 2x — 3 Subtract 2xfrom each side.
y = 2x + 3 Divide each side by -1. m = 2 and b = 3.
ANSWER ► The slope is 2. The y-intercept is 3.
4.7 Graphing Lines Using Siope-lntercept Form
2 Graph an Equation in Slope-Intercept Form
Graph the equation y = —3x + 2.
O Find the slope, —3, and the j-intercept, 2.
© Plot the point (0, b) when b is 2.
© Use the slope to locate a second point on the line.
_ —3 _ rise move 3 units down
1 run move 1 unit right
O Draw a line through the two points.
Graph an Equation in Slope-intercept Form
1. v = —2x + 3 2 . y — Ax — 5
3. v = -jx + 2
3 Use a Linear Model
PRODUCTION COSTS Chai has a small business making decorated hats.
Based on data for the last eight months, she calculates her monthly cost y of
producing x hats using the function y = 1.9x + 350.
a. Explain what the y-intercept and slope mean in this model.
b. Graph the model. Then use the graph to estimate the cost of 35 hats.
Solution
a. The y-intercept is 350. This means that her initial cost was $350. The slope
is 1.9. This means that her cost will increase at a rate of $1.90 for each hat
she makes.
b. Graph the line by using the slope to find a second point.
ANSWER ► From the graph, the cost of making 35 hats will be about $416.50.
Use a Linear Model
4. Use the graph in Example 3 above to find the cost of making 60 hats.
Chapter 4 Graphing Linear Equations and Functions
PARALLEL LINES Parallel lines are different lines in the same plane that never
intersect. Two nonvertical lines are parallel if they have the same slope and
different y-intercepts. Any two vertical lines are parallel.
Parallel
vertical
lines
Student ttadp
p More Examples
More examples
are available at
www.mcdougallittell.com
4 Identify Parallel Lines
Which of the following lines are parallel?
line a\ ~x + 2y = 6 line b\ ~x + 2 y = —2
Solution
Q Rewrite each equation in slope-intercept form,
line a\ y — ^x + 3 line b: y = ^x — l
line c: x + 2y = 4
line c\ y — — ^x + 2
© Identify the slope of each equation.
The slope of line a is The slope of line b is The slope of line
c is —-
© Compare the slopes.
Lines a and b are parallel because each has a slope of
Line c is not parallel to either of the other two lines because it has a
slope of —
CHECK / The graph gives you a
visual check. It shows that line
c intersects each of the two
parallel lines.
ANSWER ^ Line a and line b are parallel.
Identify Parallel Lines
5. Which of the following lines are parallel?
line a: 3x + 2y = 6 line b: 3x — 2y = 6
line c: 6x + 4y = 6
4.7 Graphing Lines Using Siope-lntercept Form
Exercises
Guided Practice
Vocabulary Check 1 . Complete: In the equation y = 5x — 7, ? is the ^-intercept.
2. Complete: Two nonvertical lines are parallel if they have the same ? and
different ? .
Skill Check In Exercises 3-8, find the slope and /-intercept of the equation.
3. y = 2x + 1 4. y = 8.5x 5. 5x — y = 3
6. y - x = 3 7.y + x = 15 8. y = ^x + 6
9. Which equation best represents the
graph at the right?
A. y = — 2
B. y — 2x — 2
C. y = 2x
10. Are the graphs of y = x + 2 and y = x — 4 parallel lines?
Practice and Applications
SLOPE-INTERCEPT FORM Rewrite the equation in slope-intercept form.
11-— x + y = 9 12. 3x + y = — 11 13. lOx — 5y = 50
14. y — 4x — 9 15. 2y + 12 = x 16. 3x — 6y = 18
Student HeCp
► Homework Help
Example 1: Exs. 11-25
Example 2: Exs. 26-48
Example 3: Exs. 56-62
Example 4: Exs. 49-54,
63, 64
SLOPE AND ^-INTERCEPT Find the slope and /-intercept of the graph of
the equation.
17. y = 6x + 4
18. y = 3x - 7
19. y = 2x — 9
20. y = ^x - 3
IS)
1
II
o
22. y = —2
23. 12x + 4y = 24
24. 3x + 4y= 16
25. ly - 14x = 28
GRAPHING LINES
Graph the equation.
26. y = x + 3
27. y = 2x - 1
28. y = x + 5
29. y = ~~x + 4
30. y = 6 — x
31 . y = 3x + 1
32.y = 4x + 4
33. y = x + 9
34. y = 2 X
Chapter 4 Graphing Linear Equations and Functions
GRAPHING LINES Write the equation in slope-intercept form. Then graph
the equation.
35. x + y = 0 36. 3x — 6y = 9
37. 4x + 5y = 15 38. 4x - y - 3 = 0
39. x — y + 4 = 0 40. 2x — 3y — 6 = 0
41. 5x + 15 + 5_y = lOx 42. 2x + 2y — 4 = x + 5
MATCHING EQUATIONS AND GRAPHS Match the equation with
its graph.
43. y = j^x + 1 44. y = j^x — 1 45. y = x + 2
Link to
transportation
.. jfi&i t*i = ,
CABLE CARS The first cable
car debuted in 1873, on Nob
Hill in San Francisco. It rolled
down the slope at a stately
pace of 9 miles an hour. By
1890, San Francisco had
8 cable car companies,
running 600 cars over more
than 100 miles of track.
INTERPRETING A GRAPH Identify the slope and /-intercept of
the graph.
o
/
J
/
/
/
-1
]
L
5 X
PARALLEL LINES Determine whether the graphs of the two equations
are parallel lines. Explain your answer.
49. line a: y — — 3x + 2
50. line a\ 2x — 12 — y
line b\ y + 3x = —4
line b\ y = 10 + 2x
51. line a: y — x + 8
52. line a\ 2x — 5y — —3
line b: x — y = —1
line b\ 5x + 2y = 6
53. line a: y + 6x — 8 = 0
54. line a: 3y — 4x = 3
line b\ 2y = 12x — 4
line b\ 3 y = ~4x + 9
55, CABLE CARS In the 1870s, a cable car system was built in San Francisco
to climb the steep streets. Find the steepness of the street sections shown
below by calculating each labeled slope from left to right in the diagram.
(cross section of side view)
4.7 Graphing Lines Using Siope-lntercept Form
SNOW In Exercises 56-58, use the following information.
Snow fell for 9 hours at a rate of ^ inch per hour. Before the snowstorm began,
there were already 6 inches of snow on the ground. The equation y = ^x + 6
models the depth y (in inches) of snow on the ground after x hours.
56. What is the slope of y = + 6? What is the y-intercept?
57. CRITICAL THINKING Explain what the slope and y-intercept represent in
the snowstorm model.
58. Graph the amount of snow on the ground during the storm.
SNOW Snow crystals form
on microscopic particles,
called ice nuclei, which are
present in clouds with below-
freezing temperatures.
SAVINGS ACCOUNT In Exercises 59 and 60, use the following
information. You have $50 in your savings account at the beginning of the
year. Each month you save $30. Assuming no interest is paid, the equation
5 = 30m + 50 models the amount of money s (in dollars) in your savings
account after m months.
59. Explain what the y-intercept and slope represent in this model.
60. Graph the model. Then use the graph to predict your total savings after
18 months.
PHONE CALL In Exercises 61 and 62, the cost of a long-distance
telephone call is $.87 for the first minute and $.15 for each additional
minute.
61. Let c represent the total cost of a call that lasts t minutes. Plot points for the
costs of calls that last 1, 2, 3, 4, 5, and 6 minutes.
62. CRITICAL THINKING Draw a line through the points you plotted in
Exercise 61. Find the slope. What does the slope represent?
63. PARALLEL LINES Which of the following lines are parallel?
line a: — 2x + y = 10 line b : ~6x + 3y = 13 line c : — 2x — y = 6
64. PARALLEL LINES Write an equation of a line that is parallel to
y — 4x — 5 but has a y-intercept of 3.
CHALLENGE A parallelogram is a quadrilateral with opposite sides
parallel. Determine whether the figure is a parallelogram by using slopes.
Explain your reasoning.
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 65-66 is available at
www.mcdougallittell.com
Chapter 4 Graphing Linear Equations and Functions
Stsndsfdiz&d Test 67. MULTIPLE CHOICE What is the slope of the graph of the equation
Practice y + 8 = o?
(a) Undefined CD 1 CD 0 (D — 1
68. MULTIPLE CHOICE Write the equation 6x - 9y + 45 = 0 in
slope-intercept form.
CD y = j-x - 5 <D j = -|x + 5 (H) y = jx + 5 CD y = \x - 5
69. MULTIPLE CHOICE What is the slope of a line parallel to the graph of the
equation 16.v — 32v = 160?
(A) 2 d| ©5 ® -5
SOLVING EQUATIONS In Exercises 70-78, solve the equation.
(Lessons 3.1, 3.2, and 3.3)
70. x + 6 = 14 71.9 -y = 4 72.76 = 21
73. | = 3 74. \h ~2 = 1 75. 3x - 12 = 6
76. 2(v+l) = 4 77. 3(x - 1) =-18 78. 5(w - 5) = 25
79. Science £ " : You are studying the atomic numbers and weights of
elements. You record several pairs in a table. Make a scatter plot. Then
describe the relationship between the atomic numbers and the atomic
weights. (Lesson 4.1)
Element
H
He
Li
Be
B
C
N
O
Atomic Number
1
2
3
4
5
6
7
8
Atomic Weight
1.0
4.0
6.9
9.0
10.8
12.0
14.0
16.0
Note: The abbreviations above are for the following elements: Hydrogen,
Helium, Lithium, Beryllium, Boron, Carbon, Nitrogen, and Oxygen.
80. COIN COLLECTION You have 32 coins in ajar. Each coin is either copper
or silver. You have 8 more copper coins than silver coins. Let c be the
number of copper coins. Which equation correctly models the situation?
(Lesson 3.5)
A. (c — 8) + c = 32 B. c + (c + 8) = 32
Maintaining Skills
ADDING FRACTIONS Add. Write the answer as a fraction or as a mixed
number in lowest terms. (Skills Review p. 764)
1 , 1
81 -8 + 5
“■f + !
M 2 , 4
82 3 + 5
“•i + f
83 '4 + 9
87 2 + A
87 ‘ 3 + 21
84 — + —
11 33
88 ‘ 24 + 12
4.7 Graphing Lines Using Slope-Intercept Form
Lesson 4.7 ^-intercept. With a graphing calculator or a computer, you can graph a linear
equation and find solutions.
Student HeCp
► Keystroke Help
$ ee keystrokes for
severa | models of
calculators at
www.mcdougallittell.com
Sampte
Use a graphing calculator to graph the equation 2x — 3y = 33.
Solution
© Rewrite the equation in
function form.
e
iaiia
ENTER
X
2x-3y~ 33
-3y = -2x -+-33
Student HeCp
► Study Tip
Xmin means the
minimum x-value,
Xmax means the
maximum x-value, and
Xscl is the number of
units between the tick
marks.
k j
© Press
of the graph.
to set the size
© Press to graph the
equation. A standard viewing
window is shown.
0 To see the point where the graph crosses the x-axis, you can adjust the
viewing window. Press flJjESI and use the arrow to enter new
values. Then press to graph the equation.
Chapter 4 Graphing Linear Equations and Functions
p
Student HeCp
► Study Tip
You can continue to
use zoom until the
/-coordinate is to the
nearest tenth,
hundredth, or any
other decimal place
you need.
Samplt
2 45
Estimate the value of y when x = —7 in the equation y = —x —
J o
Solution
Q Graph the equation y
2 45
-x —— using a
viewing window that will show the graph
when x ~ —7.
WINDOW
Xmin = -10
Xma x = 5
Xsc 1 = 1
Ymin = -15
Yma x = 5
Ysc 1 = 1
Q Press
j and a flashing cursor
appears. The x-coordinate and /-coordinate
of the cursor’s location are displayed at the
bottom of the screen. Press the right and left
arrows to move the cursor. Move the trace
cursor until the x-coordinate of the point is
approximately —7.
1 1 1 1 1 1 1 1 1
1 X=-7.02 Y=-
10.305
© Use the feature to get a more
accurate estimate. A common way to zoom is
and select Zoom In. You
to press |
now have a closer look at the graph at that
point. Repeat Step 2.
ANSWER ^ When x = -7, y « -10.3.
TryTfcest
Use the standard viewing window to graph the equation.
2 . y = 2x + 2 3 - x + 2y = — 1 4 . x — 3y = 3
1. y = —2x — 3
Use the indicated viewing window to graph the equation.
5 - y = x + 25
Xmin = -10
Xmax = 10
Xscl = 1
Ymin = -15
Ymax = 35
Yscl = 5
6 ■ y = O.lx
Xmin = -10
Xmax = 10
Xscl = 1
Ymin = -15
Ymax = 1
Yscl = 0.1
7 . y = lOOx + 2500
Xmin = 0
Xmax = 100
Xscl = 10
Ymin = 0
Ymax = 15000
Yscl = 1000
Determine a viewing window appropriate for viewing both intercepts
of the equation.
8. y = x - 330 9 . y = 120x 10 . y = 40,000 - 1500x
Using a Graphing Calculator
Functions and Relations
Goal
Decide whether a
relation is a function and
use function notation.
Key Words
• relation
• function
• vertical line test
• function notation
• linear function
How far does a Monarch Butterfly fly during its
migration?
Some real-life situations can be modeled
by functions. In Exercises 56-58 you
will see that the distance traveled by a
monarch butterfly during its migration is
a function of the traveling time.
Recall that a function is a rule that establishes a relationship between two
quantities, called the input and the output , where for each input, there is exactly
one output. There are other algebraic rules that associate more than one output
with an input. For example, an input-output table corresponding to x = y 2 might
include the following entries.
Input x
0
1
4
4
Output y
0
1
2
-2
Notice that the input v = 4 corresponds to two different outputs, y = 2 and
y = —2. In this case the ordered pairs (0, 0), (1, 1), (4, 2), (4, —2) represent a
relation , one that does not satisfy the requirements for a function. A relation is
any set of ordered pairs. A relation is a function if for every input there is exactly
one output.
Student HaCp
^
p Look Back
For help with domain
and range, see p. 49.
k _ )
1 Identify Functions
Decide whether the relation is a function. If it is a function, give the domain
and the range.
a. Input Output b. Input Output
Solution
a. The relation is a function. For each input there is exactly one output.
The domain is 1,2, 3, and 4. The range is 2, 4, and 5.
b. The relation is not a function because the input 1 has two outputs: 5 and 7.
Chapter 4 Graphing Linear Equations and Functions
When you graph a function or relation, the input is given by the horizontal axis
and the output is given by the vertical axis.
Student MeCp
► Study Tip
You can use your
pencil to check. Keep
your pencil straight to
represent a vertical
line and pass it across
the graph. If it touches
the graph at more than
one point, the graph is
not a function.
^ _ )
Vertical Line Test for Functions
A graph is a function if no vertical line intersects the graph at more
than one point.
v l
i
not a function
c
1 J
_ i
--—=—
T
f 1
not a function
2 Use the Vertical Line Test
Use the vertical line test to determine whether the graph represents a function.
Solution
a. No vertical line can intersect the
graph more than once. So, this graph
does represent y as a function of x.
b. It is possible to draw a vertical line
that intersects the graph twice. So,
this graph does not represent a
function.
Use the Vertical Line Test
Use the vertical line test to determine whether the graph represents
a function.
4.8 Functions and Relations
Student HeCp
>
► Reading Algebra
The symbol f(x) is read
as "the value of fat x"
or simply '7 of x". It does
not mean f times x.
\ _ )
FUNCTION NOTATION When a function is defined by an equation, it is often
convenient to name the function. Just as x is commonly used as a variable, the
letter/is commonly used to name a function. To write a function using function
notation, you use/(x) in place of y.
x-y notation: y = 3x + 2 function notation:/(x) = 3x + 2
3 Evaluate a Function
Evaluate/(x) = 2x — 3 whenx = — 2.
Solution
You can evaluate a function for a given value by substituting the given value for
the variable and simplifying.
fix) = 2x — 3 Write original function.
/(— 2) = 2(— 2) — 3 Substitute -2for*.
= — 7 Simplify.
ANSWER ^ Whenx = — 2,/(x) = — 7.
Evaluate a Function
3. Evaluate/(x) = 4x + 5 when x = 2. 4. Evaluate g(x) = x 2 when x = — 3.
A function is called a linear function if it is of the form/(x) = mx + b. For
instance, the function in Checkpoint 3 is linear. But, the function in Checkpoint 4
is not linear. To graph a linear function, rewrite the function using x-y notation.
Student HeCp
—V
► Study Tip
You don't have to use the
letter f to name a
function. Just as you can
use any letter as a
variable, you can use any
letter to name a function.
\ _ >
B3ZJ2EM 4 Graph a Linear Function
Graph/(x) = — / + 3.
Solution
Q Rewrite the function as y = — ^x + 3.
© Find the slope and the y-intercept.
m = —^ and b = 3
© Use the slope to locate a second point.
© Draw a line through the two points.
{ _
Graph the linear function.
5 -f(x) = 4x — 3 6- /z(x) = —3x + 1
7 . g(x) = ~x + 2
Chapter 4 Graphing Linear Equations and Functions
A,o Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Complete: A relation is any set of ? .
2. Complete: The function/(x) = 6x is a ? function.
Evaluate the function f(x) = —5x— 2 for the given value of x.
3. x = 4 4. x = 0 5. x = —2 6- x
Determine whether the relation is a function. If it is a function, give the
domain and the range.
7. Input Output
8. Input Output
9.
; ioo'
20-
'200
30-
-300
40-
400
&
- 500 j
<
M
k_
15
J
i
b
1
1
A
1 1
¥
5
4
1
1
¥
, T
3
5 x
Determine whether the graph represents a function. Explain your reasoning.
10 .
Practice and Applications
Student HeCp
► Homework Help
Example 1: Exs. 13-18
Example 2: Exs. 19-24
Example 3: Exs. 25-33
Example 4: Exs. 37-45
RELATIONS AND FUNCTIONS Determine whether the relation is a
function. If it is a function, give the domain and the range.
13. Input Output
-a
rr
2—
- 4
3-
- 3
r 2 J
Input
Output
0
2
1
4
2
6
3
8
14. Input Output
[ 4^J ^
0
Input
Output
0
1
2
2
4
3
3
4
15. nput Output
!0 < ^[
9
8
10
18.
Input
Output
1
1
3
2
5
3
7
1
4.8 Functions and Relations
GRAPHICAL REASONING Determine whether the graph represents a
function. Explain your reasoning.
EVALUATING FUNCTIONS Evaluate the function when x = 2, x = 0
and x = -2.
25- /(x) = 3x
28. g(x) — —x — 6
31. h(x) = 8x + 7
26. g(x) = x + 4
29. /(x) = 5x + 1
32. f{x) = —4x + 15
27. /z(x) = 3x — 5
30. /(x) = -x - 3
33. g(x) = 5x — 6
GRAPHICAL REASONING Match the function with its graph.
34. f{x) = 3x — 2
35./(x) = 2x + 2
36 ./(*) = - 2
GRAPHING FUNCTIONS Graph the function.
37. g(x) = 2x — 3 38. /z(x) = 5x — 6
40. /z(x) = 9x + 2 41. /z(x) = — x + 4
43./(x) = — 3x — 2 44. g(x) = —4x — 5
39. f(x) = 4x + 1
42. g(x) = — 2x + 5
45. h(x ) = —~x + 1
Student MeCp
HoMiEin/omc Help
^D‘ v ' Extra help with
problem solving in
Exs. 46-53 is available at
www.mcdougallittell.com
FINDING SLOPE Find the slope of the graph of the linear function f.
46. /(2) = —3,/(—2) = 5 47./(0) = 4,/(4) = 0
48./(-3) = —9,/(3) = 9 49./(6) = -l,/(3) = 8
FINDING DOMAIN AND RANGE Determine whether the relation is a
function. If it is a function, give the domain and range.
50. (1, 3), (2, 6), (3, 9), (4, 12) 51. (-4, 4), (-2, 2), (0, 0), (-2, -2)
52. (3,0), (3, 1), (3,2), (3,-1)
53. (-2,-2), (0, 0), (1, 1), (2, 2)
Chapter 4 Graphing Linear Equations and Functions
Linkup
Music
ZYDECO MUSIC blends
elements from a variety of
cultures. The accordion has
German origins. The rub
board was invented by
Louisiana natives whose
ancestors were French.
Standardized Test
Practice
54. ZYDECO MUSIC The graph shows the
number of people who attended the Southwest
Louisiana Zydeco Music Festival for different
years, where t is the number of years since
1980. Is the number of people who attended
the festival a function of the year? Explain.
► Source: Louisiana Zydeco Music Festival.
Music Festival
55. MASTERS TOURNAMENT The table shows the scores and prize money
earned for the top 7 winners of the 1997 Masters Tournament at Augusta
National Golf Club. Graph the relation. Is the money earned a function of the
score? Explain. If it is a function, give the domain and range. ►Source: Golfweb
Score
270
282
283
284
285
285
286
Prize ($)
486,000
291,600
183,600
129,600
102,600
102,600
78,570
BUTTERFLIES In Exercises
56-58, use the diagram
and caption about monarch
butterfly migration at the
right.
56. Write a linear function
that models the distance
traveled by a migrating
monarch butterfly.
57. Use the model to estimate
the distance traveled after
30 days of migration.
58. Graph your model and
label the point that
represents the distance
traveled after 30 days.
V
Monarch butterflies migrate from the northern United
States to Mexico. The 2000 mile trip takes about 40 days.
59. Scie nce Link y It takes 4.25 years for starlight to travel 25 trillion miles.
Let t be the number of years and let fit) be trillions of miles traveled. Write a
linear function f(t) that expresses the distance traveled as a function of time.
60.
61 -
MULTIPLE
(£>/(*) =
C©/(*) =
MULTIPLE
CD 6
CHOICE Write the equation 3x + y = 5 in function notation.
y + 5 CD fix) ~ ~3x + 5
3x - 5 CD fix) = -y - 5
CHOICE Evaluate the function/(x) = — x + 8 whenx = —2.
CD 10 CD 16 CD -16
4.8 Functions and Relations
Mixed Review
Maintaining Skills
Quiz 3
SOLVING EQUATIONS Solve the equation if possible. Check your
solution. (Lesson 3.4)
62. 4x + 8 = 24 63. 3n = 5n - 12 64. 9 - 5 z = ~8z
65. -5y + 6 = 4y + 3 66. 3b + 8 = 9b - 7 67. ~lq - 13 = 4 - Iq
FINDING SLOPE Find the slope of the line that passes through the
points. (Lesson 4.5)
68 . (0, 3) and (2, 1)
71. (2, 4) and (4, -4)
74. (0, —6) and (8, 0)
69. (2,-3) and (-2, 1)
72. (0, 6) and (8, 0)
75. (2, 2) and (-3, 5)
70. (-1, -3) and (-3, 3)
73. (4, 1) and (6, 1)
76. (0, 0) and (4, 5)
MODELING FRACTIONS Write the fraction that represents the shaded
portion of the figure. (Skills Review p. 768)
Rewrite the equation in slope-intercept form. Identify the slope and the
/-intercept. (Lesson 4.7)
1. v — 4 = 3x 2. x = —y + 2 3. 2x + y = 6
4. 5x + 8y = 32 5. 4x — 3y = 24
6. -27 + 9y + 18 = 0
Graph the equation. (Lesson 4.7)
7. 2x + 4y = 8 8. — 6x — 3y = 21 9. —5x + y = 0
Determine whether the two lines are parallel. (Lesson 4.7)
10. line a: y = —lx + 3 11 . line a: 4x — 8y + 6 = 0
line b\ y — lx — 10 line b\ — 12x + 6 y = 2
Evaluate the function when x = 3, x = 0, and x = -4. (Lesson 4.8)
12. h(x) = —8x 13. g(jt) = 5x — 9 14./(jc) = —4x + 3
15. g(x) = —3x — 12 16. /z(x) = 1.4x 17. f(x) =
Graph the function. (Lesson 4.8)
18. f(x) = —5x 19. /z(x) = 4x — 7 20. g(x) = —6x + 5
Chapter 4 Graphing Linear Equations and Functions
A Chapter Summary
" and Review
• coordinate plane, p. 203
• origin, p. 203
• x axis, p . 203
• y-axis, p. 203
• ordered pair, p. 203
• x-coordinate, p . 203
• y-coordinate, p. 203
• quadrant, p. 204
• scatter plot, p. 205
• linear equation, p. 210
• solution of an equation, p. 210
• function form, p. 211
• graph of an equation, p.211
• constant function, p. 218
• x intercept, p. 222
• y-intercept, p. 222
• slope, p. 229
• direct variation, p. 236
• constant of variation, p. 236
• slope-intercept form, p . 243
• parallel lines, p. 245
• relation, p. 252
• function notation, p. 254
• linear function, p. 254
The Coordinate Plane
Examples on
pp. 203-205
a. What are the coordinates of the point (4, —2)?
b. Plot the point (4, —2) in the coordinate plane.
c. Name the quadrant the point (4, —2) is in.
Solution
a. The point (4, —2) has an x-coordinate of 4 and a ^-coordinate of —2.
b_ To plot the point (4, —2), start at the origin.
Move 4 units to the right and 2 units down.
c. (4, —2) is in Quadrant IV.
!
:
L 4
“1
j
l ■
(
4, -
-2)
In Exercises 1-4, plot the ordered pair in a coordinate plane. Then name
the quadrant the point is in.
1. (4, 6) 2. (0, -3) 3. (-3.5, 5) 4. (-2, -2)
5_ Make a scatter plot of the data in the table.
Time t (hours)
1
1.5
3
4.5
Distance d(miles)
20
24
32.5
41
Chapter Summary and Review
Chapter Summary and Review continued
Graphing Linear Equations
Examples on
pp. 210—212
Use a table of values to graph 3y = 9x — 6.
To graph 3y = 9x — 6, rewrite the equation in function
form, make a table of values, and plot the points.
3 y = 9x — 6
y = 3x — 2
Graph the equation.
6. y = 2x + 2 l. y = 1 — x 8. y = —4(x +1) 9. x — 10 = 2y
Graphing Horizontal and Vertical Lines
Examples on
pp. 216-218
Graph the equation y = —3.
The y-value is always —3, regardless of the value
of x. Here are three solutions of the equation:
(—2, —3), (0, —3), and (2, —3). So, the graph of
y = —3 is a horizontal line 3 units below the x-axis.
Graph the equation.
10. y = 5 11. jc = —6 12.y=l| 13.x = 0
Graphing Lines Using Intercepts
Examples on
pp. 222-224
Graph the equation y + 2x = 10.
To graph y + 2x = 10, first find the intercepts.
y + 2x = 10 y + 2x = 10
0 + 2x = 10 y + 2(0) = 10
x = 5 y = 10
The x-intercept is (5, 0).
The ^-intercept is (0, 10).
Chapter 4 Graphing Linear Equations and Functions
Chapter Summary and Review continued
Graph the equation. Label the intercepts.
14, — x + 4y = 8 15. 3x + 5y = 15 16. 4x — 5y = —20 17. 2x + 3y = 10
The Slope of a Line
Examples on
pp. 229-232
Find the slope of the line that passes through the points (—2, 5) and (4, —7).
To find the slope of the line passing through the points (—2, 5) and (4, —7),
let = (-2, 5) and (x 2 , y 2 ) = (4, -7).
m =
yi~yi
Write formula for slope.
-7-5
Substitute values.
in —
4 - (-2)
in =
-12
6
Simplify.
m =
-2
Divide. Slope is negative.
Find the slope of the line that passes through the points.
18.(2,-1), (3, 4) 19. (0,8), (-1,8) 20. (2, 4), (5, 0) 21. (3, 4), (3,-2)
Direct Variation
Examples on
pp. 236-238
If x and y vary directly and x — 3 when y — 18, write an equation that
relates x and y.
If x and y vary directly, the equation that relates x and y is of the form y = kx.
y = kx Write model for direct variation.
18 = k{ 3) Substitute 3 for x and 18 for y.
6 — k Divide each side by 3.
An equation that relates x and y is y = 6x.
In Exercises 22-25, the variables x and y vary directly. Use the given
values of the variables to write an equation that relates x and y.
22.x = 7, y = 35 23. x = 12, y = -4 24. x = 4, y = -16 25. x = 3, y = 10.5
26. The distance traveled by a truck at a constant speed varies directly with the
length of time it travels. If the truck travels 168 miles in 4 hours, how far will
it travel in 7 hours?
Chapter Summary and Review
Chapter Summary and Review continued
4.7
Graphing Lines Using Slope-Intercept Form
Examples on
pp. 243-245
Graph the equation 4x + y = 0.
O Write the equation in slope-intercept form: y = —4x.
0 Find the slope and the ^-intercept: m = —4,b = 0.
© Plot the point (0, 0). Draw a slope triangle to locate
a second point on the line. Draw a line through the
two points.
3 x
Rewrite the equation in slope-intercept form.
27. 2x + y = 6 28. y — 4x = — 1 29. 2x — 3y = 12 30. 5y — 2x = —10
Graph the equation.
31. y = — x — 2 32. y — 5x = 0 33. x — 4y — 12 34. —x + 6y = —24
4.Z Functions and Relations
Examples on
pp. 252-254
Evaluate the function/(x) = — —x + 1 when x = 5.
fix)
zX + 1
Write original function.
/(5) = ~ (5) + 1
/( 5 ) = 0
Substitute 5 for x.
Simplify.
Evaluate the function for the given value of x. Then graph the function.
35. /(x) — x — 1 when x = —2 36. /(x) = — x + 4 when x — 4
37. f{x) — 2x — 5 whenx = 8 38. /(x) = ^x + 3 whenx — —24
In Exercises 39-42, determine whether the relation is a function. If it is a
function, give the domain and range.
39. nput Output
1
r — \
^ 9
1
n
^ 4
i-
_ J
^_ U
40. Input Output
- 1 -
0 -
PI
pf
1 -
^5
41. Input Output
42. (-2, -3), (-1, -2), (0, -1), (1, 0), (2, 1), (3, 2), (4, 3)
Chapter 4 Graphing Linear Equations and Functions
u.
fiaprzr
Chapter Test
Plot and label the points in a coordinate plane.
1. A(2, 6), B(- 4, - 1), C(-1, 4), D(3, -5) 2. A(-5, 1), B(0, 3), C(-1, -5), D(4, 6)
3. A(7, 3), B(-2, -2) C(0, 4), D(6, -2) 4. A(0, -1), B( 0, 3), C(7, -2), D(2,4)
Without plotting the point, name the quadrant the point is in.
5.(5, -2) 6. (-1,4) 7. (-3,-4) 8. (6, 0)
Use a table of values to graph the equation.
9. y = —x + 3 10. y = 4 11.y=-(5-x)
Graph the equation. Tell which method you used.
13.y = 3x 14. y = 2x — 3 15. 2x + y — 11=0
Find the slope of the line that passes through the points.
17.(0, 1), (-2,-6) 18. (-4,-1), (5,-7) 19. (-3, 5), (2,-2) 20. (-3, 1), (2, 1)
The variables x and y vary directly. Use the given values of the variables
to write an equation that relates x and y.
21. x = —2, y = —2 22. x = 2, y = 28 23.x=-3,y= 15 24. x= 13,y = 39
Rewrite the equation in slope-intercept form.
25. -lx - y = -49 26. 18 - y - 4x = 0 27. |x + y - 9 = 0 28. x - 2y = 10
Determine whether the graphs of the two equations are parallel lines.
Explain your answer.
29. y — Ax + 3 and y — —Ax — 5 30. lOy + 20 = 6x and = 3x + 35
In Exercises 31-33, evaluate the function when x = 3, x = 0, and x = -4.
31./(x) = 6x 32. g(x) = 3x + 8 33. f(x) = ~(x — 2)
34. SHOE SIZES The table below shows how foot length relates to women’s
shoe sizes. Is shoe size a function of foot length? Why or why not? If it is a
function, give the domain and range.
Foot length x(in inches)
9—
y 4
9—
z
9—
8
9—
16
<
<
Shoe size y
4
7
7
8
8
4
4
12. x = 6
16. y — 4x = 1
Chapter Test
Chapter Standardized Test
Tip
Read all of the answer choices before deciding
which is the correct one.
1, What is the equation of the line shown?
jy
/ (2,0)
(
X
— 2
-6
1
1
MO,
-9)
(A) 9x - 2y = -18
Cg) — 9x — 2 v = 18
(g) 9x + 2y = 18
(g) —9x + 2y — —18
2 . What is the ^-intercept of the line
—Ax — ^-y = 10?
(A) -20 (g) -4
Cg) 20 (g) 5
3. Write the equation 3x — Ay = 20 in slope-
intercept form.
(A) j = |x + 5
CD J = ~\x + 5
CD y = - 5
CD J = - 5
4. Find the slope of the line passing through
the points (1,2) and (2, 1).
(3)1 CD -2 CD 2 CD-l
Chapter 4 Graphing Linear Equations and Functions
5. What is the slope of the line shown?
CD None of these
6 . What is the slope of the graph of the
equation 5 x — y = —2?
(A) -5 CD 5
CD 1 CD -2
7. Which point does not lie on the graph of
x= -12?
(A) (-12, 0)
CD (-12, -12)
CD (-12,1)
CD (-1, -12)
8 . What is the x-intercept of
— 13 x-y= -65?
(A) 5 CD -5
CD 65 CD -65
9. Which point is in Quadrant III?
(3) (2, -3) CD (-4, -5)
CD (-6, 4) CD (3, 2)
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
i Dividing Fractions
7 21
Find the quotient of — -r- —
Solution
7
8
21 = 7 16
16 8 * 21
= j_12
168
= 2 » 2 » 2 * 2»7
2*2 m 2* 3 •/
2
3
Multiply by the reciprocal of the second fraction.
Multiply the numerators and the denominators.
Factor numerator and denominator.
Simplify fraction to simplest form.
Try These
Find each quotient. Write each answer in lowest terms.
1.
1
4
J.
2
_ 5 15
4 -l2^l6
'■ 10 * 25 9 * 27 10 ■
Student HeCp
► Extra Examples
More examples
gnc j p ract j ce
exercises are available at
www.mcdougallittell.com
£212219 2 Order of Operations
Evaluate the expression 36 -r- (8 — 5) 2 — (—3)(2).
Solution
36 (8 — 5) 2 - (-3)(2)
= 36 h- (3) 2 - (-3)(2)
Do operations within parentheses first.
= 36 - 9 - (— 3)(2)
Evaluate power.
= 4 - (-6)
Do multiplication and division.
= 10
Add.
Try These
Evaluate the expression.
9. 4 — 8 h- 2
10. 2 2 • 3 - 3
11. 2(3 - 4) - (-3) 2
12. 2 2 + 4[16 - 5 - (3 - 5)]
13. 3 — 2[8 — (3 — 2)]
14. 6+^2 t ~ 2
2 2 + 2
Maintaining Skills
f How can you figure out how old
an object is?
.•It,
APPLICATION: Archaeology
1. What is the radiocarbon age of an object whose
actual age is about 1000 years?
2 . What is the actual age of an object whose
radiocarbon age is about 5000 years?
Learn More About It
You will learn more about radiocarbon dating in
Exercises 33 and 34 on p. 303.
Archaeologists study how people lived in past
times by studying the objects those people left behind.
They often use a method called radiocarbon dating to
estimate the age of certain objects.
Think & Discuss
In Exercises 1 and 2, use the graph below.
Radiocarbon Dating
8000
£
re
^ 6000
o
re
c
o
! 4000
u
o
re
2000
0
2000 4000 6000
Actual age (years)
8000
Study Guide
PREVIEW
What’s the chapter about?
• Writing linear equations in slope-intercept form, point-slope form, and
standard form
• Using a linear model to solve problems
• Writing an equation of a line perpendicular to another line
Key Words
• point-slope form, p. 278 • linear model, p. 298 • perpendicular, p. 306
• standard form, p. 291 • rate of change, p. 298
PREPARE
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to p . 222 )
1. What are the x-intercept and ^-intercept of the line shown in the graph?
(A) x-intercept: —6
^-intercept: —2
CD x-intercept: —2
^-intercept: —6
(Tp x-intercept: 2
^-intercept: 6
CD x-intercept: 6
y-intercept: 2
Skill Check (refer to pp. 155 - 157 , 201 )
2 . Solve the equation 4(x + 8) = 20x.
(A) x — 2 CD x = f CD x = 2
3. What are the coordinates of point R7
(5) (-3,-1) CD (-4,-1)
CD (-4,1) CD (-1,-4)
-u
lz 10 1
-6 \
X
+
*
CD
2y= -12\
□
1
V
CD x = 3
l
4
/?
5
l
-l
]
1 X
STUDY TIP
Create a
Practice Test
Exchange practice tests
with a classmate. After
taking the tests, correct
and discuss the answers.
Practice Test foi
e ' IH tetceptf orni
^Chapters
z - 1 Write in
form the
s iope-intercept
Ration of
Chapter 5 Writing Linear Equations
Slope-Intercept Form
Goal
Use slope-intercept form
to write an equation of a
line.
Key Words
• slope
• /-intercept
• slope-intercept form
How have Olympic hurdling times decreased?
A graph can describe a trend, such as the decrease in Olympic men’s winning
hurdling times. In Exercise 40 you will write the equation of the line that models
this trend.
You can write an equation of a line if you know the slope and the y-intercept.
SLOPE-INTERCEPT FORM
The slope-intercept form of the equation of a line with slope m
and /-intercept b is
/ = mx + b
i Equation of a Line
Write the equation of the line whose slope is 3 and whose y-intercept is —
Solution
Q Write the slope-intercept form.
y = mx + b
© Substitute slope 3 for m and —4 for b.
y = 3x + (-4)
© Simplify the equation.
y = 3x - 4
ANSWER ► The equation of the line is y = 3x —
4.
Write the equation of the line described below.
1. The slope is —2 and the y-intercept is 7.
2
2 . The slope is ^ and the y-intercept is —6.
5.1 Slope-In tercep t Form
EQUATIONS FROM GRAPHS When a graph clearly indicates the y-intercept and
another point on the line, you can use the graph to write an equation of the line.
Student HeCp
^
► Study Tip
Recall that the
/-intercept is the
/-coordinate of the
point where the line
crosses the /-axis.
2 Use a Graph to Write an Equation
Write the equation of the line
shown in the graph using
slope-intercept form.
Solution
0 I/I /rite the slope-intercept form y = mx + b.
0 Find the slope m of the line. Use any two points on the graph.
Let (0, 2) be (x v y x ) and let (5, 6) be (x 2 , y 2 ).
m =
rise
run
y i — y i
6-2
5-0
= 4
5
© Use the graph to find the
/-intercept b. The graph of
the line crosses the y-axis at
(0, 2). The y-intercept is 2.
0 Substitute slope ^ for m and 2 for b in the equation y = mx + b.
4 4-0
j = 5 x + 2
4
ANSWER The equation of the line is y = — x + 2.
Chapter 5 Writing Linear Equations
Link to
Science
SPACE SHUTTLE
LANDING The space shuttle
lands as a glider with no
power. The shuttle begins its
approach at an altitude of
12,000 feet.
More about space
shuttles is available at
www.mcdougallittell.com
Student HeGp
► Study Tip
Notice that the scales
on the axes are different.
In this case you cannot
calculate slope by
counting squares.
Instead you must use
the formula.
i- J
3 Model Negative Slope
SPACE SHUTTLE LANDING
The graph at the right models
the negative slope of the space
shuttle as it descends from
12,000 to 2000 feet. Write the
equation of the line in slope-
intercept form.
Solution
Q Write the slope-intercept form y = mx + b.
0 Find the slope m of the line. Use any two points on the graph.
Let (0, 12) be (x v y x ) and let (28, 4) be (jt 2 , y 2 ).
m =
y 2 -yi
4-12
28-0
-8
28
2
7
© Use the graph to find the y-intercept b. The graph of the line crosses the
y-axis at (0, 12). The y-intercept is 12.
2
0 Substitute slope — j for m and 12 for b in the equation y = mx + b.
y = —jx + 12
ANSWER ► The equation of the line is _y = — jx +12.
5.1 Slope-Intercept Form
Exercises
Guided Practice
Vocabulary Check 1. What is the name used to describe an equation in the form y = mx + bl
2. Identify the slope of the line that has the equation y = —4x + 15.
3. Name the y-intercept of the line that has the equation y = 10x — 3.
Skill Check Determine whether the equation is in slope-intercept form.
4 . y = —Sx — 11 5 . y — 4 = 5(x + 3) 6 - x + 23y = —15
Write in slope-intercept form the equation of the line described below.
7. Slope = 1, y-intercept = 0 8. Slope = — 7, y-intercept = — 2
9 . Slope = — 1, y-intercept = 3 10 , Slope = 0, y-intercept = 4
11, Slope = 5, y-intercept = 5
12 , Slope = 14, y-intercept = —6
Practice and Applications
WRITING EQUATIONS Write in slope-intercept form the equation of the
line described below.
13 - m — 3, b — 2
16 - m = 10, b = 0
19 - m — ~\,b = — -|
14 . m — 1, b — — 1
17 . m = j t b = l
20.m = 0,b = 0
15 . m — 0, b — 6
18 . m = —4, b = — j
O* 1 7 2
21 . m = — y, b = —
Identify the slope and y-intercept of the line.
Student HaCp
► Homework Help
Example 1: Exs. 13-21
Example 2: Exs. 22-39
Example 3: Exs. 22-39
25 .
Chapter 5 Writing Linear Equations
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 28-33 is available at
www.mcdougallittell.com
GRAPHICAL REASONING Write in slope-intercept form the equation of
the line shown in the graph.
MATCHING Give the letter of the equation that matches the graph.
A.y = x + 2
B. y = — x + 2
C. y = x - 2
D. y = x + 1
E.y= 1
F. y = x
35.
I Student HeCp
^ ^
► Study Tip
When calculating the
slope in Exercise 40,
notice that the scale
on the y -axis
represents 0.2 seconds
and the scale on the
x-axis represents
4 years.
x _>
HURDLING The graph approximates winning times in the Olympic men's
110 meter hurdles. The /-intercept is 13.64 seconds.
v DATA UPDATE of Olympic mens 110 meter hurdling times atwww.mcdougallittell.com
40. Write the equation of the
line shown in the graph.
41. Use the equation from
Exercise 40 to estimate
the winning time in 1984.
42. Use the graph to predict the
winning time in 2004.
43. How realistic do you think
your prediction is? Explain.
5.1 Slope-In tercep t Form
Student Hedp
►Vocabulary Tip
This diagram will help
you remember how to
spell parallel:
p a r a(fj)e I
two parallel lines
Link to
Science
OLD FAITHFUL geyser in
Yellowstone National Park
has erupted every day at
intervals of less than two
hours for over 100 years.
More about Old
4®”' Faithful is available at
www.mcdougallittell.com
Geometry Link s The graph at the
right shows three parallel lines.
44. Write the equation of each
line in slope-intercept form.
45. Compare the equations.
What do you notice?
46. Write the equation for a line
with y-intercept — 1 that is
parallel to the three lines
shown.
The graph shows a
regulation-sized baseball diamond.
The units are in feet.
47. Write the equation of each solid
line in slope-intercept form.
48. Compare the equations. What
do you notice?
49. Write the equation for the
dashed line.
CHALLENGE In Exercises 50 and 51,
use the following information.
You walk home from school at a rate
of 4 miles per hour. The graph shows
your distance from home. Notice the
units on the axes.
50. Write the equation of the line.
51. Use your equation to find how far
3
from home you are after — hour.
o
Science Link y In Exercises
52 and 53, use the following
information.
The time y until the next
eruption of Old Faithful
depends on the length
x of each eruption.
The y-intercept is 32.
52. Write the equation of the
line shown in the graph.
53. The eruption that just ended
had length 5 minutes. Estimate
the time until the next eruption.
Eruptions of Old Faithful
Chapter 5 Writing Linear Equations
Standardized Test
Practice
Mixed Review
Maintaining Skills
The United States Bureau of the Census predicts that the population of
Florida will be about 17.4 million in 2010 and then will increase by about
0.22 million per year until 2025.
54. MULTIPLE CHOICE Choose the equation that predicts the population y of
Florida (in millions) in terms of x, the number of years after 2010.
(A) y = 17.4x + 0.22 Cb) y = —0.22x + 17.4
Cc) y = 0.22x + 17.4 Cg) y = —11 Ax + 0.22
55. MULTIPLE CHOICE According to the prediction, about how many millions
of people will live in Florida in 2011?
CD 17.18 CD 17.5 CD 17.62 CD 19.91
EVALUATING EXPRESSIONS Evaluate the expression when
X =
-3 and y =
6. (Lesson 2.8)
56.
3x
57. *
58.
x • y
x + y
x + 2
59.
2x
60. x 2 y
61.
— 8r
y
-4 y
FINDING SOLUTIONS Find three solutions of the equation. (Lesson 4.2)
62. y = 6x + 3
65. y = —5x + 7
63. y — x 4
66 . x + y = 1
„ 1
64. y = -x
67. x + 3y = 9
GRAPHING LINEAR EQUATIONS Find the slope and the /-intercept of the
graph of the equation. Then graph the equation. (Lesson 4.7)
68. y + 2x = 2 69. 3x — y = — 5 70. 9x + 3 y = 15
71. 4x + 2y = 6
72. 4 y + 12x = 16
73. 25x - 5y = 30
PERCENTS AND FRACTIONS Write the percent as a fraction or as a
mixed number in simplest form. (Skills Review p. 768)
74. 50%
75. 75%
76. 1%
77. 62%
78. 100%
79. 0.5%
80. 5%
81. 128%
82. 501%
83. 6%
5.1 Slope-In tercep t Form
DEVELOPING CONCEPTS
For use with
Lesson 5.2
Goal
Develop the point-slope
form of the equation of
a line.
Materials
• pencil
• ruler
• graph paper
Question
tigs- 1 ^
How can you write an equation of a line
given the slope and a point on the line?
Given a point on a line and the slope of the
line, you have enough information to write
the equation of the line. The steps below
show how to find the equation of the line
that passes through the point (2, 1) with
slope
Explore
Q On a coordinate grid, draw the line
that passes through the point (2, 1)
with slope Use a slope triangle
and label the second point you find
with the ordered pair (. x , y).
Q Draw the slope triangle that shows
the rise and run of the line through
(2, 1) and (x, y).
0 Explain why the rise is given by y — 1.
Explain why the run is given by x — 2.
Q Use these values of rise and run to express
the slope of the line.
I = y 1 1
3 x-2
© Clear the denominator on the right-hand side of the equation by multiplying
both sides by (x — 2). Then y — 1 = }r(x — 2) is the point-slope form of the
1
equation of the line passing through (2, 1) with slope —.
Think About It
Follow Steps 1-5 above for the line described below.
1. Passes through (2, 3) with slope —
4
2 . Passes through (—4, —2) with slope —.
Chapter 5 Writing Linear Equations
GENERAL FORMULA The following steps lead you to a general formula for
writing the equation of a line given a point on the line and the slope of the line.
The formula is called the point-slope form of the equation of a line. We use
(x p yf) as the given point and m as the given slope.
Explore
O Sketch the line passing through (x p yf) and (x, y ), where
(x, y) represents any other point on the line.
Q Draw the slope triangle that shows the rise and run of the
line through (x p y x ) and (x, y).
© Explain why the rise is given by y — y v Explain why the
run is given by x — x r
Q Use the values of rise and run to express the slope of the line.
y-yi
m = -
x — x x
© Clear the denominator on the right-hand side of the equation by multiplying
each side of the equation by (x — xf). You get y — y 1 = m(x — xf), the general
formula for the point-slope form of the equation of a line.
Think About It
Use the general formula from Step 5 above to write the equation in
point-slope form of the line that passes through the given point and has
the given slope.
4. (3, 2), m = 5
Developing Concepts
Point-Slope Form
Goal
Use point-slope form to
write the equation of a
line.
Key Words
• slope
• point-slope form
How much pressure is on a diver?
As you saw in Developing
Concepts 5.2, page 276, you
can write an equation of a line
given the slope and a point on the
line. In Exercise 44 you will write
an equation that models the
pressure on a diver.
POINT-SLOPE FORM
[ The point-slope form of the equation of the line through (x y yj
with slope m is y - = m(x - x 1 ).
Student HaCp
► Study Tip
Remember that you
can calculate slope as
1 Point-Slope Form from a Graph
Write the equation of the line in the
graph in point-slope form.
Solution
Use the given point (1,2). From the
2
graph, find m = —.
y — y l — m(x — x^} Write point-slope form.
2 2
y — 2 = —{x — 1) Substitute y for m, 1 for x v and 2 for y v
ANSWER ^ The equation y
(x — 1) is written in point-slope form.
Point-Slope Form from a Graph
Write the equation of the line in point-slope form.
3
\2
t (-3, iN
*5
I x
N.
2
Chapter 5 Writing Linear Equations
I Student HeCp
^ ^
► Study Tip
The point-slope form
y - y 1 = m(x - x } )
has two minus signs.
Be sure to account for
these signs when the
point (x y y 1 ) has
negative coordinates.
I J
J 2 Write an Equation in Point-Slope Form
Write in point-slope form the equation of the line that passes through the point
(1, —5) with slope 3.
Q Write the point-slope form. y — y x = m(x — x { )
© Substitute 1 for jc p —5 fory p and 3 form. y — (—5) = 3(x — 1)
0 Simplify the equation. y + 5 = 3(x — 1)
ANSWER ► The equation in point-slope form of the line is y + 5 = 3{x — 1).
Write an Equation in Point-Slope Form
4. Write in point-slope form the equation of the line that passes through the
point (2, 2) with slope
Student HeGp
► Study Tip
Notice that the equation
in Step 3 of Example 3
is in point-slope form.
Steps 4 and 5 convert
the equation to slope-
intercept form.
\ _ J
3 Use Point-Slope Form
Write in slope-intercept form the equation of the line that passes through the
point (—3, 7) with slope —2.
Solution
O Write the point-slope form. y — y x = m(x — x { )
0 Substitute —2 form, —3 forx p and 7 for y v y — 1 = — 2[x — (—3)]
© Simplify the equation. y — 1 = — 2(x + 3)
0 Distribute the —2. y — 7 = —2x — 6
© Add 7 to each side. _y = —2x + 1
ANSWER ► The equation of the line in slope-intercept form is _y = — 2x + 1.
CHECK y In general, you can use
a graph to check whether your
answer is reasonable.
In the graph at the right, notice
that the line y = — 2x + 1 has a
slope of —2 and passes through
the point (—3, 7).
Use Point-Slope Form
5. Write in slope-intercept form the equation of the line that passes through the
point (2, 4) with slope 3. Check your answer by graphing.
5.2 Point-Slope Form
Student HeCp
p More Examples
M°r e examples
are available at
www.mcdougallittell.com
4 Write an Equation of a Parallel Line
Write in slope-intercept form the equation of the line that is parallel to the line
y = 2x — 3 and passes through the point (3, — 1).
Solution
The slope of the original line is m = 2. So, the slope of the parallel line is also
m = 2. The line passes through the point (jt p y x ) = (3, — 1).
y-y i
= m(x — Xj)
Write point-slope form.
j-(-1)
= 2(x - 3)
Substitute 2 for m, 3 for x v and -1 for y,
y+ i
= 2(x - 3)
Simplify.
y+ i
1
II
Use distributive property.
y
1
II
Subtract 1 from each side.
ANSWER ► The equation of the line is y = 2x — 7.
CHECK y You can check
your answer graphically.
The line y = 2x — 1 is
parallel to the line y = 2x — 3
and passes through the point
(3,-1).
v*.
Write an Equation of a Parallel Line
6 . Write in slope-intercept form the equation of the line that is parallel to
the line y = — 2x + 1 and passes through the point (3, —2). Check your
answer graphically.
CHOOSING A FORM Now you know two ways to write linear equations - in
slope-intercept form or in point-slope form. Depending on the information you
are given, sometimes it is easier to write a linear equation in one form rather than
the other. The following summarizes when to use each form.
Writing Equations of Lines
1 . Use slope-intercept form
y = mx + b
if you are given the
slope m and the
/-intercept b.
2 . Use point-slope form
y - /, = rn{x - x,)
if you are given
the slope m and a
point (x v /,).
Chapter 5 Writing Linear Equations
Exercises
Guided Practice
Vocabulary Check 1 . Write the point-slope form of an equation of a line.
Skill Check Write in point-slope form the equation of the line that passes through the
given point and has the given slope.
2. (2, -1), m = 3 3. (3, 4), m = 4 4. (-5, -7), m = -2
Write the equation of the line in point-slope form.
Write in slope-intercept form the equation of the line that passes through
the given point and has the given slope.
8. (—4, 2), m = 2 9. (—1, —3), m = ^ 10. (2, —3), m = 0
Write in point-slope form the equation of the line that is parallel to the
given line and passes through the given point.
11. y = x + 5, (— 1, — 1) 12. y = —3x + 1, (2, 4) 13. y = — 6, (3, 3)
Practice and Applications
Student HeCp
► Homework Help
Example 1: Exs. 14-19
Example 2: Exs. 20-25
Example 3: Exs. 26-34
Example 4: Exs. 35-43
v _ 4
USING A GRAPH Write the equation of the line in point-slope form.
3
/ 1 <i \ _
2
1
*
(i, 1L
3
*
-1
i
5
4
> X
5.2 Point-Slope Form
WRITING EQUATIONS Write in point-slope form the equation of the line
that passes through the given point and has the given slope.
20. (—1, —3), m = 4 21. (—6, 2), m = — 5 22. (—10, 0), m = 2
23. (-8, -2), m = 2 24. (-4, 3), m = -6 25. (-3, 4), m = 6
Student HeCp
► Homework Help
Extra help with
v problem solving in
Exs. 26-34 is available at
www.mcdougallittell.com
COMPARING FORMS Write in point-slope form the equation of the line.
Then rewrite the equation in slope-intercept form.
26.(12, 2), m — -7
29. (1, 4), m = 2
32. (6, 2 ),m = j
27. (8, —1), m = 0
30. (—2, 4), m = 3
33. (—1, 1), m = -|
28. (5, -12), m = -11
31. (—5, —5), m = —2
34. (4, -2), m = |
WRITING EQUATIONS OF PARALLEL LINES Write in slope-intercept form
the equation of the line that is parallel to the given line and passes
through the given point.
35. y = 2x — 11, (3, 4) 36. y = —+ 6, (—2, 7) 37. y = yx + 4, (—4, —4)
38. y = lx — 1, (8, 0) 39. y = ~9x - 3, (0, -5) 40. y = jx, (8, -10)
PYROTEUTHIDS are
also known as fire squids
because of their brilliant
bioluminescent flashes. They
live in ocean depths ranging
from 0 to 500 meters.
EQUATIONS FROM GRAPHS Write in slope-intercept form the equation
of the line that is parallel to the line in the graph and passes through the
given point.
Sci ence Link y As a diver descends, the pressure in the water increases
by 0.455 pound per square inch (psi) for each foot of descent. At a depth
of 40 feet, the pressure of the water on the diver is 32.5 pounds per
square inch.
44. Using the point (40, 32.5)
and the slope 0.455, write the
equation in point-slope form
that models this situation.
Then rewrite the equation
in slope-intercept form.
45. Use the equation you wrote in
Exercise 44 to determine the
pressure at a depth of 90 feet.
Chapter 5 Writing Linear Equations
CHALLENGE As shown in the graph below, between 1988 and 1998,
the number of non-business trips taken by Americans increased by
about 11 million per year. In 1993, Americans took about 413 million
such trips.
46. Write the equation in slope-intercept form that gives the number of
non-business trips y (in millions) in terms of the year x. Let x represent
the number of years after 1988.
47. According to the equation you wrote in Exercise 46, about how many
non-business trips did Americans take in 1996?
48. Assuming the trend continues, estimate the number of non-business trips
Americans will take in 2005.
Standardized Test
Practice
49. MULTIPLE CHOICE Which equation is in point-slope form?
(A) y — 5x — 9 CD y + 4 = 3(— 2x + 2)
Cg) jc = 8(y - 1) CD j + 4 = 3(x - f)
50. MULTIPLE CHOICE What is the point-slope form of the equation of the line
in the graph?
CD y — 3 = 3(x ~ 0)
CG) y = 3x + 3
(E)y-(- 1) = 3(* + 3)
®r 3 = 3[JC - c-1)]
51. MULTIPLE CHOICE What is the slope-intercept form of the equation of the
line parallel to the line in the graph that passes through the point (—1, 1)?
(a) y = 2x — 3
CD y ~ 3 = 2(x - 1)
CD y — ~ 2x + 3
CD y — 2x + 3
5.2 Point-Slope Form
Mixed Review
Maintaining Skills
Quiz 1
CHECKING SOLUTIONS OF INEQUALITIES Check whether the given
value of the variable is a solution of the inequality. (Lesson 1.4)
52. 2x < 24; x = 8 53. ly + 6 > 10; .y = 3 54. 16p - 9 > 71; p = 5
55. 12a < a — 9\ a = —2 56. 4x < 28; x = 1 57. 6c — 4 > 14; c = 3
GRAPHING FUNCTIONS Graph the function. (Lesson 4.8)
58. g(x) = 3x + 1 59. h(x) = 4x — 4 60. f(x) = 2x + 10
61. f(x) = —3x + 4
62. g(x) — —x — 7
63. g(x) = —x + 5
SUBTRACTING FRACTIONS Subtract. Write the answer as a fraction or
as a mixed number in simplest form. (Skills Review p. 764)
65.
18
5
8 1
9
3
Write the equation of the line in slope-intercept form. (Lesson 5.1)
1. Slope = —2, ^-intercept =1 2 . Slope = 5, ^-intercept = 0
Write the equation of the line in slope-intercept form. (Lesson 5.1)
Write in point-slope form the equation of the line that passes through the
given point and has the given slope. (Lesson 5.2)
6. (7, 7), m = —2 7. (—8, —2), m = 3 8. (0, 0), m = — ^
Write in slope-intercept form the equation of the line that passes through
the given point and has the given slope. (Lesson 5.2)
9. (2, 3), m = 1 10. (—6, 4), m = 0 11. (1, —4), m = —4
Write in slope-intercept form the equation of the line that is parallel to
the given line and passes through the given point. (Lesson 5.2)
12 . y = 4x+l, (1, 0)
14. y = —2x + 3, (0, 5)
13. y = ~x ~ 2, (-3, -3)
15. y = |x, (2, -1)
Chapter 5 Writing Linear Equations
Writing Linear Equations
Given Two Points
Goal
Write an equation of a
line given two points on
the line.
Key Words
• slope
• slope-intercept form
• point-slope form
How steep is the mountain?
In this lesson you will learn
to write an equation of a line
given any two points on the
line. In Example 1 you will
write a linear equation that
models a snowboarder’s
descent down a mountain.
Student HeCp
►Study Tip
In Example 1, notice
that the graph shows
the /-intercept.
Because you know the
/-intercept, use the
slope-intercept form to
write the equation.
k _ )
i Use a Graph
The line at the right models a
snowboarder’s descent down a
C
mountain. Write the equation
(-4,
,5)
of the line in slope-intercept form.
3
(0,2)
Solution
i
© Find the slope.
X
-7
-5
-3
-
i ,
r 1
■v
<
5-2 3
m — -= —--— = ——
x 2 ~ Xi -4-0 4
© Write the equation of the line. From the graph, you can see the /-intercept
is b = 2. Use slope-intercept form.
y — mx + b Write slope-intercept form.
y — — ^x + 2 Substitute for m and 2 for b.
ANSWER ► The equation of the line is y =
3 JL O
~ 4 X+ 2 -
Use a Graph
The graph at the right
i
c
models a car’s ascent up a
hill. Write the equation of
(10,4)
the line in slope-intercept
3
form.
D
, 1
'i
5
7
c
)
11 X
S3
Writing Linear Equations Given Two Points
POINT-SLOPE FORM When you are given two points, but do not know the
y-intercept, you should first use the point-slope form to write the equation of the
line that passes through the points, as in Example 2.
2 Write an Equation of a Line Given Two Points
Write in slope-intercept form the equation of the line that passes through the
points (3, —2) and (6, 0).
Solution
O Find the slope. Use (Xpj^) = (3, —2) and (x 2 ,y 2 ) = (6, 0).
m = — -- Write formula for slope.
x 2 ~ x i
0 -(-2) c , .
= —p - ~z~ Substitute.
o — 5
= ~ Simplify.
0 Write the equation of the line. Use point-slope form, because you do not
know the y-intercept.
y-y l = m(x- x,)
y ~ (-2) = |(x - 3)
y + 2 = |x - 2
2
y = 3 x - 4
Write point-slope form.
2
Substitute j for m, 3 for x y -2 for y v
Simplify and use distributive property.
Subtract 2 from each side.
ANSWER ► The equation of the
line is y = —v — 4.
CHECK /
You can check your
answer by graphing.
Notice that the graph of
2
y = — x — 4 passes
through (3, —2) and (6, 0).
You can also check your
answer using substitution.
Write an Equation of a Line Given Two Points
Write in slope-intercept form the equation of the line that passes through
the given points. Check your answer.
2. (2, 3) and (4, 7) 3. (-4, 5) and (2, 2) 4. (1, -1) and (4, -4)
Chapter 5 Writing Linear Equations
Student MeCp
p More Examples
More examples
l/ are available at
www.mcdougallittell.com
3 Decide Which Form to Use
Write the equation of the line in slope-intercept form.
N
‘V
(0,2)
-1
1
5
5
X
-1
\
.(4, -2)
%
Solution
a. Find the slope.
2 -(-2)
m = -= A A = ~
X 2~ X 1 0_4
Since you know the y-intercept,
use slope-intercept form.
The y-intercept is b = 2.
y = mx + b
y — (— l)^c + 2
y = —x + 2
1
b. Find the slope.
^2-^1
m
(-D
2~ X 1 4-3
Since you do not know the
y-intercept, use point-slope form.
y-y 1 = m(x- xj
y-(- l) = 3(x-3)
y + 1 = 3x — 9
y = 3x — 10
Decide Which Form to Use
Write the equation of the line in slope-intercept form.
5
(4,5)
(0,
3)
1
~
1 |
f 1
3
5
X
Writing Linear Equations Given Two Points
O Find the slope m
y 2 - y.
0 Write the equation of the line.
• Use the slope-intercept form if you know the y-intercept.
y = mx + b
• Use the point-slope form if you do not know the y-intercept.
y — y 1 — m(x - xj
5.3 Writing Linear Equations Given Two Points
5.3 Exercises
Guided Practice
Vocabulary Check
1 - When writing an equation of a line given two points, which form should you
use if you do not know the y-intercept?
2 . Write the slope-intercept form of an equation of a line.
Skill Check
Write the equation of the line in slope-intercept form.
Write in slope-intercept form the equation of the line that passes through
the given points.
6. (-1, 1) and (2, 5) 7. (3, -2) and (-6,4) 8. (4, 3) and (1, 6)
Practice and Applications
POINT-SLOPE FORM Write in point-slope form the equation of the line
that passes through the given points.
9. (2, 3) and (0, 4) 10. (0, 0) and (-6, -5) 11. (0, -10) and (12, 4)
12. (0, 9) and (8, 7) 13. (1, 1) and (0, 2) 14. (-7, 2) and (0, 1)
15. (-8, 6) and (-13, 1) 16. (11, -2) and (17, 6) 17. (-4, 5) and (4, 5)
USING A GRAPH Write the equation of the line in slope-intercept form.
Student HeCp
► Homework Help
Example 1: Exs. 9-23
Example 2: Exs. 24-32
Example 3: Exs. 33-35
Chapter 5 Writing Linear Equations
I
Student HeCp
► Homework Help
Extra help w ' t * 1
problem solving in
Exs. 24-32 is available at
www.mcdougallittell.com
SLOPE-INTERCEPT FORM Write in slope-intercept form the equation of
the line that passes through the given points.
24. (-5, 7) and (2, -7) 25. (2, 0) and (-2, 6) 26. (1, -5) and (3, 4)
27. (-1, -2) and (2, 6) 28. (l,4)and(-l, -4) 29. (2, -3) and (-3, 7)
30. (2, 2) and (-7, -7) 31. (6, -4) and (2, 8) 32. (1, 1) and (7, 4)
DECIDING WHICH FORM Decide which form of a linear equation to use.
Then write the equation of the line in slope-intercept form.
36. AIRPLANE DESCENT The graph below models an airplane’s descent
from 12,500 to 2500 feet. Write in slope-intercept form the equation of
the line shown.
CfiiihttcC
ENGLAND ^Folkestone
terminal
r
Chunnel
N
4
English
Channel
Coquelles
terminal
FRANCE
THE CHUNNEL is a railroad
tunnel under the English
Channel, connecting England
and France. It is one of the
most ambitious engineering
feats of the twentieth century.
CHALLENGE In Exercises 37-39, use the diagram of the Chunnel below.
37. Write the equation of the line from point A to point B. What is the slope?
38. Write the equation of the line from point C to point D. What is the slope?
39. Is the Chunnel steeper on the English side or on the French side?
5.3 Writing Linear Equations Given Two Points
40, Scienc^Lmk^ At sea level, the speed of sound in air is linearly related to
the air temperature. If the temperature is 35°C, sound will travel at a rate of
352 meters per second. If the temperature is 15°C, sound will travel at a rate
of 340 meters per second. Given the points (35, 352) and (15, 340), write in
slope-intercept form the equation of the line that models this relationship.
Standardized Test
Practice
41. MULTIPLE CHOICE What is the equation of the line that passes through the
points (7, 4) and (—5, —2)?
CS) y = 2 x ~2 (3D y = ~ 2 X + 2
_ li _ l , l
®y=- 2 x ~i ®y = 2 x + 2
Mixed Review
42. MULTIPLE CHOICE What is the equation of the line shown in the graph?
CD y = 5x + y
(G) y = 5x - -j-
CS) y = ~fx + 2
„ l , li
CD y = 5 X + y
SOLVING EQUATIONS Solve the equation. (Lesson 3.3)
43. Ax- 11 = -31 44. 5x-7 + x= 19 45. ly = 9y - 8
46. 20x = 3x + 17 47. 3p + 10 = 5p — 7 48. 12x + 10 = 2x + 5
49. ROOF PITCH The center post of a roof is 8 feet high. The horizontal
distance from the center post to the outer edge of the roof is 24 feet.
Find the slope, or pitch , of the roof. (Lesson 4.5)
24 ft
Maintaining Skills
ADDING MIXED NUMBERS Add. Write the answer as a fraction or as a
mixed number in simplest form. (Skills Review p. 765)
so -4 +i l
51 - 5 f + 2 i
52. 3 + |
53. 3-1 + 54
O 0
M. li + 2f
55. 17j +
3 19
56 ' 7 ~6 + 3 25
2 13
57. 1 j + 5^
58. 2| + 2o|
Chapter 5 Writing Linear Equations
Standard Form
Goal
Write an equation of a
line in standard form.
Key Words
• standard form
• slope-intercept form
• point-slope form
• integer
• coefficient
How much birdseed can you buy?
In this lesson you will learn
about another form of linear
equation. In Exercises 57 and 58
you will use this form to model
different amounts of birdseed
that you can buy.
STANDARD FORM
Student tteCp
►Vocabulary Tip
Recall that a coefficient
can be thought of as
"the number in front of
a variable." For
example, | is the
coefficient in the
variable expression |x. *
In standard form, the variable terms are on the left side and the constant term is
on the right side of the equation.
i Convert to Standard Form
2
Write y = jx — 3 in standard form with integer coefficients.
*
Solution
0 Write the original equation.
0 Multiply each side by 5 to clear
the equation of fractions.
© Use the distributive property.
0 Subtract 2x from each side.
>’ = f x - 3
5 >’ = 5 (f x - 3 )
5y = 2x - 15
—2x + 5y = —15
ANSWER ^ In standard form, an equation is — 2x + 5 y = —15.
Convert to Standard Form
Write the equation in standard form with integer coefficients.
1 - y = — x + 5 2■ y = — ^x + 7 3. y = jx + 4
5.4 Standard Form
! Student HeGp
► Study Tip
In standard form, a
linear equation can be
written in different
ways. For instance,
another way to write
the equation in Step 6
of Example 2 is
-lx - y = 5.
I _ >
2 Write an Equation in Standard Form
Write in standard form an equation of the line passing through (—4, 3) with a
slope of —2. Use integer coefficients.
Solution
© Write the point-slope form.
© Substitute —2 for m, —4 for x v and 3 for y v
© Simplify the equation.
© Use the distributive property.
© Add 3 to each side. (Slope-intercept form)
© Add 2x to each side. (Standard form)
y-y 1 = m(x- x x )
y - 3 = -2[x - (-4)1
y — 3 = — 2{x + 4)
y — 3 = — 2x — 8
y — ~ 2x — 5
2x + y = — 5
Write an Equation in Standard Form
4. Write in standard form an equation of the line passing through (3, —5) with a
slope of —3. Use integer coefficients.
Student HeGp
- s -
► Study Tip
Example 3 uses slope-
intercept form (instead
of point-slope form)
because the second
point shows that the
/-intercept is 3.
V J
(E232EBB 3 Write an Equation in Standard Form
A line intersects the axes at (4, 0) and (0, 3). Write an equation of the line in
standard form. Use integer coefficients.
Solution
Q Find the slope. Use (x p >’,) = (4, 0) and (x,, v 2 ) = (0, 3).
_ 3^2 _ 3 — 0 3
m x 2 — x 1 0 — 4 4
0 Write an equation of the line, using slope-intercept form.
y =
mx + b
Write slope-intercept form.
y =
3 , ,
~4 X + 3
Substitute — | for m and 3 for b.
4y =
i~r + 3 )
Multiply each side by 4.
4y =
-3x + 12
Use distributive property.
3x +4y =
12
Add 3x to each side.
ANSWER ^ The equation 3x + 4 y = 12 is in standard form.
Write an Equation in Standard Form
5_ Write in standard form an equation of the line that intersects the axes at (2, 0)
and (0, 5). Use integer coefficients.
Chapter 5 Writing Linear Equations
HORIZONTAL AND VERTICAL LINES Recall from Chapter 4 that the
slope of a horizontal line is zero and the slope of a vertical line is undefined.
In Example 4 you will learn how to write equations of horizontal and vertical
lines in standard form.
Student HeCp
► Study Tip
Notice that there is no
x-term in the standard
form of a horizontal
line and no y-term in
the standard form of a
vertical line.
V _ J
4 Equations of Horizontal and Vertical Lines
Write an equation of the blue line in standard form.
1
L ;y
\
-
1
-1
]
5 X
3
i
i
1
J
-
1
-1
]
[
5 X
i
Solution
a. Each point on this horizontal
line has a y-coordinate of —3. So,
the equation of the line is y = —3.
Both equations are in standard form.
b. Each point on this vertical line
has an x-coordinate of 2. So, the
equation of the line is x = 2.
Equations of Horizontal and Vertical Lines
Write an equation of the line in standard form.
3
5
-
1
-1
1
} x
LINEAR EQUATIONS You have now studied all of the commonly used forms of
linear equations. They are summarized in the following list.
r -
Equations of Lines
SLOPE-INTERCEPT FORM: / = mx + b
POINT-SLOPE FORM: / 9 y y = m(x ~ xj
VERTICAL LINE (Undefined Slope): x = a
HORIZONTAL line (Zero Slope): y=b
standard form: Ax + By = C, where A and B are not both zero.
5.4 Standard Form
Exercises
Guided Practice
Vocabulary Check 1. Name the following form of an equation of a line: y = mx + b. What does
m represent? What does b represent?
2. Name the following form of an equation of a line: Ax + By = C. Give an
example of an equation in this form.
Skill Check Write the equation in standard form with integer coefficients.
3. y = 2x — 9 4. y — ^x + 8 5- y — ^x
Write in standard form an equation of the line that passes through the
given point and has the given slope. Use integer coefficients.
6- (3, 4), m — —4 7. (1, —2), m — 5 8- (—2, —5), m — 3
Write in standard form an equation of the line that passes through the
two points. Use integer coefficients.
9. (3, 1), (4, -2) 10. (1, 6), (1, -5) 11. (5, 0), (0, 3)
Write an equation of the line in standard form.
Practice and Applications
CONVERTING TO STANDARD FORM Write the equation in standard form
with integer coefficients.
15. y = —5x + 2 16. v = 3x — 8
18. y = jx
19. y = -fx
17. y = -9 + 4x
20. y = 9x + ^
Student HeCp
► Homework Help
Example 1: Exs. 15-20
Example 2: Exs. 21-29
Example 3: Exs. 30-38
Example 4: Exs. 39-44
WRITING EQUATIONS Write in standard form an equation of the line that
passes through the given point and has the given slope.
21 . (-8, 3),m = 2
24. (—6, —7), m = — 1
27. (2, 9), m = |
22 . (-2,7), m= -4
25. (3, —2), m = 5
28. (5, -8), m = \
23. (— 1, 4), m = -3
26. (10, 6), m = 0
29. (7, 3), m=~
BmI
Chapter 5 Writing Linear Equations
WRITING EQUATIONS Write in standard form an equation of the line that
passes through the two points. Use integer coefficients.
31- (-3,0), (0, 2)
Student HeGp
► Study Tip
In Exercises 30-38,
find the slope first,
use point-slope form,
then convert to
standard form.
I J
30. (4, 0), (0, 5)
33. (0, 1), (1, -1)
36. (9,-2), (-3, 2)
34. (-4, 0), (0,-5)
37. (-3, 3), (7, 2)
32. (0, 0), (2, 0)
35. (-4, 1), (2, -5)
38. (4, -7), (5,-1)
HORIZONTAL AND VERTICAL LINES Write an equation of the line in
standard form.
41.
u
3
K°) 1
-1
1 X
-1
If
The names of different sports are hidden in the first
quadrant of a coordinate plane, as shown on the grid below. Write an
equation in standard form of each line containing the given sport. For
example, an equation for "softball" is —x + y=2.
45. Basketball
46. Lacrosse
47. Skiing
48. Football
49. Golf
50. Rugby
51. Hockey
Ttr
F X Y
LON
A A O
H G C
B OF
C 0 C
JS D S
E 1 Y
B R U
I N
Z F
L B
P
T I
R B
E O
K U
Y E
G B
O X
/
E Z
Hr
TT
N G
E H
O G
Ti
S L
4-4-
i
Student HeCp
► Homework Help
Extra help with
p ro k| em solving in
Exs. 52-54 is available at
www.mcdougallittell.com
WRITING EQUATIONS FROM GRAPHS Write in standard form an equation
of the line. Use integer coefficients.
5.4 Standard Form
Link_
Birds
BIRDSEED Thistle seed
attracts goldfinches. Before
a storm, goldfinches greatly
increase the amount they eat
in order to gain weight.
► Source: Canadian Wildlife Service
Standardized Test
Practice
ERROR ANALYSIS In Exercises 55 and 56, find and correct the error.
56.
BIRDSEED MIXTURE In Exercises 57 and 58, use the following information.
You are buying $24 worth of birdseed that consists of two types of seed. Thistle
seed costs $2 per pound. Dark oil sunflower seed costs $1.50 per pound. The
equation 2x + 1.5y = 24 models the number of pounds of thistle seed x and the
number of pounds of dark oil sunflower seed y that you can buy.
57. Graph the line representing the possible seed mixtures.
58. Copy and complete the table. Label the points from the table on the graph
created in Exercise 57.
Pounds of thistle seed, x
0
3
6
9
12
Pounds of dark oil sunflower seed, y
?
?
?
?
?
CHALLENGE The equation below represents the intercept form of
the equation of a line. In the equation, the x-intercept is a and the
/-intercept is b.
x y
- + -r = 1
a b
59. Write the intercept form of the equation of the line whose x-intercept is 2 and
y-intercept is 3.
60. Write the equation from Exercise 59 in standard form.
61. MULTIPLE CHOICE Which is an equation of the line in standard form?
(A) —3x + 2y — 2
CD y = -x + 1
Cep 3x + 2y = 1
CD y ~ 1 = 2 X
62. MULTIPLE CHOICE Choose an equation in standard form of the line that
passes through the point (—1, —4) and has a slope of 2.
CD — 2x + y = —2
CH) -x-y = 9
Chapter 5 Writing Linear Equations
CD — 3y = x — 9
CD x - 3y = -9
Mixed Review
Maintaining Skills
Quiz 2
SOLVING EQUATIONS Solve the equation. (Lesson 3.3)
63. 8 + y = 3 64. y — 9 = 2 65. 6 (q + 22) = —120
66. 2(x + 5)= 18 67. 7 - 2a =-14 68. -2 + 4c = 19
GRAPHING EQUATIONS Use a table of values to graph the equation.
Label the x-intercept and the /‘intercept. (Lesson 4.2)
69.y = x + 5 70. y = 4x — 4 7*1^= -x +8
ROUNDING Round to the nearest whole dollar. (Skills Review p. 774)
72. $14.76 73. $908.23 74. $4,573.70 75. $14,098.15
76. $99.99 77. $0.05 78. $0.51 79. $12,345.67
Write in slope-intercept form the equation of the line that passes through
the points. (Lesson 5.3)
1. (10, -3) and (5, -2) 2. (6, 2) and (7, 5) 3. (4, 4) and (-7, 4)
Write in slope-intercept form the equation of the line that passes through
the two points. (Lesson 5.3)
Write the equation in standard form with integer coefficients. (Lesson 5.4)
6. v = — 3x + 9 7. y = ^x + 4 8. y = jix — 1
Write in standard form an equation of the line that passes through the
point and has the given slope. (Lesson 5.4)
9. (6, 8),m = 2 10. (4, 1 ), m = — ^ 11. ( 1 , 5 ),m = -g
Write an equation of the line in standard form. (Lesson 5.4)
1
-
1
-1
]
5 x
5.4 Standard Form
Modeling with Linear
Equations
Goal
Write and use a linear
equation to solve a
real-life problem.
Key Words
• linear model
• rate of change
How many movie theaters will there be in 2005?
In Example 1 you will see that
the number of movie theaters in
the United States increased at a
constant rate from 1985 through
1997. In Example 2 you will use
this linear pattern to predict the
number of movie theaters in the
year 2005.
A linear model is a linear function that is used to model a real-life situation. A
rate of change compares two quantities that are changing. Slope is often used to
describe a real-life rate of change.
Student tteCp
► Study Tip
Because you are given
the slope and a point
on the line, use
point-slope form.
v J
M i Write a Linear Model
From 1985 through 1997, the number of movie theaters in the United States
increased by about 750 per year. In 1993, there were about 26,000 theaters.
Write a linear model for the number of theaters y. Let t = 0 represent 1985.
DATA UPDATE of the number of movie theaters is available at www.mcdougallittell.com
Solution
The rate of increase is 750 per year, so the slope is m = 750. The year 1993 is
represented by t = 8. Therefore, (t v y x ) = (8, 26,000) is a point on the line.
© Write the point-slope form. y — y 1 = m(t —
© Substitute 750 for m, 8 for t v y — 26,000 = (750 )(t — 8)
and 26,000 for y v
© Use the distributive property. y — 26,000 = 750 1 — 6000
© Add 26,000 to each side. y = 750 1 + 20,000
ANSWER ► The linear model for the number of theaters in the United States is
y = 150t + 20,000, where t = 0 represents 1985.
Write a Linear Model
1. From 1985 through 1997, movie attendance in the United States
increased by about 25 million per year. In 1994, movie attendance
was about 1300 million. Write a linear model for movie attendance
y (in millions). Let t = 0 represent 1985.
Chapter 5 Writing Linear Equations
PREDICTING WITH LINEAR MODELS Once you have written a linear model,
you can use it to predict unknown values. When you do this, you are assuming
that the pattern established in the past will continue into the future.
! Student MeCp
► Study Tip
Graphs describing past
behavior can be used
to estimate future
trends.
k _/
2 Use a Linear Model to Predict
Use the linear model in Example 1
y = 750 1 + 20,000
to predict the number of theaters in the year 2005.
Recall that t = 0 represents the year 1985.
Solution
Method 1 Use an algebraic approach.
Because t = 0 represents the year 1985, 2005 is represented by t = 20.
Q Write the linear model. y = 750 1 + 20,000
© Substitute 20 for t. y = 750(20) + 20,000
© Simplify. y = 15,000 + 20,000 = 35,000
ANSWER ^ You can predict that there will be about 35,000 theaters in 2005.
Method 2 Use a graphical approach.
A graph of the equation y = 7501 + 20,000 is shown below.
ANSWER ^ From the graph, you can see that when t = 20 (which represents the
year 2005), y = 35,000. Therefore, you can predict that there will be
about 35,000 theaters in the year 2005.
Use a Linear Model to Predict
Use the linear model you wrote for Checkpoint 1 on page 298 to predict
the movie attendance in the year 2005. Let t = 0 represent
the year 1985.
2. Use an algebraic approach.
3. Use a graphical approach.
5.5 Modeling with Linear Equations
777
Student HeCp
p Look Back
For help with algebraic
modeling, refer to
pp. 36-38.
v _ 4
J 3 Write and Use a Linear Model
You are buying hamburger and chicken for a barbecue. The hamburger costs
$3 per pound and the chicken costs $4 per pound. You have $60 to spend.
a. Write an equation that models the different amounts (in pounds) of
hamburger and chicken you can buy.
b. Use the model to complete the table that illustrates several different
amounts of hamburger and chicken you can buy.
Hamburger (lb), x
0
4
8
12
16
20
Chicken (lb), y
?
?
?
?
?
?
Solution
a. Model the possible combinations of hamburger and chicken.
Verbal
Model
Price of
Weight of
_I—
Price of
Weight of _
Total
hamburger
hamburger
[
chicken
chicken
cost
Labels Price of hamburger = 3
(dollars per pound)
Weight of hamburger = x (pounds)
Price of chicken = 4 (dollars per pound)
Weight of chicken = y
Total cost = 60
(pounds)
(dollars)
Algebraic 3 x +4 y =60
Model
Linear model
b. Complete the table by substituting the given values of x into the equation
3x + Ay = 60 to find y.
Hamburger (lb), x
0
4
8
12
16
20
Chicken (lb), y
15
12
9
6
3
0
Note that as the number of pounds of hamburger increases, the number of
pounds of chicken decreases and as the number of pounds of hamburger
decreases, the number of pounds of chicken increases.
Write and Use a Linear Model
You are buying pasta salad and potato salad for the barbecue. The pasta
salad costs $4 per pound and the potato salad costs $5 per pound. You
have $60 to spend.
4. Write an equation that models the different amounts (in pounds) of potato
salad and pasta salad you can buy.
TTT
Chapter 5 Writing Linear Equations
Exercises
Guided Practice
Vocabulary Check Complete the sentence.
1 _ A ? is a linear function that is used to model a real-life situation.
2 . Slope is often used to describe a real-life ? .
Skill Check Match the description with its graph. In each case, tell what the slope of
the line represents.
3. An employee is paid $12.50 per hour plus $1.50 for each unit produced
per hour.
4. A person is paying $10 per week to a friend to repay a $100 loan.
5_ A sales representative receives $20 per day for food, plus $.32 for each
mile driven.
Practice and Applications
6. COMPANY PROFITS Between the years of 1990 and 2000, the annual profit
for the Alpha Company increased by about $70,000 per year. In 1998, the
company had an annual profit of $2,000,000. Write the equation in slope-
intercept form that gives the annual profit P for the Alpha Company in terms
of t. Let t = 0 represent the year 1990.
I Student HeCp
► Homework Help
Example 1: Exs. 6-9,
12-14, 18-20
Example 2: Exs. 10, 11,
15-17, 21-23
Example 3: Exs. 24-32
v___/
MOUNTAIN CLIMBING In Exercises 7-11, a mountain climber is scaling a
400-foot cliff. The climber starts at the bottom at t = 0 and climbs at a
constant rate of 124 feet per hour.
7. What is the slope in the linear model for the situation?
8. The y-intercept represents the height at which the climber begins scaling the
cliff. What is the y-intercept in the linear model?
9. Use the slope and y-intercept to write the linear model for the distance y
(in feet) that the climber climbs in terms of time t (in hours). Use slope-
intercept form.
10, After 3 hours, has the climber reached the top of the cliff?
11. Use the equation from Exercise 9 to determine the time that the climber will
reach the top of the cliff.
5.5 Modeling with Linear Equations
T
CANOE RENTAL In Exercises 12-17, use the following information.
Renting a canoe costs $10 plus $28 per day. The linear model for this situation
relates the total cost of renting a canoe, y, with the number of days rented, x.
12, What number corresponds to the slope in the linear model?
13, What number corresponds to the y-intercept in the linear model?
14, Use the slope and y-intercept form to write the linear model.
15. Graph the linear model from Exercise 14.
16. Use the linear model to find the cost of renting a canoe for 3 days.
17. If you had $66 to spend, for how many days could you rent a canoe?
Link to
Careers
AUTO MECHANICS, often
called automotive service
technicians, inspect, maintain,
or repair automobiles and
light trucks.
CAR COSTS In Exercises 18-23, use the following information.
From 1994 through 1997, the cost of owning and operating a car per mile,
which includes car maintenance and repair, increased by about 2.2 cents per
year. In 1995, it cost about 48.9 cents per mile to own and operate a car.
Let t = 0 represent the year 1994.
► Source: American Automobile Manufacturers Association, Inc.
18. Find the slope of the linear equation that models this situation.
19. Name one point on the line.
20. Use the slope from Exercise 18 and the point from Exercise 19 to write a
linear model for the cost C of owning and operating an automobile in terms
of time t.
21 . Use an algebraic approach to predict the cost of owning and operating a car
in 2003.
22 . Graph the linear model from Exercise 20.
23. Use your graph to estimate the cost of owning and operating a car in 1996.
TICKET PURCHASE In Exercises 24-26, use the following information.
A school club visits a science museum. Student tickets cost $5 each. Non¬
student tickets cost $7 each. The club paid $315 for the tickets. Use the verbal
model below.
Cost of
Number of
Cost of
Number of
student
•
student
+
non-student
•
non-student
ticket
tickets
ticket
tickets
24. Let x represent the number of student tickets. Let y represent the number of
non-student tickets. Finish assigning labels.
25. Write the algebraic model from the verbal model.
26. Copy the table. Then use the algebraic model to complete the table.
Number of student tickets, x
7
14
28
35
56
63
Number of non-student tickets, y
?
?
?
?
?
?
Chapter 5 Writing Linear Equations
Student HeCp
► Homework Help
Extra help with
~^P~' problem solving in
Exs. 27-30 is available at
www.mcdougallittell.com
BASKETBALL GAME In Exercises 27-30, use the following information.
A basketball team scored 102 points in a playoff game. Each field goal is
2 points and each free throw is 1 point. The team scored no 3-point field goals.
27. Write a linear model for the number of points the team scored in terms of
field goals x and free throws y.
28. Write the equation from Exercise 27 in slope-intercept form.
29. Copy the table. Then use the linear equation to complete the table.
Number of field goals, x
20
25
30
35
40
Number of free throws, y
?
?
7
7
7
30. Plot the points from the table and sketch the line.
BUYING VEGETABLES In Exercises 31 and 32, use the following
information. You are buying vegetables to make a vegetable tray for a party.
You buy $10 worth of cauliflower and broccoli. The cauliflower costs $2 per
pound and the broccoli costs $1.25 per pound.
31. Write an equation in standard form that represents the different amounts
(in pounds) of cauliflower C and broccoli B that you could buy.
32. Copy the table. Then use the linear equation to complete the table.
Pounds of cauliflower, C
0
1
2
3
4
5
Pounds of broccoli, B
?
7
?
?
7
?
ARCHAEOLOGY In Exercises 33 and 34, use the graph and the following
information. Radiocarbon dating is a method of estimating the age of ancient
objects. The radiocarbon age and the actual age of an object are nearly the
same for objects that are less than 2000 years old. As you can see in the graph,
the radiocarbon age of objects that are more than 2000 years old does not agree
with the actual age determined by other methods.
33. Use the graph to
estimate the
radiocarbon age of an
object that is actually
5000 years old.
34. Now use the equation
y = TjX + 285.7, where
2000 < x < 9000
to estimate the
radiocarbon age of
the same 5000-year-
old object.
5.5 Modeling with Linear Equations
Standardized Test
Practice
Mixed Review
Maintaining Skills
35, MULTIPLE CHOICE You and a friend have $30 to spend at a health center.
It costs $10 an hour to use the racquetball court and $5 an hour to use the
tennis court. Which equation represents the number of hours you can spend
on each court? Let x represent the number of hours on the racquetball court
and y represent the number of hours on the tennis court.
(a) 5x + 10y = 30 Cb) IOx + 5y = 30
CcT) 5y = IOx — 30 (S) y = 5x + 6
36. MULTIPLE CHOICE Your basketball team scores 84 points with no 3-point
baskets. Each free throw x is worth 1 point. Each field goal y is worth 2
points. Which equation relates the number of free throws with the number of
field goals?
(T) y ~ 2x + 1 Cg) x + y = 84
CEP 2x + y = 84 CD x + 2y = 84
ORDER OF OPERATIONS Evaluate the numerical expression. (Lesson 1.3)
37. 6 - 3 • 2 38. 12 -s- 3 - 3 • 1 39. 4 2 - 6 • (4 + 7)
RATIOS Convert the units. (Lesson 3.8)
40. 5 days to hours 41. 36 inches to feet 42. 12 years to months
43. SLOPE What is the slope
of the ramp in the photo
at the right? Explain how
you arrived at your answer.
(Lesson 4.5)
SLOPE-INTERCEPT FORM Write in slope-intercept form the equation of
the line described below. (Lesson 5.1)
44. m = 0, b = 1 45. m = —2, b = 3 46. m = b = 0
WRITING EQUATIONS Write in slope-intercept form the equation of the
line that passes through the given points. (Lesson 5.3)
47. (0, -3) and (6, 5) 48. (7, 4) and (-3, 0) 49. (5, 2) and (8, 2)
COMPARING PERCENTS AND DECIMALS Compare using <, >, or =.
(Skills Review pp.
50.25% ? 0.25
53.0.065 ? 65%
56.0.017 ? 17%
v 770)
51.0.3 ? 3%
54. 12% ? 1.2
57.5% ? 0.05
52.0.01 ? 1%
55. 160% ?| 1.6
58.0.889 ? 89%
Chapter 5 Writing Linear Equations
DEVELOPING CONCEPTS
For use with
Lesson 5.6
Goal
Describe the relationship
between the slopes of
perpendicular lines.
Question
What is the relationship between the slopes of perpendicular lines?
Materials
• pencil
• ruler
• graph paper
• protractor
Lines that intersect at a right, or
90°, angle are called perpendicular
lines.
Explore
eg*— 1 -f>
0 The line at the right has a
2
slope of —. Copy the line onto
a piece of graph paper.
© Use a protractor to draw a line on the graph
paper that is perpendicular to the given line.
Center the protractor on a point on the line
with integer coordinates.
© Find the slope of the perpendicular line
rise
using m — -.
° run
Q Find the product of the slopes of the two lines.
Think About It
Follow the steps above for the following lines. What do you notice about
the relationship between the slopes of perpendicular lines?
4. Based on your observations, make a general statement about the product of
the slopes of perpendicular lines.
Developing Concepts
Perpendicular Lines
Goal
Write equations of
perpendicular lines.
Key Words
• perpendicular
What is the shortest flight path?
As you saw in Developing
Concepts 5.6, page 305, the
product of the slopes of
perpendicular lines is — 1.
When you take geometry
you will see a proof of this
relationship. In Example 3
you will use perpendicular
lines to plan the path of a
helicopter flight.
Two lines in a plane are perpendicular if they intersect at a right, or 90°, angle.
PERPENDICULARLINES
In a coordinate plane, two
nonvertical lines are
perpendicular if and only if the
product of their slopes is -1.
Horizontal and vertical
lines are perpendicular
to each other.
Student MeCp
► Mohh Examples
More examples
are available at
www.mcdougallittell.com
03Z&EI9 ’■J Identify Perpendicular Lines
Determine whether the lines are perpendicular.
Solution
3 4
The lines have slopes of — and — — Because
1, the lines are perpendicular.
Determine whether the lines are perpendicular.
1- y = 3x + 2, y = ~3x — 1 2 . y = ^x + 1, y = —jx + 1
Chapter 5 Writing Linear Equations
Graphing can be used to check whether your answer is reasonable. Graphing
cannot be used to show that two lines are perpendicular.
Student HeCp
► Study Tip
Examples of
perpendicular lines:
IL
X
Examples of non-
perpendicular lines:
2 Show that Lines are Perpendicular
a. Write in slope-intercept form the equation of the line passing through
(2, 5) and (4, 4).
b. Show that the line is perpendicular to the line y = 2x + 1.
Solution
a. O Find the slope. Let (x p y x ) = (2, 5) and (x 2 , y 2 ) = (4, 4).
y 2 "X _ 4-5 _ _ 1
m
x 2 - x 1
4-2
0 Write the equation of the line using point-slope form.
y - y x = m(x - x x )
1
y - 5 = — 2 <X - 2)
y-5
y
Write point-slope form.
Substitute for m, 2 for x v and 5 for y v
-\x + 1
1
2 X+6
Use distributive property.
Add 5 to each side.
ANSWER ► The equation of the line is y = —^x + 6.
b. The lines have slopes of — ^ and 2. Because ( — • (2) = — 1, the lines
are perpendicular.
CHECK /
You can check that your
answer is reasonable by
graphing both lines. From
the graph, you can see that
the lines appear to be
perpendicular.
Show that Lines are Perpendicular
3. Write in slope-intercept form the equation of the line passing through (1,3)
and (3, 6). Show that this line is perpendicular to the line y = —~x — 5.
4. Write in slope-intercept form the equation of the line passing through (0, 0)
and (1,2). Show that this line is perpendicular to the line y = — + 7.
5.6 Perpendicular Lines
HELICOPTER SEARCH
AND RESCUE CREWS can
save people from sinking
ships, burning buildings,
floods, car and plane
crashes, and other dangers.
Student HeCp
► Study Tip
Here's a shortcut for
finding the slope of a
perpendicular line,
applied to Ex. 3.
2
Original slope:
6 3
Find the reciprocal: —«
3 z
Take the opposite: ^
The slope of the
perpendicular line is
V. ___ j
3 Write an Equation of a Perpendicular Line
HELICOPTER PATH You are in a helicopter as shown in the graph below. The
shortest flight path to the shoreline is one that is perpendicular to the shoreline.
Write the equation for this path.
Solution
2 2
The slope of the shoreline is ——. Solve the equation — — X m = — 1 to find the
3 3
slope of the perpendicular path. Multiply both sides by — — to get m = —. The
helicopter’s current location is (x v y x ) = (14, 4).
Q Write the point-slope form.
1
Vi
II
1
© Substitute — for m, 14 for x v and 4 for y v
y — 4 = f(x - 14)
© Use the distributive property.
y - 4 = |x - 21
© Add 4 to each side.
3 n
y = 2 x ~ 11
3
ANSWER The equation for the path of the helicopter is y = —x — 17.
Write an Equation of a Perpendicular Line
5- You are in a ship as shown in the graph below. The shortest path to the shore
is one that is perpendicular to the shoreline. Write the equation for this path.
Chapter 5 Writing Linear Equations
e
Exercises
Guided Practice
1. Complete: Perpendicular lines intersect at a ? angle.
2. Two lines are perpendicular. If the slope of one of the lines is —y, then what
is the slope of the other line?
Determine whether the lines are perpendicular.
3. y = yx — 3, j = — 5x + 3 4. y = — 4x + 8, y = yx + 7
5. y = -^x + 1, y = fx — 2 6. y = 3, x = 4
o J
Write the equation of the line passing through the two points. Show that
this line is perpendicular to the given line.
7. (-3, 0), (3, 6);y = -x-2 8. (-4, -4), (-2, 2); y = - 1
Write the equation of the line passing through the point and
perpendicular to the given line.
9. (5, 2), y = — yx + 4 10. (6, 0), y = — 2x + 7
Practice and Applications
Student HeCp
^
► Homework Help
Example 1: Exs. 11-19
Example 2: Exs. 20-25
Example 3: Exs. 26-39
i _>
IDENTIFYING PERPENDICULAR LINES Determine whether the lines are
perpendicular.
11-y = x + 4, y = x - 4
12 . y = — yx + 1, y = —3x + 3
13. y = —x — 1 ,y = —2x
14. y = -|x + 2, y = -|x - 2
15. y =yx + 2, Ay = -lx - 16
16. y = —5, x = 5
GRAPHICAL REASONING Write the equation of each line in the graph.
Determine whether the lines are perpendicular.
17 .
1
A
,
1
/
X
K
/
/
\
19.
1
5
-
1
-1
i
L x
Vocabulary Check
Skill Check
5.6 Perpendicular Lines
PERPENDICULAR LINES Write in slope-intercept form the equation of the
line passing through the two points. Show that the line is perpendicular
to the given line. Check your answer by graphing both lines.
20. (8, 5), (5, -1 );y = -\x + 4 21. (-2, -2), (1, -3); y = 3x - 1
22. (-3, 6 ), (3, 0)-y = x + 2 23. (4, -7), (7, 5); y = -jx
24. (1,9), (9,9);*= 1
25. (- 6 , -4), (0, 0); y = ~x - 3
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 26-31 is available at
www.mcdougallittell.com
USING GRAPHS Write in slope-intercept form the equation of the line
passing through the given point and perpendicular to the given line.
WRITING EQUATIONS Write in slope-intercept form the equation of the
line passing through the given point and perpendicular to the given line.
32. (2, 6 ), y = ~x + 4 33. (0, 3), y =
34. (0, 0), >■ =-\x~l 35. (—2, 2),y = 1
36. (—3, — 1), y = —2x + 8
38. (5, 0), y = x — 2
37. (2, — 1), j = f% — 1
39. (-4, -7), y = —4x - 7
LOGICAL REASONING Complete the statement with always , sometimes,
or never.
40. A horizontal line is ? perpendicular to a vertical line.
41 . The product of the slopes of two nonvertical perpendicular lines is ? — 1.
42. The line y = 2x + 3 is ? perpendicular to a line with slope —2.
43. The line y = — -^x + 5 is ? perpendicular to a line with slope 3.
T
Chapter 5 Writing Linear Equations
History
BENJAMIN BANNEKER
(1731-1806)
An astronomer, farmer,
mathematician, and
surveyor, Banneker is
credited for having
developed the layout of
Washington, D.C.
44. History M At the end of the eighteenth century Benjamin Banneker
was recommended by Thomas Jefferson to help lay out the new capital,
Washington, D.C. As you can see in the map below, the city is laid out in a
grid system of perpendicular streets. Assuming the x-axis is F Street and the
y-axis is 16th Street, what is the equation of the line that passes through the
point (—4, 1) and is perpendicular to 13th Street (x = 3)?
45. CONSTRUCTION The city water department is proposing the construction
of a new water pipe. The new water pipe should be perpendicular to the old
pipe. Use the graph below to write the equation for each water pipe.
Standardized Test
Practice
46. CHALLENGE Do the three points (12, 0), (0, 16), and (12, 25) form the
vertices of a right triangle? Explain your answer.
47. MULTIPLE CHOICE Choose which lines are perpendicular.
Line p passes through (4, 0) and ( 6 , 4).
Line q passes through (0, 4) and ( 6 , 4).
Line r passes through (0, 4) and (0, 0).
(A) line p and line q CeD line p and line r
Cg) line q and line r Cp) None of these
48. MULTIPLE CHOICE Which are not slopes
(T) 1 and -1 (G)
CED and -31 GD
of perpendicular lines?
11 , 16
T6 al,d TT
5.6 Perpendicular Lines
Mixed Review
Maintaining Skills
Quiz 3
SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 2.7)
49. 2k — 8 — 8k 50. -5c + 10 + 8 c - 3 51. 12x + I2y - 6x + 2
SOLVING EQUATIONS Solve the equation. (Lesson 3.3)
52. Ax — 11 = —31 53- 5x — 7 + x = 19 54. 2x — 6 = 20
HORIZONTAL AND VERTICAL LINES Determine whether the line is
horizontal or vertical. Then graph the line. (Lesson 4.3)
55. y = —2 56. x = 1 57. x = 4 58. y = 3
DIVIDING FRACTIONS Divide. Write the answer as a fraction or as a
mixed number in simplest form. (Skills Review p. 765)
59.
63.
67.
6
7
9
6
11
8
60.
64.
68 .
61.
65. 2
1 1
69. y - 3
62.
66 .
70.
FOOTBALL SCORE You are playing football. Each touchdown is worth
7 points (assuming the extra point is scored) and each field goal is worth
3 points. Your team scored 42 points. (Lesson 5.5)
1 . Write a linear model for the number of points that your team scored in terms
of touchdowns x and field goals y.
2 . Write the equation in slope-intercept form. Then copy and complete
the table.
Number of touchdowns, x
0
3
6
Number of field goals, y
?
?
?
3. Plot the points from the table and sketch the line.
Determine whether the lines are perpendicular. (Lesson 5.6)
4. y = yx + 6 , y = —2x + 6 5. y = jc — 5, y = —x
Write the equation of the line passing through the two points. Show that
the line is perpendicular to the given line. (Lesson 5.6)
6. (5, 6), (0, l);y = -x + 2 7. (-3, 0), (0, -4); y = \x~l
8 . Write the equation of the line passing through (2, 7) and perpendicular to the
line y — yx + 3.
Chapter 5 Writing Linear Equations
Chapter Summary
and Review
• point-slope form, p. 278
• linear model, p. 298
• perpendicular, p . 306
"V
• standard form, p. 291
L -
• rate of change, p. 298
_J
Slope-Intercept Form
Examples on
pp. 269-271
Write in slope-intercept form the
equation of the line shown in the graph.
Q Write the slope-intercept form,
y = mx + b
0 Find the slope of the line. Use any two
points on the graph. Let (2, 0) be (x v y 1 )
and let ( 0 , 2 ) be (x 2 , y 2 ).
y 2 ~y i = 2 - o
X 2~ X 1 0 — 2
2
-2
= -1
© Use the graph to find the y-intercept b. The line passes through the
point (0, 2) so the y-intercept is b = 2.
0 Substitute slope — 1 for m and 2 for b into the equation y = mx + b.
y = — lx + 2
ANSWER ► The equation of the line is y = — x + 2.
Write the equation of the line in slope-intercept form.
1- m = 6 , b = —4 2. m = 1, b = ^
4. m = 12, b =
5. m
3■ m = —8, b = 8
6- m = 0, b — 10
Write in slope-intercept form the equation of the line shown in the graph.
i
,
\
-
1
-1
L x
\
\
\
\
Chapter Summary and Review
Chapter Summary and Review continued
Point-Slope Form
Examples on
pp. 278-280
Write in slope-intercept form the equation of the line that passes through
(5, —2) and that has slope —3.
0 Write the point-slope form.
0 Substitute —3 for m, 5 for x x and —2 for y y
© Use the distributive property and simplify.
© Subtract 2 from each side.
y-y 1 = m(x- x { )
y - (-2) = — 3(x - 5)
y + 2 = —3x + 15
y = ~3x + 13
Write the equation of the line in point-slope form. Then write the
equation in slope-intercept form.
13, Write the equation of the line that is parallel to the line y = 5x — 2 and
passes through the point (—2, 3).
Writing Linear Equations Given Two Points
Examples on
pp. 285-287
Write the equation of the line that passes through the points (5, —4) and (2, 2).
© Find the slope.
m —
yi~yi
X 2 — X l
2 - (-4) = 2 + 4
2-5 -3
_6_
-3
-2
Write the formula for slope
Substitute (5, -4) for {x v y,) and (2, 2) for (x 2 , y 2 );
then simplify.
0 Write the equation, using point-slope form. Use m = — 2 and (x p y x ) = (5, —4).
y — y 1 = m(x — x^) Write point-slope form.
y — (—4) = — 2(x — 5) Substitute -2 for m, 5 for x v and -4 for y v
y + 4= — 2x+ 10 Simplify and use distributive property.
y = ~ 2x + 6 Subtract 4 from each side.
Chapter 5 Writing Linear Equations
Chapter^ Summary and Review continued
Write in slope-intercept form the equation of the line that passes through
the given points.
14.(4,-9) and (-3, 2) 15. (1, 8) and (-2,-1) 16. (2, 5) and (-8, 2)
17. (1,4) and (2, -4) 18. (0, 8) and (2, 8) 19. (9, 16) and (-9,-16)
Standard Form
Examples on
pp. 291-293
a. Write in standard form an equation of the line passing through (3, 4)
with slope ——.Use integer coefficients.
b. Write in standard form an equation
of the line shown.
1
-
1
-1
1
X
a -y ~y l = m(x - x x )
2 ,
>’
f(* “ 3 )
Write point-slope form.
2
y — 4 = —jx + 2
Substitute -j for m, 3 for x 1 and 4 for y v
Use distributive property.
j* + y = 6
Add 4 to each side. Then add jx to each side.
2x + 3y = 18
Multiply each side by 3 to clear equation of fractions.
b. Each point on this vertical line has an x-coordinate of 3. Therefore, the
equation of the line is x = 3. This equation is in standard form.
Write in standard form an equation of the line that passes through the
given point and has the given slope. Use integer coefficients.
20 . (-2, — l),m = 3 21 . (6, -l),m = 0 22 . (2, 3), m =
-4
Write in standard form equations of the horizontal line and the vertical
line that pass through the point.
23. (-1,7) 24. (9, 11) 25. (-8, -6)
Write in standard form an equation of the line that passes through the
two points. Use integer coefficients.
26. (-1, 0) and (3, 10) 27. (0, 7) and (1, 5) 28. (4, 9) and (-2, -6)
Chapter Summary and Review
Chapter Summary and Review continued
Modeling With Linear Equations
Examples on
pp. 298-300
Between 1995 and 2001, a company’s profits decreased by about $1200 per
year. In 1997, the company had an annual profit of $1,500,000. Write an equation that gives the
annual profit P in terms of the year t. Let t = 0 represent 1995.
Because the profit decreased by 1200 per year, the slope is m — — 1200. Because 1997 is
represented by t = 2, you know (t v P { ) = (2, 1,500,000) is a point on the line.
Q Write the point-slope formula. P — P x = m(t — t x )
© Substitute — 1200 for m, 2 for t v and P — 1,500,000 = —1200 (t — 2)
1,500,000 for P v
© Use the distributive property. P — 1,500,000 = —1200^ + 2400
© Add 1,500,000 to each side and simplify. P = 1,502,400 —1200^
29. Use the linear model in the example, P = 1,502,400 — 1200f, to predict the
total profit for the company in 2006.
30. You have $36 to spend on posters for your bedroom. You can buy a large
poster i for $6.00 and a small poster s for $4.00. Write an equation that
models the different amounts of small and large posters you can buy.
31. Use the equation from Exercise 30 to fill in the table.
Number of small posters, s
0
3
6
9
Number of large posters, i
?
?
?
?
Perpendicular Lines
Examples on
pp. 306-308
Determine whether the line y = 3x — 6 is perpendicular toy = — -^x + 2.
Recall that two lines are perpendicular if the product of their slopes is — 1.
The lines have slopes 3 and —Because 3 • ( —| j = — 1, the lines are perpendicular.
Determine whether the lines are perpendicular.
32. y = — -^rx, y — ~^x — 6 33. 5x + lOy = 3, y = 2x — 9
Write in slope-intercept form the equation of the line that passes through
the given point and is perpendicular to the given line.
34. (4, -6), y = + 17 35. (0, 0), y = \x - 1 36. (-2, 1), y =3x + |
Chapter 5 Writing Linear Equations
Write in slope-intercept form the equation of the line with the given slope
and /-intercept.
1. m = 2, b = — 1 2. m = — b = 3 3- m = 61, b — 9
4. m = b = — 3 5. m = —3, b = 3 6. m = 0, /? = 4
Write in slope-intercept form the equation of the line that passes through
the given point and has the given slope.
7. (2, 6 ), m = 2 8. (3, —9), m — — 5 9- (—5, — 6 ), m — — 3
10. (1, 8 ), ra = —4 11 . (4, —2), ra = ^ 12. — 5^, m = 8
Write in slope-intercept form the equation of the line that passes through
the given points.
13. (-3, 2), (4, -1) 14. ( 6 , 2), ( 8 , -4) 15. (-2, 5), (2, 4)
16. (-2, - 8 ), (-1, 0) 17. (-5, 2), (2, 4) 18. (9, -1), (1, -9)
Write the equation in standard form with integer coefficients.
19. —8j = 20 + |x 20. 5y = 25x 21.-2j + = 4
Write in slope-intercept form the equation of the line that passes through
the given point and is perpendicular to the given line.
22. (3, 5), y = —5x + 4 23. (-2, -2), y = x + 1 24. (9,-4), y = ~3x - 2
25. (0, 0), y = + 6 26. (-7, 3), y = ~x 27. (4, 4), y = -2 +
TICKET PURCHASE In Exercises 28-30, use the following information.
The math club goes to an amusement park. Student tickets cost $15 each. Non¬
student tickets cost $25 each. The club paid $315 for the tickets.
28. Write in standard form an equation that relates the number of student tickets x
with the number of non-student tickets y.
29. Write the equation in slope-intercept form and complete the table.
Number of student tickets, x
1
6
11
16
21
Number of non-student tickets, y
?
?
?
?
?
30. Plot the points from the table and sketch the line.
Chapter Test
Chapter Standardized Test
Tip
Ca^c£!DC^Cj£>
Spend no more than a few minutes on each question. Return to
time-consuming questions once you’ve completed the others.
1 . What is the slope-intercept form of the
equation of the line that has slope 2 and
2
y-intercept —1
(a) y = ~x + 2 CD y = 2 x + y
(Cp y = yx + 2 CD — 14x + y = 7
2 . What is the equation of the line that passes
through the point (4, —5) and has slope
(A) y = 4x — 5 CD y ~ ~\ x + ^
C© y = + 7 (© y = - 1
3. An equation of the line parallel to the line
3
y = —2x — 3 with a y-intercept of — — is
?
(a) >’ = —2 x + | C© y = -2x - |
C© V = 2x - | CD)y = ^x-|
4. What is the equation of the line that passes
through the points ( 8 , —4) and ( 6 , 4)?
(A) y = ~x + 28
CD y = — 4x + 28
C© y = ~\x + 7
CD y = — 4x — 7
5_ What is the slope-intercept form of the
equation of the line whose x-intercept is
3 and whose y-intercept is 5?
(a) y = -|x + 5 CD y = — + 5
C© y = + 5 c© y = -|x + 5
6 . Which equation is in standard form with
integer coefficients?
(a) x ~\y = | C© y = 2x - 5
CD y = — 5 + 2x CD — 2x + y = — 5
7. What is the standard form of an equation
of a line that passes through the point
(— 6 , 1 ) and has a slope of — 2 ?
(A) 2x + y = 13
CD 2x — y = 11
CD 2x + y = — 11
CD \x + y = -13
CD None of these
8 . Which is the slope-intercept form of the
equation of a line that is perpendicular to
the line y = 2x — 7 and passes through the
point (—5, 6 )?
(A) x + 2 y = -7 CD y = ~\x - 1 -
CD y = ~ 2 X + 2 ® y = ~ 2 X + 2
Chapter 5 Writing Linear Equations
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
MMS3SM 1 Number Lines
Draw the number line with a low number of negative ten and a high number of
twenty-five using intervals of five.
Solution
0 Subtract the low number from the 25 — (—10) = 35
high number.
© Divide the difference by the interval distance. 35 -r- 5 = 7
© Create a number line with
seven equal parts, plus m —|-1-1-1-1-1-1-1—►
two sections at the ends.
© Label the number line. —|-1-(-1-1-1-1-1—►
-10 -5 0 5 10 15 20 25
Try These
Draw the number line described below.
1 _ A low number of negative twenty-one and a high number of twenty-eight
with intervals of seven.
2 . A low number of zero and a high number of sixteen with intervals of four.
3. A low number of zero and a high number of one hundred fifty with
intervals of six.
4. A low number of ten and a high number of forty-six with intervals of nine.
Student HcCp
► Extra Examples
More examples
anc j p rac tj ce
exercises are available at
www.mcdougallittell.com
2 Compare Decimals
Compare 0.045 and 0.0449.
Solution
0.045 is changed to 0.0450 Add zeros to make the two numbers
end in the same place value.
0.0450 is larger than 0.0449 Compare the numbers.
0.045 > 0.0449 Use a greater than sign.
Try These
Compare the two numbers.
5. 0.033 and 0.0332
8 . 0.006 and 0.00576
11. 0.01 and 0.001
6 . 0.005 and 0.0045
9. 0.01278 and 0.01
12 . 0.0005 and 0.003
7. 0.0292 and 0.029
10. 0.007 and 0.065
13. 0.0548 and 0.00549
Maintaining Skills
Solving and Graphing
Linear Inequalities
Think & Discuss
1 . Estimate the frequency range of each instrument.
2 . Which of these instruments has the greatest
frequency range?
Learn More About It
Musical instruments produce vibrations in the
air that we hear as sound. The frequency of a sound
determines its pitch, that is, how high or low it sounds.
When frequencies are measured as "cycles per second,"
we are using a unit called a hertz. The table below
shows the frequency ranges of three different musical
instruments.
Frequency Ranges of Instruments
Flute
Soprano
clarinet
Soprano
saxophone
0 600 1200 1800 2400
Frequency (hertz)
APPLICATION: Music
You will write inequalities to describe frequency
ranges in Exercises 23-26 on page 346.
V
APPLICATION LINK More information about music is
available atwww.mcdougallittell.com
y
Study Guide
PREVIEW
What’s the chapter about ?
• Solving and graphing inequalities
• Solving and graphing absolute-value equations and absolute-value inequalities
Key Words
• graph of an inequality,
p. 323
• addition property of
inequality, p. 324
• subtraction property
of inequality, p. 324
multiplication property
of inequality, pp. 330,
331
division property of
inequality, pp. 330, 331
compound inequality
p. 342
• absolute-value
equation, p. 355
• absolute-value
inequality, p. 361
• linear inequality in
two variables, p. 367
PREPARE
Chapter Readiness Quiz
STUDY TIP
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to p. 26)
1. Which of the following is not an inequality?
(A) x — 2 < 1 (ID 6/7 — 4 > 26 Cc) 5^ +1 = 11 (Ip y < — 5
Skill Check (refer to pp. 26, 151, 212)
2. Which inequality has 5 as one solution?
(A) 1 + 2x > 12 (ID 3x — 2 < 13 Cc) 8 + x < 12 Cp) 4x > 28
3- Which number is a solution of the equation 6x + 8 = 36 + 2x?
(a) 3 CM) 6 CM) V CM) 12
4. Which is the equation that represents
the graph shown at the right?
(A) y = x + 1 CID y = x — 1
(C) y = —x + 1 Cp) y = —x — 1
Check Your Work
Showing all your steps
when you do your
homework helps
you to find errors.
Chapter 6 Solving and Graphing Linear Inequalities
Solving Inequalities Using
Addition or Subtraction
Goal
Solve and graph one-step
inequalities in one
variable using addition
or subtraction.
Key Words
• graph of an inequality
• equivalent inequalities
• addition property of
inequality
• subtraction property of
inequality
How far away are the stars?
The star that appears brightest in the
night sky is Sirius. Sirius is very far
from Earth. It takes nearly 9 years for
its light to reach us. In Example 4 you
will write an inequality to describe
even greater distances.
The graph of an inequality in one variable is the set of points on a number line
that represent all solutions of the inequality. If the endpoint on the graph is a
solution, draw a solid dot. If it is not a solution, draw an open dot. Draw an
arrowhead to show that the graph continues on indefinitely.
Student HeCp
► Reading Algebra
An open dot in a
graph represents
< or > inequalities. A
solid dot represents
< or > inequalities.
V
*1
J
Graph an Inequality in One Variable
Write a verbal phrase to describe the inequality. Then graph the inequality.
a. x<2 b.a>—2 c. z<l d. d>0
Solution
INEQUALITY VERBAL PHRASE GRAPH
a. v < 2
All real numbers less
than 2
Frequency Ranges o
b. a > —2
All real numbers greater
—i—*• :
_ i _ _i
i i
f 1
— >—
than —2
-3 -2 -1
0 1
2
3
c. z < 1
All real numbers less
^ a_i —i—
'i i i
-+-
than or equal to 1
-3 -2 -1
o i
2
3
d. d>0
All real numbers greater
-iii
JL .1
. 4
_ JL
than or equal to 0
-3 -2 -1
0 1
2
3
__
Graph an Inequality in One Variable
Write a verbal phrase to describe the inequality. Then graph the inequality.
1.f>l 2. x> —1 3. n<0 4. y < 4
6.1 Solving Inequalities Using Addition or Subtraction
A solution of an inequality in one variable is a value of the variable that makes
the inequality true. To solve such an inequality, you may have to rewrite it as a
simpler equivalent inequality. Equivalent inequalities have the same solutions.
Adding the same number to, or subtracting the same number from, each side of
an inequality in one variable produces an equivalent inequality.
3 < 7 + 2 +2
3 + 2 < 7 + 2 +—I-1-1-1-1-*-1-1-1-+-I—►
0 1 2 3 4 5 6 7 8 9 10
5 < 9
Student HeCp
\
► Study Tip
The properties are
stated for > and <
inequalities. They are
also true for > and <
inequalities.
\ _ )
PROPERTIES OF INEQUALITY
Addition Property of Inequality
For all real numbers a, b, and c: If a > b , then a + c > b + c.
If a < b, then a + c < b + c.
Subtraction Property of Inequality
For all real numbers a, b, and c: If a > b , then a - c > b - c.
If a < b, then a - c < b - c.
Student MeCp
► Study Tip
To check solutions,
choose numbers that
make the arithmetic
easy. For Example 2
you could check zero
as a value of x. ******
0 + 5>3
5>3*/
| 2 Use Subtraction to Solve an Inequality
Solve v + 5 > 3. Then graph the solution.
Solution
v + 5 > 3
x + 5 — 5 > 3 — 5
. ► x > —2
Write original inequality.
Subtract 5 from each side.
(Subtraction Property of Inequality)
Simplify.
ANSWER ► The solution is all real numbers greater than or equal to —2. Check
several numbers that are greater than or equal to —2 in the original
inequality. The graph of the solution is shown below.
h—: —i—f—+
-3 -2-10 1
1 i
2 3
You cannot check all the solutions of an inequality. Instead, choose several
solutions. Substitute them in the original inequality. Be sure they make it true.
Then choose several numbers that are not solutions. Be sure they do not make the
original inequality true.
Use Subtraction to Solve an Inequality
Solve the inequality. Then graph the solution.
5. x + 4 < 7 6. n + 6>2 7. 5>a + 5
Chapter 6 Solving and Graphing Linear Inequalities
3 Use Addition to Solve an inequality
Solve — 2 > n — 4. Then graph the solution.
Student HeCp
Solution
- , \
►Writing Algebra
The inequality 2 > n
can also be written
as n < 2 and has the
same solution as n < 2. * h
\ _
—2 > n
—2 + 4 > n
.► 2 > n
ANSWER ^ The
— 4 Write original inequality.
— 4 + 4 Add 4 to each side.
(Addition Property of Inequality)
Simplify.
solution is all real numbers less than 2. Check the solution.
The graph of the solution is shown below.
-3
-2
-1
+
3
Use Addition to Solve an Inequality
Solve the inequality. Then graph the solution.
8. x - 5 > 2 9- p— 1 < —4 10. -3 < y - 2
Link to
Science
ASTRONOMY You can use
the constellation Orion to
locate Sirius. Orion's belt
points southwest to Sirius.
4 Write and Graph an Inequality in One Variable
ASTRONOMY A light year is the distance light travels in a year. One light
year is about 6,000,000,000,000 miles. The star Sirius is about 8.8 light years
from Earth. Write an inequality that describes distances to points in space that
are farther from Earth than Sirius is. Then graph the inequality.
Solution Let d be the distance in light years of any point in space that is
farther from Earth than Sirius is.
Write the inequality in words: The distance is greater than 8.8.
Translate into mathematical symbols: d > 8.8
ANSWER ► The inequality is d > 8.8. The graph of the inequality is shown below.
8.8
-—I-1-1-i—at-1-1—►
5 6 7 8 9 10 11
Write and Graph an Inequality in One Variable
11. In Example 4, suppose that d represented distances to points in space whose
distance from Earth is greater than or equal to the distance from Earth to
Sirius. How would the inequality change? How would the graph change?
12. Deneb is about 1600 light years from Earth. Write an inequality that
describes the distances to points in space that are farther from Earth than
Deneb is. Then graph the inequality.
6.1 Solving Inequalities Using Addition or Subtraction
Exercises
Guided Practice
1. Describe the graph of an inequality of the form x>a.
2. Explain why x — 6 > 10 and x > 16 are equivalent inequalities.
Vocabulary Check
Skill Check Decide whether you would use an open dot or a solid dot to graph
the inequality.
3. a <3 4.10 <k 5.;>-l
6. m + 5<4 7.x —3 >12 8.-1 <3 + 1
Tell whether the arrow on the graph of the inequality points to the right
or to the left.
9. x < 8 10. _y > 20 11.7 + a<28
12. f+8<12 13. -6>r-5 14. fc-8>-l
Practice and Applications
DESCRIBING INEQUALITIES Write a verbal phrase to describe the
inequality.
15. z < 8 16. t< —3 17. p>2l 18. ra > 0
CHECKING SOLUTIONS Check to see if the given number is or is not a
solution of the inequality graphed below.
-- 1 - 1 - 1 - 1 - 1 - 1 - 1 --
-5 -4 -3 -2-10 1
19. 3 20. -3 21. 0 22. 1
GRAPHING Match the inequality with its graph.
23. n > —2 24. y < — 2 25.x>2
26. w < —2
27.2 >z
28. 2 < c
A.
H-
-2
+
0
+
2
+
0
+
2
+
0
+
2
Student tteCp
► Homework Help
Example 1: Exs. 15-28
Examples 2 and 3:
Exs. 29-52
Example 4: Exs. 53-56
" _y
D. —4 -1-h— E. —|-1-h— F. <«—|-1-t—►
-2 0 2 -2 0 2 -2 0 2
USING INVERSE OPERATIONS Tell which number you would add to or
subtract from each side of the inequality to solve it.
29. k + 11 < -3
30. h ~ 2 > 14
31. r + 6< -6
32. 31 < —4 + y
33. -7 > -3 + x
34. 17 + z < -6
Chapter 6 Solving and Graphing Linear Inequalities
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 41-55 is available at
www.mcdougallittell.com
Unkjt^
Science
MERCURY The melting
point of mercury is the
temperature at which
mercury becomes a liquid.
Mercury is the only metal
that is a liquid at room
temperature.
More about mercury
is available at
www.mcdougallittell.com
SOLVING AND MATCHING Solve the inequality. Then match the solution
with its graph.
35. d + 4 < 6
36. x — 3 > 2
37. q + 12 > 4
38. h + 6 < -2
39. s ~ 5 > -5
40. v - 3 < 2
1, 1 _
R - 1 1 i 1 JL-
C . i i i ■ >
1 1 w — 1 —
-1 0 1
! 1
2 3
* 1 1 . 1 1
-10 -9 -8 -7 -6
* 1 1 ! i i
3 4 5 6 7
D iii
j. i „
E i i i i r „
E ^ i i i i i _
-1 0 1
1 *
2 3
i i T 1 1 1
3 4 5 6 7
* f | | ►
-10 -9 -8 -7 -6
SOLVING AND GRAPHING Solve the inequality. Then graph the solution.
41. x + 6 < 8
42. -5 <4 +/
43. -4 +/< 20
44. 8 + w< -9
45. p — 12 > —1
46. -2 > b - 5
47. —8 < jc — 14
48. m + 7 > -10
49. -6 > c - 4
50. -2 + z < 0
51. — 10 > a — 6
52. 5 + r> -5
53. x - 5 > 7
54. 14 < 8 + n
55. c + 11 < 25
56. CRITICAL THINKING
Jesse finished a 200 meter dash in 35 seconds. Let r
represent any rate of speed in meters per second faster than Jesse’s.
a. Write an inequality that describes r. Then graph the inequality.
b. Every point on the graph represents a rate faster than Jesse’s. Do you think
every point could represent the rate of a runner?
57. WALKING RACE A racer finished a 5 kilometer walking race in
45 minutes. Let r represent any faster rate in kilometers per minute.
Write an inequality that describes r.
58. Scien ce Link / Mercury has the lowest melting point of any metallic
element, — 38.87°C. Let p represent the melting point in degrees Celsius of
any other metallic element. Write an inequality that describes p.
59. ASTRONOMICAL DISTANCES The star Altair is about 5 parsecs from
Earth. (A parsec is about 3.26 light years.) Let d represent the distance
from Earth of any point in space that is more distant than Altair. Write an
inequality that describes d in light years. Then graph the inequality.
60. SHARKS On July 27, 1999, a mako shark
weighing 1324 pounds was caught off the coast of
Massachusetts by a fisherman named Kevin Scola.
It was the biggest mako shark ever caught using
a rod and reel. Let w represent the possible weight
in pounds of any mako shark caught before that time
using a rod and reel. Write and graph an inequality
that shows all possible values of w.
Photo by Kevin Scola of the
fishing vessel Survival
6.1 Solving Inequalities Using Addition or Subtraction
ERROR ANALYSIS In Exercises 61 and 62, find and correct the error.
Standardized Test
Practice
Mixed Review
Maintaining Skills
63. MULTIPLE CHOICE Which statement about the inequality x — 3 > 2 is true?
(A) The arrow on its graph points to the left.
QD “I is a solution.
CcT) The dot on its graph is solid.
CD 5 is not a solution.
64. MULTIPLE CHOICE Which number is not a solution of the inequality
—5 + f<7?
(T) 12 (3D -12 (H) o GD 5
SOLVING EQUATIONS Solve the equation. (Lesson 3.3)
65. 4x - 3 = 21 66. -5jc + 10 = 30 67. -3s - 2 = -44
68. jx + 5 = —4 69. ^(a + 4) = 18 70. y(x — 5) = 6
71. n + 2n + 5 = 14 72. 3(x - 6) = 12 73. 9 = -3(x - 2)
74. BIKE RIDING You ride an exercise bike each day. The table below shows the
time t in minutes and the distance d in miles that you rode on each of four
days. Write a model that relates the variables d and t. (Lesson 4.5)
Time t
5
10
12
15
Distance d
0.60
1.20
1.44
1.80
WRITING EQUATIONS Write in slope-intercept form the equation of the
line that passes through the given points. (Lesson 5.3)
75. (1, 2), (4, - 1) 76. (2, 0), (-4, -3) 77. (1, 1), (-3, 5)
78. (-1,4), (2, 4) 79. (-1,-3), (2, 3) 80. (8, 1), (5,-2)
81. (-2,4), (4, 2) 82. (1,-5), (6, 5) 83. (-3, 6), (2, 8)
RECIPROCALS Find the reciprocal. (Skills Review p. 763)
85. -f
24
86 ‘ 25
87 -
«8.f
90. 3
91. -1
92.4
93. 9
§
94. n
__ 5
95. 8
Chapter 6 Solving and Graphing Linear Inequalities
DEVELOPING CONCEI
Ju ya^jxjii'ijuxj JuixjiJiiJj-iJi:
For use with
Lesson 6.2
Goal
Use reasoning to
determine whether
operations change an
inequality.
Question
How do operations affect an inequality?
Explore
■■ ■ ■LL.J— | ,
Materials
• paper
• pencil
Q Write a true inequality by choosing two different numbers and placing the
appropriate symbol < or > between them.
Q Copy and complete the table. Apply the given rule to each side of your
inequality. Write the correct inequality symbol between the resulting numbers.
Original
inequality
Rule
Resulting
inequality
Did you have to
reverse the inequality?
?
Add 4.
?
?
?
Add -4.
?
?
?
Subtract 4.
?
?
?
Subtract -4.
?
?
?
Multiply by 4.
?
?
?
Multiply by -4.
?
?
?
Divide by 4.
?
?
?
Divide by —4.
?
?
© Repeat Step 2 using different positive and negative integers.
Think About It
Using your results from Steps 2 and 3, predict whether the inequality
symbol will change when you apply the given rule. Check your prediction.
1, 4 < 9; add 7. 2. 15 > 12; subtract —5. 3- 4 > —3; multiply by 5.
4. 1 < 8; multiply by —10. 5. —6 < 2; divide by —3. 6. 0 < 8; divide by 2.
7- LOGICAL REASONING Copy and complete the table.
Must you reverse the inequality?
a positive number
a negative number
Add
?
?
Subtract
?
?
Multiply by
?
?
Divide by
?
?
Developing Concepts
Solving Inequalities Using
Multiplication or Division
Goal
Solve and graph one-step
inequalities in one
variable using
multiplication or division.
Key Words
• multiplication property
of inequality
• division property of
inequality
Should you rent or buy ice skates?
Aisha wants to learn to figure
skate. Should she rent skates or
buy them? In Exercise 56 you
will solve an inequality to help
her decide.
The results of Developing Concepts 6.2, page 329, suggest that you can solve an
inequality by multiplying or dividing each side by the same positive number, c.
r Student HeCp
► Study Tip
The properties are
stated for > and <
inequalities. They are
also true for > and <
inequalities.
v j
PROPERTIES OF INEQUALITY
r
Multiplication Property of Inequality (c > 0)
For all real numbers a, b, and for c > 0: If a > b, then ac > be.
If a < b, then ac < be.
Division Property of Inequality (c > 0)
3 b
For all real numbers a, b, and for c > 0: If a > b, then — > —.
c c
If a < b, then — < —.
c c
f<10
f<4
10
Multiply by a Positive Number
Original inequality.
Multiply each side by 4.
(Multiplication Property of Inequality)
a <40
Simplify.
ANSWER ► The solution is all real numbers less than or equal to 40. The graph
of the solution is shown below.
-10
10
20
30
40
V
Chapter 6 Solving and Graphing Linear Inequalities
2 Divide by a Positive Number
4x > 20 Original inequality.
4x 20 , .....
Divide each side by 4.
(Division Property of Inequality)
x > 5 Simplify.
ANSWER ► The solution is all real numbers greater than 5. The graph of the
solution is shown below.
4-t-1-1-1-4-1-1-t-1-h
0 1 2 3 4 5 6 7 8 9 10
Multiply or Divide by a Positive Number
Solve the inequality. Then graph the solution.
1 .|<| 2 . 18 <2 k 3.6<y 4 . —21 < 3_y
The results of Developing Concepts 6.2, page 329, suggest that you must reverse ,
or change the direction of, the inequality when you multiply or divide each side
by the same negative number, c.
Student HeCp
►Writing Algebra
To reverse an inequality:
< becomes >
< becomes >
> becomes <
> becomes <
PROPERTIES OF INEQUALITY
Multiplication Property of Inequality (c < 0)
For all real numbers a, b, and for c < 0:
If a > b, then ac < be.
If a < b, then ac > be.
Division Property of Inequality (c < O)
c? b
For all real numbers a, b, and for c < 0: If a > b, then — < —.
c c
If a < b, then — > —.
c c
3 Multiply by a Negative Number
h><5
Original inequality.
-2(-^)>-2(5)
Multiply each side by -2 and reverse the inequality.
y>—10 Simplify.
ANSWER ► The solution is all real numbers greater than or equal to — 10.
The graph of the solution is shown below.
H-1-h
5 10 15
-15 -10
6.2 Solving Inequalities Using Multiplication or Division
Student HeCp
► More Examples
More exam Pl es
are available at
www.mcdougallittell.com
4 Divide by a Negative Number
Solve the inequality. Then graph the solution,
a. —12m > 18 b. — 8x < 20
Solution
a. —12m > 18 Write original inequality.
< Divide each side by -12 and reverse the inequality.
m<—1.5 Simplify.
ANSWER ► The solution is all real numbers less than — 1.5. The graph of the
solution is shown below.
-1.5
;— i i o i —i—i—►
-4 -3 -2-10 1
b. — 8x < 20
-Sx 20
-8 “ -8
v > —2.5
Write original inequality.
Divide each side by -8 and reverse the inequality.
Simplify.
ANSWER ► The solution is all real numbers greater than or equal
to —2.5. The graph of the solution is shown below.
- 4 -
-4
Multiply or Divide by a Negative Number
Solve the inequality. Then graph the solution.
^ 1 , _ 2 .
5. — —/? >1 6- —~x < —5
8. — 14z> -70 9. -24 < 6 1
7. -\k<-4
10 . 12 > - 5 n
3llPti
Properties of Inequality
For all real numbers a, b, and c:
• If a > b, then a + c > b + c and a - c > b - c.
•If a < b, then a + c < b + c and a - c < b - c.
a b
• If a > b and c > 0, then ac > be and — > —.
c c
a b
•If a > b and c < 0, then ac < be and — < —.
c c
Chapter 6 Solving and Graphing Linear Inequalities
Exercises
Guided Practice
Vocabulary Check 1 . Explain what “reverse the inequality” means.
2 . Are the inequalities —x<2 and 2< —x equivalent? Explain.
Skill Check Describe the first step you would use to solve the inequality. Then tell
whether you would reverse the inequality.
3.f>3
4 . -9 < | 5. 4 w > 48
6 . -56 > 8 d
7 . ~b <3 8.-4 >-d
6
Tell whether the inequalities are equivalent. Explain your reasoning.
9 . —k > 42, k > —42
10.-|<- s , s <|
11 . 4 > —yc, c > —28
12 . 5 z < ~ 75, z > —15
13 . — llx > 33, x > —3
14 . —y < —5, w> 15
Practice and Applications
SOLVING INEQUALITIES Describe the first step you would use to solve
the inequality. Then tell whether you would reverse the inequality.
15. |>6 16. 81 <9? 17. | >-26 18. 2r<-2
19. —Ik > —56 20. 4>-|y 21. 48 <-3* 22. -|<-6
GRAPHING INEQUALITIES Tell whether the graph below is the graph of
the solution of the inequality.
--1-1-1-1-1-1-1-1-i—-
0 1 2 3 4 5 6 7 8
23. -f < - 1 24. ~lz < 2 25. 5z > 30 26. 2z > 12
6 3
LOGICAL REASONING Tell whether the inequalities are equivalent.
Explain your reasoning.
27. 12_y > —24; y < —2 28. -\m > -3; m > 24
O
29 - 15 < — b\ —15 > b 30 . < —2; n > —6
31 . 8 < ~m\ -16 >m 32 . 20b>~2;b< - jj
Student HeCp
► Homework Help
Examples 1 and 2:
Exs. 15-18, 23-55
Examples 3 and 4:
Exs. 19-22, 23-52
k __ J
6.2 Solving Inequalities Using Multiplication or Division
ERROR ANALYSIS In Exercises 33 and 34, find and correct the error.
FARM AID Concerts are
often used to raise funds for
charity. The Farm Aid concerts
have raised over $14.5 million
for American farmers
since 1985.
■
33 .
SOLVING INEQUALITIES Solve the inequality. Then graph the solution.
35. 15/7 < 60
36. 6k > —120
37- \j - —12
38. -a> -100
39. ~n < 12
40. 20 y > 50
41. 11 > —
42. — 18x > 9
43 — — < -2
10 “
44. | z > 24
45. — 12r > -18
46. —4 f< 14
ESTIMATION Estimate the solution and explain your method.
47. 10 > 1.999 d 48. \r< -50.1155 49. -|o>5.91
LOGICAL REASONING
or never.
Complete the statement with always, sometimes,
50 . If k is greater than 0, then kx is ? greater than 0.
51 . Ifk is greater than 0 and x is greater than zero, then kx is ? greater than 0.
52 . If* is less than 0, then kx is ? greater than 0.
53 . Ifk is less than 0 and v is greater than zero, then kx is ? greater than 0.
54 . FOSTERS You want to buy some posters to decorate your dorm room.
Posters are on sale for $5 each. Write and solve an inequality to determine
how many posters you can buy and spend no more than $25.
55 . FUNDRAISING Musicians are planning a fundraiser for local farmers. The
admission fee will be $20. Write and solve an inequality to determine how
many tickets must be sold to raise at least $25,000.
56 . FIGURE SKATING Aisha plans to take figure skating lessons. She can rent
skates for $5 per lesson. She can buy skates for $75. For what number of
lessons is it cheaper for Aisha to buy rather than rent skates?
57 . SUBWAY You can ride the subway one-way for $.85. A monthly pass costs
$27.00. For what number of rides is it cheaper to pay the one-way fare than
to buy the monthly pass?
4
58 . CHALLENGE Solve the inequality — > 2 by multiplying each side by v.
HINT: Consider the cases v > 0 and v < 0 separately.
Chapter 6 Solving and Graphing Linear Inequalities
Standardized Test
Practice
Mixed Review
Maintaining Skills
59. MULTIPLE CHOICE Which inequality is represented by the graph?
-i-1-1-1-1-1-4-1-1-►
-2-1012345
(A) 3x>9 Cb) 24 > 8_y ®f-° C®-6<-|
60. IVIULTIPLE CHOICE Which inequality is not equivalent to k < —3?
C T)3>-k (G) -3k>9 CED A: + 4 < 1 Q)|ifc<-2
61. IVIULTIPLE CHOICE Solve -5 jc < -10.
(A) x < — 2 (b)x<2 (© x> -2 CD) x > 2
FINDING DIFFERENCES Find the difference. (Lesson 2.4)
62.12- 19 63. -6- 8 64. 3 -(-1)
65. -7 - (-7) 66. -9 - 9 67. 0 - (-2)
FINDING QUOTIENTS Find the quotient. (Lesson 2.8)
68. 52 h- (-26) 69. -8 + 2 70. -10 - (-2)
71.72. 23 + (~) 73. —15 + ( —1|)
SOLVING EQUATIONS Solve the equation. Check your solution in the
original equation. (Lesson 3.5)
74. 2(x + 5) = 5(x - 1) 75. ~4(y + 3) = -(6 - 2 y)
76. 8 - (c + 7) = 6(11 - c) 77. 3(-jc - 2) = 2x + 2(4 + x)
SOLVING FORMULAS Solve the formula for the indicated variable.
(Lesson 3.7)
78. d = — 79. A = \bh 80. P = a + b + c
v 2
Solve for m. Solve for b. Solve for c.
81. IDENTIFYING ORDERED PAIRS
Write the ordered pairs that
correspond to the points labeled
A, B , C, and D in the coordinate
plane at the right. (Lesson 4.1)
FACTORS List all the factors of the number. (Skills Review p. 761)
82.98 83.140 84.114
85. 144 86. 289 87. 425
88. 1064 89. 2223 90. 5480
6.2 Solving Inequalities Using Multiplication or Division
Solving Multi-Step
Inequalities
Goal
Solve multi-step
inequalities in one
variable.
Key Words
• multi-step inequality
Can you make a profit selling fishing flies?
In fishing, a fly is a lure that is made to look
like a real insect. In Example 5 you will use
an inequality to figure out how you can make
a profit selling fishing flies.
The inequalities in Lessons 6.1 and 6.2 could
be solved in one step using one operation.
A multi-step inequality requires more than
one operation.
Solve a Multi-Step Inequality
Solve 2y — 5 < 7.
Solution
2y - 5 <1
Write original inequality.
2y < 12
Add 5 to each side.
y < 6
Divide each side by 2.
ANSWER ► The solution is all real numbers less than 6.
2 Solve a Multi-Step Inequality
Solve 5 — x> 4.
Solution
5 — x> 4 Write original inequality.
—v > — 1 Subtract 5 from each side,
x < 1 Multiply each side by -1 and reverse the inequality.
ANSWER ^ The solution is all real numbers less than 1.
Solve a Multi-Step Inequality
Solve the inequality.
1 _ 3x — 5 > 4
2 . 10 - n<5
Chapter 6 Solving and Graphing Linear Inequalities
Student HeQp
► More Examples
More examples
are available at
www.mcdougallittell.com
3 Use the Distributive Property
Solve 3(x + 2) < 7.
Solution
3(x + 2) < 7 Write original inequality.
3x + 6 < 7 Use distributive property.
3x < 1 Subtract 6 from each side.
1
X< 3
Divide each side by 3.
ANSWER ► The solution is all real numbers less than —.
Use the Distributive Property
Use the distributive property as the first step in solving the inequality.
Then tell what the next step is and solve the inequality.
3. 3{n - 4) > 6 4. -2(x + 1) < 2
JE5HSBU 4 Collect Variable Terms
Solve 2x — 3 >4x — 1.
Solution
Method 1
2x — 3 > 4x —
Student HeGp
-\
► Study Tip
To avoid concerns
about reversing the
inequality, first collect
variable terms on the
side whose variable
term has the greater
coefficient. .
^ _ J
2x > 4x +
— 2x > 2
x< —1
Method 2
2x — 3 > 4x —
---••► — 3 > 2x —
—2 > 2x
1
2
1
1
Write original inequality.
Add 3 to each side.
Subtract 4xfrom each side.
Divide each side by -2 and reverse the inequality.
Write original inequality.
Subtract 2xfrom each side.
Add 1 to each side.
— 1 > x Divide each side by 2.
ANSWER ► The solution is all real numbers less than or equal to — 1.
v_
Collect Variable Terms
describe the steps you used.
6 . — 3z + 15 > 2z
8 . 4y — 3 < —y + 12
Solve the inequality and
5. 5n — 21 < 8 n
7. x + 3 > 2x — 4
6.3 Solving Multi-Step Inequalities
Link to
Sports
FLY-FISHING A fishing fly is
made by attaching feathers,
pieces of shiny metal, or
colored thread to a fishhook.
The process is known as
"tying flies."
Student HeGp
► Study Tip
Be sure to interpret
the solution to reflect
the real-life situation.
k _/
5 Write and Use a Linear Model
FLY-FISHING You want to start tying and selling
fishing flies. You purchase the book shown in the
advertisement. The materials for each fly cost $.20.
You plan to sell each fly for $.60. How many fishing
flies must you sell to make a profit of at least $200?
Solution
Profit is equal to income minus expenses. To find
your income, multiply the price per fly by the
number sold. Your total expenses include the cost of
materials, $.20 per fly, and the cost of the book, $15.
to tiefties
at home, fo€ profit
Illustrated
step-by-step book
shows you how!
Send $13.95 plus $1.05 for
shipping and handling to:
Wehave Nonameforit
Verbal
Price
Number of
Total
>
Desired
Model
per fly
flies sold
expenses
profit
Labels
Algebraic
Model
Price per fly = 0.60
Number of flies sold = x
Total expenses = 0.20 x + 15
Desired profit > 200
0.60 x - (0.20 x + 15) > 200
0.60x - 0.20x - 15 > 200
0.4x - 15 > 200
0.4x > 215
x> 537.5
(dollars per fly)
(flies)
(dollars)
(dollars)
Write algebraic model.
Use distributive property.
Combine like terms.
Add 15 to each side.
Divide each side by 0.4.
ANSWER ^ You cannot sell half a fishing fly. So you must sell at least 538 flies
to make a profit of at least $200.
Write and Use a Linear Model
You plan to make and sell candles. You pay $12 for instructions. The
materials for each candle cost $0.50. You plan to sell each candle for $2.
Let x be the number of candles you sell.
9. Write an algebraic expression for each quantity.
a. your income b. cost of materials
c. total expenses d. your profit
10. Write and solve an inequality to determine how many candles you must sell
to make a profit of at least $300.
Chapter 6 Solving and Graphing Linear Inequalities
6.3 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Explain why 3a + 6 > 0 is a multi-step inequality.
2 . Describe the steps you could use to solve the inequality — 3 y + 2 > 11.
Determine whether the inequality is a multi-step inequality. Then explain
how you would solve the inequality.
3. d + 2 > — 1
6 . 4y - 3 < 13
9. \b + 2 > 6
4. — a < 0
7. 5x + 12 <62
10 . 3 m + 2 < 1m
5. -4x> -12
8 . 10 - c>6
11. 2w — 1 > 6w + 2
Practice and Applications
COMPLETING THE SOLUTIONS Copy and complete the exercise to solve
the inequality.
12. 4x — 3 > 21
4x — 3+ ? > 21 + ?
4x ?
4 “ 4
> ?
13. 7 < 14 - jfc
7- ? < 14 - k - ?
2 <-k
-!(■)>-!(-£)
I I > /
Student HeCp
I ►Homework Help
Examples 1 and 2:
Exs. 12-30
Example 3: Exs. 31-43
Example 4: Exs. 36-43
Example 5: Exs. 44-48
v j
JUSTIFYING SOLUTIONS Describe the steps you would use to solve
the inequality.
14. la — A <11 15. 11 — 2/i > —5
17. 22 + 2b< —2 18. |f + 5>|?
SOLVING INEQUALITIES Solve the inequality.
20 . x + 5 > —13 21 . —6 + 5x < 19
16.|x + 5>-15
19. 6 (z - 2) < 15
22 . 4x- 1 < -17
23. —5 < 6x — 12
26. -x + 9 > 14
24. —17 > 5x - 2
27. 7 - 3x < 16
25. 15 + x>7
28. 12 > —2x - 6
MATCHING Match the inequality with its graph.
29.3x + 9>6 30.-3x- 9>6 31.-3(x-3)>6
A. -1-1-h
-10 12
B.
<—I-h
-10 -5
0
C.
-2 -1 0
■I—►
1
6.3 Solving Multi-Step Inequalities
Link_
Careers
AMUSEMENT RIDE
DESIGNERS use math and
science to ensure rides are
safe. An amusement ride
designer usually has a degree
in mechanical engineering.
More about
amusement ride
designers is available at
www.mcdougallittell.com
SOLVING INEQUALITIES Solve the inequality.
32. 2(x -4) >3
33. \(x -
35. 15 < |(x + 4)
36. -x -
38. 2x + 10 > 7x + 7
8) <2 34. —(2x + 4) > 6
4 > 3x — 2 37. 6 + v < —4x + 1
39. 9 — 3x > 5(—x + 2)
40. — 3(x + 3) < 4x — 7
41. 6(x + 2) > 3x —2
ERROR ANALYSIS In Exercises 42 and 43, find and correct the error.
42. _ a. > _ & 43. t ^
12 4f - 1
AMUSEMENT RIDES In Exercises 44 and 45, use the following information.
An amusement park charges $5 for admission and $1.25 for each ride ticket.
You have $25. How many ride tickets can you buy?
Price
•
Number
+
Admission
<
Amount of
per ticket
of tickets
price
$ you have
44. Assign labels to the verbal model above and write the algebraic model.
45. Solve the inequality and interpret the result.
PIZZA TOPPINGS In Exercises 46-48, use the following information.
You have $18.50 to spend on pizza. A cheese pizza costs $14. Each extra topping
costs $.75. How many extra toppings can you buy?
46. Write a verbal model to represent the problem.
47. Assign labels to your model and write the algebraic model.
48. Solve the inequality and interpret the result.
[ Student HeCp
► Skills Review
For help with perimeter
and area, see p. 772.
L J
ionwttY Write and solve an inequality for the values of x.
49. Perimeter > 26 meters
9m
51. Area <12 square feet
50. Perimeter < 25 meters
C
3 _
_ c
x
52. Area >144 square inches
Chapter 6 Solving and Graphing Linear Inequalities
Standardized Test
Practice
Mixed Review
Maintaining Skills
Quiz 1
53. MULTIPLE CHOICE Solve the inequality 2x - 10 > 3(-Jt + 5).
(A) x < 5 CD x > —5 (©*>5 CD * < “5
54. MULTIPLE CHOICE Which number is not a solution of 4(x + 2) > 3x — 1?
CD -10 CD -8 (jD 0 CD 10
EVALUATING EXPRESSIONS Evaluate the expression. (Lessons 1.1 and 1.2)
55. (a + 4) — 8 when a — 1 56. 3x + 2 when x — —4
57. b 3 — 5 when b — 2 58. 2(r + s) when r = 2 and s = 4
TRANSLATING SENTENCES Write the verbal sentence as an equation or
an inequality. (Lesson 1.5)
59. Sarah’s height h is 4 inches more than your height a.
60. The number c of cows is more than twice the number s of sheep.
61. SHOPPING You bought a pair of shoes for $42.99, a shirt for $14.50, and a
pair of jeans for $29.99. You used a coupon to save $10 on your purchase.
How much did you spend? (Lesson 2.3)
MIXED NUMBERS AND IMPROPER FRACTIONS Write the mixed number
as an improper fraction. (Skills Review p. 763)
62.2^ 63. 1- 64.2o| 65. 3|
Graph the inequality. (Lesson 6.1)
1. b> 12 2.j > —9 3. —8 > y
Solve the inequality. Then graph the solution. (Lessons 6. 7 and 6.2)
4. a + 2 < 7 5. — 3 + m< — 11 6. —13 > b — 1
7. \z>-2 8. — |x<-27 9. 105 > — 15k
10. RIDES A person must be at least 52 inches tall to ride the Power Tower ride
at Cedar Point in Ohio. Let h represent the height of any person who meets
the requirement. Write an inequality that describes h. (Lesson 6.1)
11 . PLAYS It costs $20 to attend a play at the playhouse. A season’s pass costs
$180. For what number of plays is it cheaper to pay the $20 price than to buy
the season’s pass? (Lesson 6.2)
Solve the inequality. (Lesson 6.3)
12. 5<-| + 4 13. -4x + 2>14
15. -(-* +8)>-10 16. -10 < -2(2* - 9)
14. — x — 4 > 3x — 12
17. x + 3 < 2(x — 7)
6.3 Solving Multi-Step Inequalities
Solving Compound
Inequalities Involving "And"
Coal
Solve and graph
compound inequalities
involving and.
Key Words
• compound inequality
Where can plants grow on
a mountain?
The types of plant life on a mountain
depend on the elevation. At lower
elevations, trees can grow. At higher
elevations, there are flowering plants,
but no trees. At very high elevations,
there are no trees or flowering plants.
In Example 2 you will use inequalities
to describe such plant-life regions.
A compound inequality consists of two inequalities connected by the word and
or the word or. You will study the first type of compound inequality in this lesson.
You will study the second type in Lesson 6.5.
Student HeCp
>
► Study Tip
A number is a solution
of a compound
inequality with and if
the number is a
solution of both
inequalities.
I _/
1 Write Compound Inequalities with And
Write a compound inequality that represents the set of all real numbers greater
than or equal to 0 and less than 4. Then graph the inequality.
Solution
The set can be represented by two inequalities.
0 < x and x < 4
The two inequalities can then be combined in a single inequality.
0 < x < 4
The compound inequality may be read in these two ways:
• x is greater than or equal to 0 and x is less than 4.
• x is greater than or equal to 0 and less than 4.
The graph of this compound inequality is shown below.
*—I- *— I - 1 - 1 — $—h—
-1 0 1 2 3 4 5
Write a verbal sentence that describes the inequality.
1. —2 < y < 0 2. 7 < £ < 8 3.4<rc<ll
Chapter 6 Solving and Graphing Linear Inequalities
MOUNTAIN PLANT LIFE
The timberline on a mountain
is the line above which no
trees grow. Alpine flowers
are flowers that grow above
the timberline.
Student HeCp
— --
►Study Tip
To perform any
operation on a
compound inequality
with and, you must
perform the operation
on all three expressions. -
\ _J
2 Compound Inequalities in Real Life
MOUNTAIN PLANT LIFE
Write a compound inequality
that describes the approximate
elevation range for the type of
plant life on Mount Rainier, a
mountain peak in Washington.
a. Trees below 6000 feet
b. Alpine flowers below
7500 feet
c. No trees or alpine flowers
at or below 14,410 feet
Solution Let y represent the
elevation in feet.
a. 2000 < y < 6000 b. 6000 < y < 7500 c. 7500 < y < 14,410
L_
*22222*3 Solve Compound Inequalities with And
Solve —2<x + 2<4. Then graph the solution.
Solution
Method 1 Separate the inequality. Solve the two parts separately.
x + 2 > — 2 and x + 2 < 4 Separate inequality.
x + 2 — 2>—2 — 2 and x + 2 — 2 < 4 — 2 Subtract 2 from each side.
x > —4 and x<2 Simplify.
—4 < x < 2 Write compound inequality.
Method 2 Isolate the variable between the inequality symbols.
— 2<x + 2<4 Write original inequality.
■*►— 2 — 2<x + 2 — 2<4 — 2 Subtract 2 from each expression.
—4<x<2 Simplify.
ANSWER ► The solution is all real numbers greater than —4 and less than or
equal to 2. The graph of the solution is shown below.
- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 --
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
14,410 ft
No trees or alpine
flowers
— 7500 ft -
finnn ft
Alpine flowers
-1 2000 ft
Trees
j LUUU 1 l
— oft |
Solve Compound Inequalities with And
Choose a method from Example 3 to solve the inequality. Then graph
the solution.
4. -1 <x + 3 <7 5. -6<3x< 12 6. 0<x - 4< 12
6.4 Solving Compound Inequalities Involving "And
4 Solve Multi-Step Compound Inequalities
Solve — 3 < 2x + 1 < 5. Then graph the solution.
Solution
Isolate the variable between the inequality symbols.
-3<2x + 1<5
— 3 — l<2x+l — 1<5 — 1
—4 < 2x < 4
-4 2x 4
2 ~ 2 ~ 2
-2 < x < 2
Write original inequality.
Subtract 1 from each expression.
Simplify.
Divide each expression by 2.
Simplify.
ANSWER ► The solution is all real numbers greater than or equal to —2 and less
than or equal to 2. The graph of the solution is shown below.
+—I-f-I-(-1-1-I-1-I—►
-1 0 1
Student HeCp
^
► Study Tip
When you multiply or
divide each expression
of a compound
inequality by a
negative number,
remember to reverse
both inequalities. .-
_ )
5 Reverse Both Inequalities
Solve —2<—2 — x<l. Then graph the solution.
Solution
Isolate the variable x between the two inequality symbols.
—2 < —2 — x < 1
— 2 + 2 < — 2 — x + 2 < 1 + 2
0 < —x < 3
1(0) > —1(—x) > —1(3)
0 > x > —3
Write original inequality.
Add 2 to each expression.
Simplify.
Multiply each expression by -1 and
reverse both inequalities.
Simplify.
ANSWER ► The solution is all real numbers greater than —3 and less than 0.
The graph of the solution is shown below.
-4-h
H-h
-4 -3 -2 -1
A compound inequality is usually written in a way that reflects the order of
numbers on a number line. In Example 5 above, the solution would usually be
written — 3 < x < 0.
Solve the inequality. Then graph the solution.
7. 3 < 2x + 3 < 7 8. -6 < -3x < 12 9. -3 < -4 - x < 2
Chapter 6 Solving and Graphing Linear Inequalities
Exercises
Guided Practice
Vocabulary Check 1. Name the two connecting words used in compound inequalities.
2 . The word compound comes from a Latin word meaning “to put together.”
Explain why 3 < x < 9 is called a compound inequality.
Skill Check Match the compound inequality with its graph.
3- — 1 < x < 3 4. — 1 < x and x < 3
A. —I 1-1-t-B. +—I-I-I-I-
-10123 -10123
Write a verbal sentence that describes the inequality.
5. 7<4 + x<8 6. — 1 < 2x + 3 < 13 7. 4< -8 - x<7
Write an inequality that represents the statement.
8. x is less than 5 and greater than 2.
9. x is greater than or equal to —4 and less than or equal to 4.
10. x is less than 7 and is greater than or equal to — 1.
Practice and Applications
R Student MeCp
Iomework Help
Example 1: Exs. 11-21
Example 2: Exs. 22-28
Example 3: Exs. 29-34
Example 4: Exs. 35-38
Example 5: Exs. 39-46
\ _ J
READING INEQUALITIES Write a verbal sentence that describes
the inequality.
11. —23 < x < —7 12. 0 < x < 18 13. —4 < x < 19
WRITING INEQUALITIES Write an inequality that describes the graph.
i 4 . - i — i — 1- ~i —i
-4 -3 -2 -1 0
16. -*-H-(-1-1- h
-7 -6 -5 -4 -3
15. ■*—|-1-1-1-h
0 12 3 4
17. ■*—|-1-1-1-h
- 2-1 0 1 2
WRITING AND GRAPHING INEQUALITIES Write an inequality that
represents the statement. Then graph the inequality.
18. x is greater than —6 and less than — 1.
19. x is greater than or equal to 0 and less than 5.
20 . x is greater than 1 and less than or equal to 8.
21. x is less than or equal to —2 and greater than —4.
6.4 Solving Compound Inequalities Involving "And
PONY EXPRESS The Pony
Express carried mail along
a trail from Missouri to
California from the spring of
1860 to the fall of 1861. It
was made obsolete by the
introduction of the telegraph.
? Student HeCp
► Homework Help
Exs. 39-46 is available at
www.mcdougallittell.com
1
22, Hist ory Link } In summer it took a Pony Express rider about 10 days to
ride from St. Joseph, Missouri, to Sacramento, California. In winter it took as
many as 16 days. Write an inequality to describe the number of days d that
the trip might have taken.
FREQUENCY RANGES In Exercises 23-26, use the following information.
Frequency is used to describe the pitch of a sound, which is how high or low it
sounds. Frequencies are measured in hertz . Write an inequality to describe the
frequency range/of the following sounds.
23, Sound of a human voice: 85 hertz to 1100 hertz
24, Sound of a bat’s signals: 10,000 hertz to 120,000 hertz
25, Sound heard by a dog: 15 hertz to 50,000 hertz
26, Sound heard by a dolphin: 150 hertz to 150,000 hertz
27, TELEVISION ADVERTISING In 1967 a 60-second television commercial
during the first Super Bowl cost $85,000. In 1998 advertisers paid
$2.6 million for two 30-second spots. Assuming those were the least and
greatest costs during that period, write an inequality that describes the
cost c of 60 seconds of commercial time from 1967 to 1998.
28, STEEL ARCH BRIDGE The longest
steel arch bridge in the world is
the New River Gorge Bridge near
Fayetteville, West Virginia. The
bridge is 1700 feet long. Write an
inequality that describes the length /
(in feet) of any other steel arch
bridge. Then graph the inequality.
SOLVING COMPOUND INEQUALITIES
the solution.
29. 6 < x - 6 < 8
30. —5 < x — 3 < 6
31. 0<x + 9< 17
32. —14 < 7x < 21
33. -4<2x< 18
34. 4<x - 7< 15
35. -3 <2x + 5 < 11
36. 7 < 3x - 8 < 19
37. 10<3x - 2< 19
38. 0 < 12x + 6 < 18
SOLVING AND GRAPHING INEQUALITIES Solve the inequality. Then
graph the solution.
39. -7 < 3 - jc < 5 40. -25 < — 5jc < 0
41. 42 < —3x < 48 42. -5 < -6 - x<3
43. -3 <5 - 2x< 1 44. -7< -1 - 6x< 11
45. -13 <2 - 5x< -3 46. -44 < 1 - 9x<55
Chapter 6 Solving and Graphing Linear Inequalities
Standardized Test
Practice
Mixed Review
Maintaining Skills
47. CHALLENGE Explain why the inequality 3 < x < 1 has no solution.
48. MULTIPLE CHOICE Which of the following is the graph of -2 < x < 3?
(A) *1 11)1 1* CD *1 1 1 1 1 1 *
-3 -2 -1
-101234
CD —I—I—I—t —y
Co) * 11)11 1
- 2-10123
- 2-10123
49. MULTIPLE CHOICE Which inequality can be solved by reversing both
inequality signs?
(T) -1 < x < 1 CD 15 > 2x + 4 > 1
CD -24 < 3x - 4 < -4 CD -5 > -x > -2
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 1.1)
50. k + 5 when k — 2 51.6 a when a — 4 52. m — 20 when m — 30
x
53. — when x = 30 54. 5 z when z — 3.3 55. 5 p when p = 4
56. 4 — n when n — 3 57. ~ when t — —18 58. 2x when x = 3
SOLVING EQUATIONS Solve the equation. (Lessons 3.1, 3.2)
59.x + 17 = 9 60. -8 = x + 2 61.x - 4 = 12 62. x - (-9) = 15
63. ^ = —6 64. —3x = —27 65. 4x = —28 66. —^x = 21
67. ROLLER SKATING A roller skating rink charges $7 for admission and skate
rental. If you bring your own skates, the admission is $4. You can buy a pair
of roller skates for $75. How many times must you go skating to justify
buying your own skates? (Lesson 3.5)
POPULATION In Exercises 68 and 69, use the following information.
In 1990 the population of the United States was about 249 million. Between
1990 and 1998 the population increased about 2.6 million per year. (Lesson 5.5)
DATA UPDATE of U.S. Bureau of the Census data at www.mcdougallittell.com
68 . Write an equation that models the population P (in millions) in terms of
time t , where t = 0 represents the year 1990.
69. Use the model to estimate the population in 1995.
FRACTIONS AND PERCENTS Write the fraction as a percent.
(Skills Review p. 769)
70 i
71 —
8
72 4
73 -i
37
74 —
50
75 3
75 - 4
76 4
77 —
"■ 25
6.4 Solving Compound Inequalities Involving "And"
Solving Compound
Inequalities Involving "Or"
Goal
Solve and graph
compound inequalities
involving or.
Key Words
• compound inequality
How fast is the baseball moving?
To practice catching pop flies, you might
throw a baseball straight up into the air. As
the ball rises, its velocity gradually decreases
until it reaches its highest point. Then the ball
begins to fall. As it falls, its velocity increases.
In Example 5 you will solve a compound
inequality dealing with the velocity of
a baseball.
In Lesson 6.4 you studied compound inequalities that involve the word and.
In this lesson you will study compound inequalities that involve the word or.
Student MeCp
— -\
► Study Tip
Recall from Lesson 6.4
that graphs of
compound inequalities
with and have only
one part.
X_ J
||^333| 1 Write a Compound Inequality with Or
Write a compound inequality that represents the set of all real numbers less
than — 1 or greater than 2. Then graph the inequality.
Solution
You can write this statement using the word or.
x< — 1 orx> 2
The graph of this compound inequality is shown below. Notice that the
graph has two parts. One part lies to the left of — 1. The other part lies to the
right of 2.
< 1 1 - 1 —
- 2-1 0 1 234
Write a verbal sentence to describe the inequality.
1.x<0<?rx>5 2. x< —10 or x > 10 3. x <2 or x>3
Write an inequality that represents the set of numbers. Tell whether the
graph of the inequality has one part or two.
4. All real numbers less than or equal to —3 or greater than 0.
5. All real numbers less than 3 or greater than 6.
6 . All real numbers greater than —2 and less than 7.
Chapter 6 Solving and Graphing Linear Inequalities
Student HeCp
► Study Tip
A number is a solution
of a compound
inequality with or if the
number is a solution of
either inequality.
L _ J
2 Solve a Compound Inequality with Or
Solve the compound inequality x — 4 < 3 or 2x > 18. Then graph the solution.
Solution
A solution of this compound inequality is a solution of either of its parts. You
can solve each part separately using the methods of Lessons 6.1 and 6.2.
x — 4 < 3 or 2x > 18 Write original inequality.
1 &
x — 4 + 4 < 3 + 4 or — > — Isolate x.
x < 7 or x>9 Simplify.
ANSWER ► The solution is all real numbers less than or equal to 7 or greater
than 9. The graph of the solution is shown below.
^- 1 - 1 - 1 - 1 - 4 - 1 - 1 —►
5 6 7 8 9 10 11
Student HeCp
► More Examples
More examples
are available at
www.mcdougallittell.com
3 Solve a Multi-Step Compound Inequality
Solve the compound inequality 3x + 1 < 4 or 2x — 5 > 7. Then graph
the solution.
Solution
Solve each of the parts using the methods of Lesson 6.3.
3x + 1 < 4
or 2x — 5 > 7
Write original inequality.
3x + 1 - 1 < 4 -
1 or 2x — 5 + 5>7 + 5
Isolate x.
3x < 3
or 2x>12
Simplify.
3x 3
2x 12
3 < 3
or T>T
Solve for x.
X < 1
or x > 6
Simplify.
ANSWER ► The solution is all real numbers less than 1 or greater than 6.
The graph of the solution is shown below.
I ! t—I-1-1-1—t !
-1 0 1 2 3 4 5 6 7
Solve a Compound Inequality with Or
Tell whether -5 is a solution of the inequality.
7- x < — 5 or x > —4 8. x< — 3 orx > 0
Solve the inequality. Then graph the solution.
9. x — 4<— 8<?rx + 3>5 1 0. 2x + 3 < 1 or 3x — 5 > 1
6.5 Solving Compound Inequalities Involving "Or'
You can use compound inequalities to describe real-life situations. Examples 4
and 5 deal with velocity. Recall that positive numbers are used to measure velocity
of upward motion and that negative numbers are used to measure velocity of
downward motion.
B2ZEH94 Make a Table
A baseball is hit straight up in the air. Its initial velocity is 64 feet per second.
Its velocity v (in feet per second) after t seconds is given by:
v = -32 t + 64
Make a table that shows the velocity of the baseball for whole-number values
of t from t = 0 to t = 4. Describe the results.
Solution
I (sec)
0
1
2
3
4
v (ft/sec)
64
32
0
-32
-64
The baseball starts with a velocity of 64 feet per second, moving upwards. It
slows down and then stops rising at t = 2 seconds when the baseball is at its
highest point. Then the baseball begins to fall downward. When t = 4, the
velocity is —64 feet per second. The negative sign indicates the velocity has
changed to a downward direction.
B2EEESB 5 Solve a Compound Inequality with Or
Find the values of t for which the velocity of the baseball in Example 4 is
Student MeCp greater than 32 feet per second or less than —32 feet per second.
P --
► Study Tip
When you multiply or
divide by a negative
number to solve a
compound inequality
with or, remember to
reverse both
inequalities. .
l _ J
Solution The velocity is given by —32 1 + 64.
-32 1 + 64 > 32
or
-32 1 + 64 < -32
■32 1 + 64 - 64 > 32 - 64
or
—32 1 + 64 - 64 < -32
—32? > —32
or
— 32t < —96
-32 1 , -32
or
—32? ^ -96
* -32 < -32
-32 > -32
t < 1
or
t> 3
ANSWER ^ The velocity is greater than 32 feet per second when t is less than
1 second and less than —32 feet per second when t is greater than
3 seconds.
Solve a Compound Inequality with Or
11. Refer to Example 5. Find the values of t for which the velocity of the
baseball is greater than 0 feet per second or less than —32 feet per second.
T
Chapter 6 Solving and Graphing Linear Inequalities
Exercises
Guided Practice
Vocabulary Check
1. Describe how the solution of a compound inequality involving or differs
from the solution of a compound inequality involving and.
Skill Check
Match the inequality with its graph.
2 . x < —2 or x> 1
4. x < — 2 or \ <x
A- <11 11 11
-10 12 3 4
C- < 1 1 1111
-3-2-1 0 1 2
3. 1 <x orx < —2
5- x < 1 or x > 2
B- < 11 1 11 1
-3-2-1 0 1 2
D. < 1 1 11 1 1
-3-2-1 0 1 2
Write a verbal sentence that describes the inequality.
6. x < —25 or x > 7 7. x < 10 or x > 13 8. x < —9 or x > 3
Write an inequality that represents the set of numbers. Then graph
the inequality.
9. All real numbers less than —6 or greater than — 1.
10, All real numbers less than 0 or greater than or equal to 5.
Practice and Applications
READING INEQUALITIES Write a verbal sentence that describes
the inequality.
11.x<15orx>31 12. x < 0 orx > 16 13. x > 11 orx < — 7
WRITING INEQUALITIES Write an inequality that describes the graph.
14. M —I- \ -1-4-t—► 15. -1-1- 1 1 1 *
- 8-4048 “4 -3 - 2-1 0 1
Student HeCp
I ►Homework Help
Example 1: Exs. 11-20
Example 2: Exs. 21-30
Example 3: Exs. 31-40
Example 4: Ex. 41
Example 5: Exs. 42-45
16. ^<—I-1-i-!-1—► 17. <—I-1-I-I-I—►
-2 -1 0 1 2 6 7 8 9 10
WRITING AND GRAPHING INEQUALITIES Write an inequality that
represents the set of numbers. Then graph the inequality.
18. All real numbers less than —6 or greater than 2.
19. All real numbers greater than 7 or less than 0.
20. All real numbers less than 3 or greater than 10.
6.5 Solving Compound Inequalities Involving "Or'
SOLVING INEQUALITIES Solve the inequality. Then graph the solution.
21.x — 1 < — 3 <?rx + 3 > 8
23.x + 3>2<?rl2x<-48
25. lx < —42 or x + 5 > 3
27.x - 4< —12 or 2x > 12
22 . —12 > 8x or 4x > 6
24. —22 > 1 lx or 4 + x > 4
26. 5 + x > 20 or 3x < —9
28. —3x <15 or 5 + x < —11
CHECKING SOLUTIONS Solve the inequality. Then determine whether
the given value of x is a solution of the inequality.
29. x — 7 < 3 or 2x > 24; x = 8 30. 5x> —15 or x + 4< — 1; x = —4
31. —2x > 6 or 2x + 1 > 5; x = 0 32. 3x < —21 or 4x — 8 > 0; x = 3
SOLVING INEQUALITIES Solve the inequality. Then graph the solution.
33. x + 10 < 8 or 3x — 7 > 5
35. 2x + 1 > 13 or —18 > 7x + 3
37. 2x + 7 < 3 or 5x + 5 > 10
39. 3x + 5 < —19 or 4x + 7 > — 1
34. — 8x > 24 or 2x — 5 > 17
36. 6 + 2x > 20 or 8 + x < 0
38. 3x + 8 > 17 or 2x + 5 < 7
40. 1 — 5x < —14 or —3x — 2 > 7
YO-YO In Exercises 41 and 42, use the following information.
A yo-yo is thrown toward the ground with an initial velocity of —4 feet per
second. Its velocity v in feet per second t seconds after being thrown is given
by v = At — 4, where t runs from 0 to 2 seconds.
41. Make a table that shows the yo-yo’s velocity for t — 0, 0.5, 1, 1.5, and
2 seconds. Describe the results.
BUS FARES Reduced bus
fares are often available for
the very young, the disabled,
senior citizens, or students
who ride the bus to get
to and from school.
42. Find the times for which the yo-yo’s velocity is greater than 2 feet per second
or less than —2 feet per second.
43. S cience Water may be in the form of a solid, a liquid, or a gas.
Under ordinary conditions at sea level, water is a solid (ice) at temperatures
of 32°F or lower and a gas (water vapor) at temperatures of 212°F or higher.
Write a compound inequality describing when water is not a liquid.
BUS FARES In Exercises 44 and 45, use the following information.
A public transit system charges fares based on age. Children under 5 ride free.
Children who are 5 or older but less than 11 pay half fare. People who are at
least 11 but younger than 65 pay full fare. Those 65 or over pay reduced fares.
44. Write a compound inequality to describe a, the ages in years of children who
pay half fare.
45. Write a compound inequality to describe y, the ages in years of those eligible
for reduced rates based on age.
46. CHALLENGE Describe the solutions of the inequality x < 2 or x > 1.
Chapter 6 Solving and Graphing Linear Inequalities
Standardized Test
Practice
Mixed Review
Maintaining Skills
47. MULTIPLE CHOICE Which of the following is the graph of the compound
inequality x < — 4 or x > 0?
® —|-1-1-1-CD —I-1-1-1-I—►
-6 -4-202
-6 -4-202
CD ■* —I— i —!—t—h
-6 -4-202
CD +—I-1—I—^-1—►
-6 -4-202
48. MULTIPLE CHOICE Which number is not a solution of the compound
inequality — 2x > 18 or 3x + 8 > 26?
CD -12 CD -9 CD 6 CD 9
INPUT-OUTPUT TABLE Make an input-output table for the function.
Use x = 0, 1, 2, 3, and 4 as values for x. (Lesson 1.8)
49. y = 3x + 2 50. y = —2x +1 51. y = 5 — x
52. y = 2x — 3 53. y = 2x — 4 54. y = 3x — 1
GRAPHING Graph the numbers on a number line. (Lesson 2.1)
55. -4, 6, -5 56. 3.2, -6.4, 3.5 57. j, |, 4
SOLVING EQUATIONS Solve the equation. Round the result to the
nearest hundredth. (Lesson 3.6)
58. 1.2* - 1.7 = 4.5 59. 1.3 + 4Ax = 6.6
60. 3.6x — 8.5 = 12.4 61. 2.3x + 3.2 = 18.5
62. 2.56 - 6.54x = -5.21 - 3.25x 63. 2.32x + 6.56 = 3.74 - 7.43x
SOLVING INEQUALITIES Solve the inequality. Then graph the solution.
(Lessons 6 .7 and 6.2)
64. x + 6 > -6
67.x - 10 > 15
70. 6x > —54
65. 16 < x + 7
68 . 2 < x - 7
71. —x>2
66 . 9 >-15 + x
69. -3x< -15
72. |x < 6
MULTIPLYING Multiply the fraction by the whole number.
(Skills Review p. 765)
73.} (84)
76.} (21,000)
79. (81,000)
74. } (375)
77. } (84,000)
80. } (31,500)
75.} (884)
78. (72,000)
81. Yj-(121,000)
6.5 Solving Compound Inequalities Involving "Or"
Goal
Use a number line to solve
absolute-value equations.
Materials
• graph paper
• colored pencils
Question
How can you use a number line to solve absolute-value
equations?
You can solve an absolute-value equation of the form \x\ = c by finding all
points on the number line whose distance from zero is c.
For example, the equation | x | = 2 means x is 2 units from zero. As shown
below, both —2 and 2 are 2 units from zero. Therefore, if \x ] = 2, then x = —2
orx = 2.
2 units 2 units
f -*-v-*-i
-4-I- 4 -I-4
- 2-1012
+
3
4
Explore
O
One way to solve the equation | x — 3 | = 2
is to use a table. Copy and complete the
table, circling those values of x for which
lx — 3 I = 2.
© Another way to solve the equation | x — 3 | = 2 is to use the number line. The
equation | x — 3 | = 2 can be read as “The distance between x and 3 is 2.” On
the number line below, find the points whose distance from 3 is 2.
H-1-1-1-1-1- + -1-1-1-h
-3 -2 -1 0 1 2 3 4 5 6 7
Think A bout It
1. LOGICAL REASONING Explain why the answers to Steps 1 and 2 are
the same.
Solve the absolute-value equation.
2. |x | = 5 3. |x — 2 | = 4 4. |x + 2 | = |x — (—2) | = 3
Chapter 6 Solving and Graphing Linear Inequalities
Solving Absolute-Value
Equations
Goal
Solve absolute-value
equations in one
variable.
Key Words
• absolute-value
equation
How tall are miniature poodles?
Breeds of dogs are often classified based
on physical traits. Poodles are divided into
classes according to height. In Example 5
you will write an equation to describe the
heights of miniature poodles.
An absolute-value equation is an equation of the form | ax + b | = c. You can
solve this type of equation by solving two related linear equations.
SOLVING AN ABSOLUTE-VALUE EQUATION
For c > 0, x is a solution of | ax + b\ = c if x is a solution of:
ax + b = c or ax + b = -c
For c< 0, the absolute-value equation | ax + b \ = c has no solution,
since absolute value always indicates a number that is not negative.
1 Solve an Absolute-Value Equation
Solve the equation.
a. |jc | = 8 b- |jc | = —10
Solution
a. There are two values of x that have an absolute value of 8.
\x | = 8
x = 8 or x = —8
ANSWER ^ The equation has two solutions: 8 and —8.
b_ The absolute value of a number is never negative.
ANSWER ^ The equation |x | = —10 has no solution.
Solve an Absolute-Value Equation
Solve the absolute-value equation.
1 - |x| = 6 2 . |x| = 0
3. 1x1= —6
6.6 Solving Absolute-Value Equations
2 Solve an Absolute-Value Equation
Solve |x — 2 | = 5.
Solution
Because | x — 2 | = 5, the expression x — 2 is equal to 5 or —5.
Student HeCp
1 ^ -
► Study Tip
Check the solutions
to an absolute-value
equation by substituting
each solution in the
original equation. **•**••*
V _
x- 2 IS POSITIVE
x — 2 — 5
x — 2 + 2 — 5 + 2
x — 1
or
or
x - 2 IS NEGATIVE
x 2 — 5
x — 2 + 2 — —5 + 2
x — — 3
f CH
ANSWER ► The equation has two solutions: 7 and —3.
CHECK 1 7 — 2 I = I 5 I = 5 | —3 — 2 I =
-5 =5
3 Solve an Absolute-Value Equation
Solve | 2x - 7 | -5 = 4.
Solution
First isolate the absolute-value expression on one side of the equation.
| 2jc — 7 | —5 = 4
I 2x — 7 I —5 + 5 = 4 + 5
2x - 7 =9
Because | 2x — 7 | =
2x - 7 IS POSITIVE
2x - 7 = 9
2x — 1 + 1 — 9 + 1
2x = 16
2x
2
16
2
9, the expression 2x — 7 is equal to 9 or —9.
or 2x - 7 IS NEGATIVE
2x-l = -9
2x-1 + 1= -9 + 1
2x= -2
2x = -2
2 2
x = 8
or
x = — 1
ANSWER ► The equation has two solutions: 8 and — 1.
CHECK / | 2(8) - 7 | — 5 = | 9 | - 5 = 9- 5 = 4
| 2( — 1) — 7 | - 5 = |-9 | —5 = 9 — 5 = 4
Solve an Absolute-Value Equation
Solve the absolute-value equation and check your solutions.
4. I x + 3 I = 5 5. lx — 3 | =5 6. | 4x — 2 I
7. 3x - 2 =0
8 . x+1 +2 — 4
9. 2x - 8
= 6
-3 = 5
Chapter 6 Solving and Graphing Linear Inequalities
You can use a number line to write an absolute-value equation that has two given
numbers as its solutions.
4 Write an Absolute-Value Equation
Write an absolute-value equation that has 7 and 15 as its solutions.
Solution
Graph the numbers on a number line and locate the midpoint of the graphs.
4 units 4 units
^-*-V-*-^
*—I—*—I—t—t—♦—I—t—I—*—t—►
6 7 8 9 10 11 12 13 14 15 16
The graph of each solution is 4 units from the midpoint, 11. You can use the
midpoint and the distance to write an absolute-value equation.
Midpoint Distance
i * i *
U- ll| =4
ANSWER ^ The equation is | x — 11 | =4. Check that 7 and 15 are solutions
of this equation.
Link]
AhitttaCs
Shoulder height
POODLES A poodle is
labeled a toy, a miniature, or
a standard based on its
shoulder height. The smallest
poodle is the toy. The largest
is the standard.
■affldliM Jj 5 Write an Absolute-Value Equation
POODLES The shoulder height of the shortest miniature poodle is 10 inches.
The shoulder height of the tallest is 15 inches. Write an absolute-value equation
that has these two heights as its solutions.
Solution
Graph the numbers on a number line and locate the midpoint of the graphs.
Then use the method of Example 4 to write the equation.
2.5 units 2.5 units
--1-» -H. l-~^- . I -^+^ 4-1-^
9 10 11 12 12.5 13 14 15 16
The midpoint is 12.5. Each solution is 2.5 units from 12.5.
Midpoint Distance
I lil
| x - 12.5 | = 2.5
ANSWER ^ The equation is | x — 12.5 | =2.5. Check that 10 and 15 are
solutions of this equation.
Write an Absolute-Value Equation
10. Write an absolute-value equation that has 4 and 12 as its solutions.
6.6 Solving Absolute-Value Equations
b Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Explain why the equation x +
5 is not an absolute-value equation.
2 . Choose the two equations you would use to solve the absolute-value equation
| x — 7 | =13. Then solve the two equations.
A.x - 7 = 13
B. x + 7 = 13
C. x — 7
■13 D.x + 7
-13
Tell how many solutions the equation has.
3. I x I =17 4. lx I = -2 5. lx - 1
-3 6. x + 1 = 1
Write the two linear equations you would use to solve the
absolute-value equation.
7. I x - 4 I = 10 8. | 2x - 3 | = 8 9. I 3x + 2 I
-1=5
Practice and Applications
SOLVING ABSOLUTE-VALUE EQUATIONS Solve the absolute-value
equation. If the equation has no solution, write no solution.
|x| =36
11. | x |
= 9
12. | x |
= -25
|x| = -15
14. | x |
= 10
15. | x |
= 100
Student HeCp
► Homework Help
Example 1: Exs. 10-15
Example 2: Exs. 16-27
Example 3: Exs. 32-40
Example 4: Exs. 41-46
Example 5: Exs. 47, 48
SOLVING ABSOLUTE-VALUE EQUATIONS Solve the equation and check
your solutions. If the equation has no solution, write no solution.
16.
X +
ll
= 3
17.
x - 2| =5
18.
4x |
=
16
19.
3x |
=
36
20.
x + 8| =9
21.
X -
4 |
=
6
22.
x +
6|
= -7
23.
OO
<N
II
H
OO
24.
X +
5|
=
65
25.
X —
3|
= 7
26.
15 + x | =3
27.
1
2 X
=
9
LOGICAL REASONING Complete the statement with always , sometimes ,
or never.
28, If x 2 = a 2 , then | x | is ? equal to \a\.
solutions.
solutions.
b | is ? equal to
| b -a \.
— 4 | — p will ?
have two
— p | = 4 will ?
have two
Chapter 6 Solving and Graphing Linear Inequalities
Student UeCp
► Homework Help
Extra help with
w* problem solving in
Exs. 32-40 is available at
www.mcdougallittell.com
ASTRONOMERS Study
energy, matter, and natural
processes throughout the
universe. Professional
astronomers need a doctoral
degree. Nevertheless,
amateur astronomers make
important discoveries as well.
More about
v astronomers at
www.mcdougallittell.com
SOLVING ABSOLUTE-VALUE EQUATIONS Solve the equation and check
your solutions. If the equation has no solution, write no solution.
32. 1 6x — 4
= 2
33.
<N
1
4?
= 22
34.
3x + 5 | =22
35. | 2x + 5
1=3
36.
6x — 3
= 39
37.
ON
II
r-*
1
<3
38. 5 - 4x
|-3 = 4
39.
2x - 4
- 8 = 10
40.
5x - 4 | +3 = 19
ABSOLUTE-VALUE EQUATIONS Match the absolute-value equation with
its graph.
41. |x + 2| = 6 42. | x — 6 | =2 43. | x — 2 | = 6
A.
6 units
a
6 units
A
i
/
L A
1 L
V
4
L 1
%
A t
* \
-12 -
\ V
-10 -8
1 T
-6 -4
w
-2
\ t
0 2
w l
4 6
r ►
8
B.
6 units
a
6 units
A
- i
i 4
1 (
- V -
4
1 1
i
i
m 1
-8
1 V
-6 -4
1 1
-2 0
W
2
1 1
4 6
W I
8 1C
I m
1 12
C.
2 units
A
2 units
A
*—i—
, — .
h—♦
-1-
V
—1—
s
■4 -h
—i—►
0123456789 10
WRITING ABSOLUTE-VALUE EQUATIONS Write an absolute-value
equation that has the given solutions.
44. 8 and 18 45. -6 and 10 46. 2 and 9
ASTRONOMY In Exercises 47 and 48,
use the following information.
The distance between Earth and the
sun is not constant, because Earth’s
orbit around the sun is an ellipse.
The maximum distance from Earth
to the sun is 94.5 million miles.
The minimum distance is about
91.4 million miles.
47. Graph the maximum and minimum distances on a number line and locate the
midpoint. Determine the distance from the midpoint to the minimum and
from the midpoint to the maximum.
48. Use your answers from Exercises 47 to write an absolute-value equation
that has the minimum and maximum distances between Earth and the sun
(in millions of miles) as its solutions.
49. CHALLENGE The highest elevation in North America is 20,320 feet above
sea level at Mount McKinley. The lowest elevation is 282 feet below sea
level in Death Valley. Find an absolute-value equation that has the highest
and lowest elevations in North America as its solutions.
Not drawn to scale E a rth
6.6 Solving Absolute-Value Equations
Standardized Test
Practice
50. (MULTIPLE CHOICE Which number is a solution of |x \ - 5 = 6?
(A) 6 CD -6 CD-l CD -ll
51. MULTIPLE CHOICE Which numbers are solutions of | 2x - 4 | + 7 = 23?
CD —6 and 10 CD —13 and 17
CED —12 and 20 CD —10 and 6
GRAPHING EQUATIONS Graph the equation on a coordinate plane.
(Lesson 4.3)
52. x = -1 53. 3y = 15 54. x + 6 = 7
SLOPE-INTERCEPT FORM Write the equation in slope-intercept form.
(Lesson 5.1)
55. 5x + y = 20 56. 3 x-y = 21 57. 12x = 3y + 36
WRITING EQUATIONS Write the slope-intercept form of the equation of
the line that passes through the given point and has the given slope.
(Lesson 5.2)
58. (0, 4), m = 3 59. (2, -5), m = -2 60. (-3, 1), m = 2
Maintaining Skills ROUNDING Round 47,509.1258 to the indicated place value.
(Skills Review p. 774)
61. thousands 62. tenths 63. hundreds
64. thousandths 65. hundredths 66. ones
Quiz 2
Solve the inequality. Then graph the solution. (Lessons 6.4 and 6.5)
1. -5<jc- 8 <4 2 . —10 < 2x + 8 < 22
3. —10 < — 4x — 18 < —2 4. 5x > 25 or 2x + 9 < — 1
5- — 3 > v + 6 or — x < 4 6- 2 — x < — 3 or 2x + 14 < 12
7_ TEMPERATURES The lowest temperature recorded on Earth was — 128.6°F
in Antarctica. The highest temperature recorded on Earth was 136°F in Libya.
Write an inequality that describes any other record temperatures T.
(Lesson 6.4) ►Source: National Climatic Data Center
Solve the equation. If the equation has no solution, write no solution.
(Lesson 6.6)
x = 14
9. \x | = —43
10.
<N
II
o\
1
H
|x + 15 | = 6
12. 3x - 18 | = 36
13.
5x + 10 | + 15 = 60
14, Write an absolute-value equation that has —3 and 18 as its solutions.
(Lesson 6.6)
Chapter 6 Solving and Graphing Linear Inequalities
Solving Absolute-Value
Inequalities
Goal
Solve absolute-value
inequalities in one
variable.
Key Words
• absolute-value
inequality
How fast does water from a fountain rise and fall?
When water is shot upward from a
fountain, it gradually slows down.
Then it stops and begins to fall with
increasing speed. In Exercise 42 you
will use an inequality to analyze the
speed of water rising and falling in
a fountain.
An absolute-value inequality is an inequality that has one of these forms:
| ax + Z? | < c | ax + Z? | < c | ax + Z? | > c | ax + Z? | > c
To solve an absolute-value inequality, you solve two related inequalities. The
inequalities for < and > inequalities are shown. Similar rules apply for < and >.
| ax + Z? | < c | ax + Z? | > c
means means
ax + b < c and ax + b> —c ax + b> c or ax + b < —c
J 1 Solve an Absolute-Value Inequality
Solve |x| >5. Then graph the solution.
Solution
The solution consists of all numbers x whose distance from 0 is greater than 5.
In other words \x \ > 5 means x > 5 or x < —5. The inequality involves > so the
related inequalities are connected by or.
ANSWER ► The solution is all real numbers greater than 5 or less than —5.
This can be written x < —5 or x> 5. The graph of the solution is
shown below.
-—I-1-1-1-1-1-1-1-1-1-1-1-H
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
CHECK y Test one value from each region of the graph.
| —6 | = 6, 6 > 5 | 0 | = 0, 0 is not greater than 5 | 6
6.7 Solving Absolute-Value Inequalities
Student HeCp
► Study Tip
The expression inside
the absolute-value
symbols can be
positive or negative.
When you rewrite the
expression for the
negative value, reverse
the inequality.
\ ___>
2 Solve an Absolute-Value Inequality
Solve | x — 4 | <3. Then graph the solution.
Solution The solution consists of all numbers x whose distance from 4 is
less than 3. The inequality involves < so the related inequalities are connected
by and.
| x — 4 | <3 Write original inequality.
x — 4 < 3 and x — 4 > — 3 Write related inequalities.
x — 4 + 4<3 + 4 and x — 4 + 4>— 3 + 4 Add 4 to each side.
x < 7 and x > 1 Simplify.
ANSWER ► The solution is all real numbers greater than 1 and less than 7.
This can be written 1 < x < 7. The graph of the solution is
shown below.
-1
2
3
6 7 8
+
9
Check the solution.
I_
Student MeCp
-\
► Study Tip
When you check your
solution, choose values
that make substitution
simple. In Example 3
you might choose -10,
-5, and 0.
L _ )
3 Solve an Absolute-Value Inequality
Solve | x + 5 | >2. Then graph the solution.
Solution The solution consists of all numbers x whose distance from —5 is
greater than or equal to 2. The inequality involves > so the related inequalities
are connected by or.
| x + 5 | >2 Write original inequality,
x + 5 > 2 or x + 5 < — 2 Write related inequalities.
x + 5
5 > 2 — 5 or x + 5 — 5 < —2 —5
x > — 3 or x < —7
Subtract 5 from each side.
Simplify.
ANSWER ► The solution is all real numbers greater than or equal to —3 or less
than or equal to —7. This can be written x < —7 or x> —3.
The graph of the solution is shown below.
-- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - V
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
Check the solution.
Solve an Absolute-Value Inequality
Solve the absolute-value inequality.
1 .
VI
2.
| x — 2 |
<5
3.
4.
| 3x | >9
5.
| x — 2 |
>7
6.
x + 1 | <4
x — 3 | >12
Chapter 6 Solving and Graphing Linear Inequalities
Solve a Multi-Step Inequality
3 >2.
Solution
First isolate the absolute-value expression on one side of the inequality.
| x + 1 | — 3 > 2 Write original inequality.
| x + 1 | — 3 + 3>2 + 3 Add 3 to each side.
|jc + 1 | >5 Simplify.
The inequality involves > so the related inequalities are connected by or.
| x + 1 |
>5
Write simplified inequality.
x + 1 > 5
or
x + 1 < —5
Write related inequalities.
jc + 1 - 1 >5
— 1 or
jc + 1 - 1 < -5 - 1
Subtract 1 from each side.
x > 4
or
SO
1
VI
Simplify.
ANSWER ► The solution is all real numbers greater than or equal to 4 or less
than or equal to —6. This can be written x < — 6 or x > 4.
Solve a Multi-Step Inequality
7. Solve the inequality | 2x — 2 | >4.
Student HeCp
V
► Study Tip
Compare Example 5 to
Example 5 on page 350.
Together the examples
show the connection
between absolute-
value inequalities and
compound inequalities.
I j
5 Use an Absolute-Value Inequality
BASEBALL A baseball is hit straight up with an initial velocity of 64 feet per
second. Its speed s (in ft/sec) after t seconds is given by = | —32 1 + 64 | .
Find the values of t for which s is greater than 32 feet per second.
Solution Solve | -32* + 64 | > 32.
The inequality involves > so the related inequalities are connected by or.
| —32; + 64 | >32
-32 1 + 64 > 32 or -32 1 + 64 < -32
—32; >—32 or —32; <—96
;< 1 or ;> 3
Write original inequality.
Write related inequalities.
Subtract 64 from each side.
Divide by -32 and reverse
the inequalities.
ANSWER ► The speed is greater than 32 ft/sec when t is less than 1 second or
greater than 3 seconds. This can be written ; < 1 or t > 3.
8. In Example 5 find the values of t for which s is greater than 48 ft/sec.
6.7 Solving Absolute-Value Inequalities
ESS? JSZ*. _ j - 0
o:J Exercises
Guided Practice
Vocabulary Check Match the phrase with the example it describes.
1 . An absolute-value inequality
A.
r-
II
o\
1
2. An absolute-value equation
B.
1 -15 |
3. An absolute-value
C.
| -x + 4 >18
4. Choose the two inequalities you would use to solve the inequality
| x — 8 | >5. Tell whether they are connected by and or by or.
A. x — 8 > — 5 B. v — 8 < — 5 C. x — 8 < 5 D. x — 8 > 5
Tell whether the given number is a solution of the inequality.
5- | x + 6 | < 4; —10 6- | x — 2 | > 9; 7 7. | 5x — 2 | > 8; 3
Practice and Applications
RELATED INEQUALITIES Write the two inequalities you would use to
solve the absolute-value inequality. Tell whether they are connected by
and or by or.
Al
H
9. | x | >1
10.
| x — 16 | ■
X
1
IA
12. 7x - 3 | <2
13.
10 + 7x
P Student HeCp
^ -\
► Homework Help
Example 1: Exs. 8, 9,
14-16
Examples 2 and 3:
Exs. 10,11,
17-25
Example 4: Exs. 12,13,
30-41
Example 5: Exs. 42-46
k _>
SOLVING ABSOLUTE-VALUE INEQUALITIES Solve the inequality. Then
graph and check the solution.
14.
X
IV
15.
x | <15
16.
| X | >5
17.
x + 5| >1
18.
8x | >20
19.
| x — 10
| >20
20.
lx | <49
21.
x — 4| >8
22.
| x + 3 |
<8
23.
— 3 + x | <18
24.
10 + x| <13
25.
| 9 + x |
<7
LOGICAL REASONING
Complete the statement with
always,
sometimes,
or never.
26- If a < 0, then | x | > a is ? true.
27- A solution to the inequality | x — 5 | < 4 will ? be negative.
28. A solution to the inequality | x — 7 | > 9 will ? be negative.
29. A solution to the inequality | x + 7 | < 6 will ? be negative.
Chapter 6 Solving and Graphing Linear Inequalities
Student UeCp
► Homework Help
Extra help with
~^P y problem solving in
Exs. 30-41 is available at
www.mcdougallittell.com
SOLVING MULTI-STEP INEQUALITIES Solve the inequality. Then graph
and check the solution.
30. 12x — 9 | <11
33. | 2x + 7 | > 23
36. | x + 2 | - 5 > 8
39. I 5x — 15 I — 4 > 21
31. | 4x + 2 |
< 6
8
OO
X
1
o
1 - 6
37. 10 + 8x
- 2> 16
40. | 3x + 2 |
- 5<0
32. | 32x - 16 | > 32
35. | 4x — 3 | <7
38. | —4 + 2x | + 5 < 23
41. | 3x — 9 | - 2 <7
WRITING AND SOLVING INEQUALITIES In Exercises 42 and 43, write and
solve an absolute-value inequality to find the indicated values.
42. WATER FOUNTAIN A stream of water
rises from a fountain straight up with
an initial velocity of 96 feet per second.
Because the speed is the absolute value
of the velocity, its speed s (in feet per
second) after t seconds is given by
s = | —32 1 + 96 |. Find the times t for
which the speed of the water is greater
than 32 feet per second.
43. CANNON BALLS A cannon ball is fired
straight up in the air with an initial velocity
of 160 feet per second. Its speed s (in feet
per second) after t seconds is given by 5 1 = | —32 1 + 160 |. Find the times t
for which the speed of the cannon ball is greater than 64 feet per second.
FIREWORKS The diagram
above shows what happens
when fireworks are launched.
More about fireworks
is available at
www.mcdougallittell.com
Sci ence Link } In Exercise 44-46, use the following information.
The color of light is determined by a property of light called its wavelength.
When a firework star bursts, the chemicals in the firework burn. The color is
determined by the wavelength of the light
given off in the fire.
44. A firework star contains a copper
compound. The absolute-value
inequality | w — 455 | < 23 describes
the wavelengths w of the light given
off by the compound when it burns.
What color is the star?
45. A firework star contains a sodium
compound. The absolute-value
inequality | w — 600 | < 5 describes
the wavelengths w of the light given
off by the compound when it burns.
What color is the star?
46. A firework star contains a strontium compound. The absolute-value equation
| w — 643 | < 38 describes the wavelengths w of the light given off by the
compound when it burns. What color is the star?
47. CHALLENGE Graph the solutions of | x — 2 | > x + 4.
Color Wavelength, w
Ultraviolet
w < 400
Violet
400 < w < 424
Blue
424 < w < 491
Green
491 < w < 575
Yellow
575 < w < 585
Orange
585 < w < 647
Red
647 < w < 700
Infrared
700 < w
6.7 Solving Absolute-Value Inequalities
Standardized Test
Practice
Mixed Review
Maintaining Skills
48. MULTIPLE CHOICE Which number is a solution of | 2x + 3 | > 17?
(A) -5 CD 0 CD 7 CD 10-5
49. MULTIPLE CHOICE Which is the graph of | 2x + 1 | < 3?
CD «-l 1—1
-2-1012
CD
CED —1—1—1—1—1—
CD
H—1—1—b
-2 -1
1
- 2-1 0 1
*—I 1-1-1-h
- 2-1 01 2
FINDING THE DOMAIN Find the domain of the function. (Lesson 2.8)
_ _ A
50 - y = e
51. y =
1
x — 4
52. y =
1
X + 1
EXCHANGE RATE Convert the currency using the given exchange rate.
Round to the nearest whole number. (Lesson 3.8)
53, Convert 55 Canadian dollars to United States dollars.
(1 United States dollar = 1.466 Canadian dollars)
54. Convert 195 United States dollars to Mexican pesos.
(1 United States dollar = 9.242 pesos)
FINDING SOLUTIONS Find three different ordered pairs that are solutions
of the equation. (Lesson 4.3)
55.x
-12
56. y = 4
57.x
VERTICAL LINE TEST Use the vertical line test to determine whether the
graph represents a function. (Lesson 4.8)
59.
v 1
4
1 X
SUBTRACTING MIXED NUMBERS Subtract. Write the answer as a
fraction or as a mixed number in simplest form. (Skills Review p. 765)
2 2
62. 6| - 5|
5 2
63. 8f - 3f
2 3
64 2— — 1 —
5 10
17 2
65. 15— - 4-
66. l\ - 3f
9 3
6 ? . 19^ - 3
Chapter 6 Solving and Graphing Linear Inequalities
Graphing Linear Inequalities
in Two Variables
Goal
Graph linear inequalities #f 0 W QQn yQU p/of) Q healthy meal?
m two variables. „ # 1 #
Key Words
• linear inequality in
two variables
Nutritionists advise that you eat a variety
of foods. Your diet should supply all the
nutrients you need with neither too few nor
too many calories. In Exercises 51 and 52
you will use inequalities to plan a meal.
A linear inequality in x and y is an inequality that can be written as follows,
ax + by < c ax + by <c ax + by > c ax + by >c
where a, b and c are given numbers. An ordered pair (x, y) is a solution of a linear
inequality if the inequality is true when the values of x and y are substituted into
the inequality.
B2HJE39 1 Check Solutions of a Linear Inequality
Check whether the ordered pair is a solution of 2x — 3y > — 2.
a. (0, 0)
b. (0, 1)
Solution
<x, V )
2x - 3 y
a. (0, 0)
2(0) - 3(0) = 0
b. (0, 1)
2(0) - 3(1) = -3
c. (2, -1)
2(2) - 3(— 1) = 7
c. (2,-1)
2x - 3y>-2
CONCLUSION
<N
1
Al
O
(0, 0) is a solution.
-3X-2
(0, 1) is not a solution.
<N
1
Al
r-
(2, —1) is a solution.
Student MeCp
-—\
►Vocabulary Tip
A line divides the
coordinate plane into
two half-planes. The
solution of a linear
inequality in two
variables is a half¬
plane.
k _/
The graph of a linear inequality in two variables is the graph of the solutions
of the inequality.
The graph of 2x — 3y > — 2 is shown at the right.
The graph includes the line 2x — 3 y = —2
and the shaded region below the line.
Every point that is on the line or in the
shaded half-plane is a solution of the
inequality. Every other point in the
plane is not a solution.
o
D
(0,1)
* z'
-1
(0,0) 3 5 x
*(2,-D
6.8 Graphing Linear Inequalities in Two Variables
GRAPHING A LINEAR INEQUALITY
Student HeGp
► Study Tip
A dashed line indicates
that the points on the
line are /7of solutions.
A solid line indicates
that the points on the
line are solutions.
K- J
step 0 Graph the corresponding equation. Use a dashed line for
> or <. Use a solid line for < or >.
step 0 Test the coordinates of a point in one of the half-planes.
step © Shade the half-plane containing the point if it is a solution
of the inequality. If it is not a solution, shade the other
half-plane.
Student HeCp
► Study Tip
You can use any point
that is not on the line
as a test point. It is
convenient to use the
origin because 0 is
substituted for each
variable.
L j
J 2 Vertical Lines
Graph the inequality x< —2.
Solution
0 Graph the corresponding equation x = — 2.
The graph of x = —2 is a vertical line. The
inequality is <, so use a dashed line.
0 Test a point. The origin (0, 0) is not a
solution and it lies to the right of the line.
So the graph of x < —2 is all points to the
left of the line x = —2.
© Shade the half-plane to the left of the line.
ANSWER ^ The graph of x < —2 is the half-plane to the left of the graph
of x = —2. Check by testing any point to the left of the line.
3 Horizontal Lines
Graph the inequality y < 1.
Solution
0 Graph the corresponding equation y = 1.
The graph of y = 1 is a horizontal line. The
inequality is <, so use a solid line.
0 Test a point. The origin (0, 0) is a solution
and it lies below the line. So the graph of
y < 1 is all points on or below the line y = 1.
0 Shade the half-plane below the line.
ANSWER ► The graph of y < 1 is the graph of y = 1 and the half-plane below
the graph of y = 1. Check by testing any point below the line.
Horizontal and Vertical Lines
Graph the inequality.
1.x>— 1 2.x<4 3. y > — 3 4. y < 2
Chapter 6 Solving and Graphing Linear Inequalities
Student HeCp
'
► Study Tip
The graph of
y> ax + b is above
the graph of
y = ax + b. The
graph of y< ax + b
is below the graph of
y = ax + b. Similar
rules apply for > and <
\ __ j
J 4 Use Slope-Intercept Form
Graph the inequality x + y > 3 using the slope-intercept form of the
corresponding equation.
Solution
Write the corresponding equation in slope-intercept form.
x + y — 3 Write corresponding equation.
y — —x + 3 Subtract x from each side.
The graph of the line has a slope of — 1 and a
y-intercept of 3. The inequality is >, so use a
dashed line.
Test the origin: 0 + 0 = 0 and 0 is not greater
than 3, so (0, 0) is not a solution. Since (0, 0)
lies below the line, shade above the line.
ANSWER ► The graph of x + y > 3 is all points
above the line. Check by testing any
point above the line.
L_
X
\
\
x+ y> 3
\
1
\
%
3
-
1
-1
(0,0)
K
X
\
X
-3
Student HeCp
^More Examples
More examples
are available at
www.mcdougallittell.com
5 Use Slope-Intercept Form
Graph the inequality 2x — y > — 2 using the slope-intercept form of the
corresponding equation.
Solution
Write the corresponding equation in slope-intercept form.
2x — y = —2 Write corresponding equation.
—y = — 2x — 2 Subtract 2xfrom each side.
y = 2x + 2 Multiply each side by -1
The graph of the line has a slope of 2 and a
y-intercept of 2. The inequality is >, so use a
solid line.
Test the origin: 2(0) — 0 = 0 and 0 is greater
than —2, so (0, 0) is a solution. Since (0, 0) lies
below the line, shade below the line.
ANSWER ► The graph of 2x — y > — 2 is all
points on and below the line. Check
by testing any point below the line.
f
3
/
/ 2x
- y
>
-2
/
/,
3
-1
(0,0)
5 x
/
/
J
f
-3
Use Slope-Intercept Form
6 , x + y < 4
7. 3x - y < 4
Graph the inequality.
5. 2x + y > — 1
6.8 Graphing Linear inequalities in Two Variables
mM Exercises
Guided Practice
Vocabulary Check 1 . Write an example of a linear inequality in two variables.
2. Decide whether (2, —3) is a solution of the inequality 5x + y > 10.
3. Describe the graph of the inequality x > 0. Use the phrase half-plane.
Skill Check In Exercises 4 and 5, use the graph at the right.
4. Choose the inequality whose solution is
shown in the graph.
A. x - y >4 B. x - y <4
C. x — y>4 D. x —y<4
5. Choose the ordered pair that is not a solution of the inequality whose graph
is shown.
A. (4,0) B. (2, —3) C.(-l, -3) D. (0, -4)
6 . Does the graph of y < —3 lie above or below the graph of y = —3?
7. Does the graph of x > —3 lie to the right or to the left of the graph of x = —3?
Check whether (0, 0) is a solution of the inequality.
8 . y < — 2 9.x>—2 10.x + y> — 1
11-x + y < -2 12. 3x - y <3 13.x - 3y > 12
Practice and Applications
CHECKING SOLUTIONS Check whether each ordered pair is a solution of
the inequality.
14. x + y > —3; (0, 0), (—6, 3) 15. 2x + 2y < 0; (0, 0), (—1, —1)
16. 2x + 5y > 10; (0, 0), (1, 2) 17. 3x - 2y < 2; (0, 0), (2, 0)
18. y — 2x > 5; (0, 0), (8, 1) 19. 5x + 4y > 6; (0, 0), (2, —4)
Student HeCp
► Homework Help
Example 1: Exs. 14-19
Examples 2 and 3:
Exs. 20-25,
36-41
Examples 4 and 5:
Exs. 26-35,
42-53
r _ )
HORIZONTAL AND VERTICAL LINES Match the inequality with its graph.
20. x<3 21.y<3 22. x>—3
B.
Chapter 6 Solving and Graphing Linear Inequalities
NUTRITIONISTS plan
nutrition programs and
supervise preparation and
serving of meals. Most
nutritionists have a degree
in food and nutrition or a
related field.
More about
nutritionists at
www.mcdougallittell.com
DASHED VS. SOLID Tell whether you would use a dashed line or a solid
line to graph the inequality.
23.y<~7 24.x >10 25.x<9
SLOPE-INTERCEPT FORM Write the equation corresponding to the
inequality in slope-intercept form. Tell whether you would use a dashed
line or a solid line to graph the inequality.
26. x + y > —15 27. x — y < 0 28. 4x + y < 9
29. x — 2y > 16 30. 6x + 3y > 9 31. — 4x — 2y < 6
GRAPHING In Exercises 32-35, consider the inequality 2x - y < 1.
32. Write the equation corresponding to the inequality in slope-intercept form.
33. Tell whether you would use a solid or a dashed line to graph the
corresponding equation. Then graph the equation.
34. Test the point (0, 0) in the inequality.
35. Is the test point a solution? If so, shade the half-plane containing the point.
If not, shade the other half-plane.
GRAPHING LINEAR INEQUALITIES Graph the inequality.
36. x > —4 37. x < 5 38.y>-l
39.x- 3> -2
42. 3x + y > 9
45.x + 2y< -10
48. 2x — y > 6
40. y + 6 < 5
43. y + 4x > -1
46. x + 6y < 12
49. — y + x < 11
41. 6y <24
44.x + y > -8
47. 4x + 3y < 24
50. -x - y < 3
NUTRITION In Exercises 51 and 52, use the following information and
the calorie counts of the breakfast foods that are in the table below.
You want to plan a nutritious breakfast. It should supply at least 500 calories or
more. Be sure your choices would provide a reasonable breakfast.
51. You want to have apple juice, eggs,
and one bagel. Let a be the number
of glasses of apple juice and e the
number of eggs. The inequality
123a + 15e + 195 > 500 models the
situation. Determine three ordered
pairs (a, e) that are solutions of the
inequality where 0 < a < 5 and 0 < e < 8.
52. You decide on cereal, milk, and one
glass of tomato juice. Let c be the
number of cups of cereal and m the
number of cups of milk. The inequality 102c + 150m + 41 > 500 models the
situation. Determine three ordered pairs (c, m) that are solutions of the
inequality where 0 < c < 8 and 0 < m < 4.
Breakfast food
Calories
Plain bagel
195
Cereal, 1 cup
102
Apple juice, 1 glass
123
Tomato juice, 1 glass
41
Egg
75
Milk, 1 cup
150
6.8 Graphing Linear inequalities in Two Variables
GOLD Most metals
deteriorate quickly in salt
water. Gold, however, is not
harmed by salt water, by air,
or even by acid. Gold does
not easily interact with other
chemicals.
More about gold
4^^ is available at
www.mcdougallittell.com
Modeling with a Linear Inequality
GOLD AND SILVER Divers searching for gold and silver coins collect the
coins in a wire basket that contains 50 pounds of material or less. Each gold
coin weighs about 0.5 ounce. Each silver coin weighs about 0.25 ounce. What
are the different numbers of coins that could be in the basket? Write an
algebraic model that models this situation.
Solution
Find the weight in ounces of the contents of the basket. There are 16 ounces in
a pound, so there are 50 • 16 or 800 ounces in 50 pounds.
Write an algebraic model.
Verbal
Model
Labels
Weight
Number
Weight
Number
Weight
per gold
•
of gold
+
per silver
•
of silver <
in
coin
coins
coin
coins
basket
Weight per gold coin = 0.5
Number of gold coins = x
Weight per silver coin = 0.25
(ounces per coin)
(coins)
(ounces per coin)
Number of silver coins = y
(coins)
Algebraic
Model
l_
Maximum weight in basket = 800
0.5 x + 0.25 y < 800
(ounces)
53. Graph the algebraic model in the example above.
54. Name and interpret two solutions of your inequality from Exercise 53.
Standardized Test
Practice
55. MULTIPLE CHOICE Choose the ordered pair that is a solution of the
inequality whose graph is shown.
(A) (0, 0)
CD (-2,0)
CD (-2,-1)
CD (2,-1)
1
*
/
A
-
1
2 x
s
/ \
56. MULTIPLE CHOICE Choose the inequality whose graph is shown.
CD 2y - 6x< -4
CD 2y — 6x< —4
CD 2y — 6x> -4
CD 2y — 6x> —4
Chapter 6 Solving and Graphing Linear Inequalities
Mixed Review
EVALUATING EXPRESSIONS Evaluate the expression. Then simplify your
answer. (Lesson 1.3)
Maintaining Skills
Quiz 3
__ 16+11 + 18 _ 0 20 + 15 + 22 + 19 __ 37 + 65 + 89 + 72 + 82
57 ■ 3 5S - 4 59> 5
CONVERTING TEMPERATURES In Exercises 60 and 61, use the
g
temperature conversion formula F = ^ C + 32, where F represents
degrees Fahrenheit and C represents degrees Celsius. (Lesson 3.7)
60. Solve the temperature formula for C.
61 . Use the formula you wrote in Exercise 60 to convert 86 degrees Fahrenheit to
degrees Celsius.
FINDING SLOPES AND ^-INTERCEPTS Find the slope and /-intercept of
the graph of the equation. (Lesson 4.7)
62. y = — 5x + 2 63. y = ^x — 2 64. 5x — 5y = 1
65. 6x + 2 y= 14 66. y = -2 67. y = 5
PERCENTS Determine the percent of the graph that is shaded.
(Skills Review p. 768)
Solve the inequality. Then graph and check the solution. (Lesson 6.7)
00
Al
H
2.
A
^|-
1
H
3.
\x + 7 | < 2
3x- 12 <9
5.
2x + 7 < 25
6.
4x+ 2 — 5 >17
7. BASEBALL A baseball is thrown straight up with an initial velocity of
48 feet per second. Its speed s (in feet per second) after t seconds is given by
s = | —32 1 + 48 |. Find the times t for which the speed of the baseball is
greater than 24 feet per second. (Lesson 6.7)
Check whether each ordered pair is a solution of the inequality.
(Lesson 6.8)
8. x + y < 4; (0, -1), (2, 2) 9. y - 3x > 0; (0, 0), (-4, 1)
10 . —2x + 5y>5; (2, 1), (-1,2) 11 . -x ~ 2y<4;(l, -1), (2, -3)
Graph the inequality. (Lesson 6.8)
12. x < —4 13. y > 3 14.y-5x>0
15. y < —2x 16. 3x + y>l 17. 2x — y > 5
6.8 Graphing Linear Inequalities in Two Variables
H L
*
USING A GRAPHING CALCULATOR
IIIMfri rflgf |ggj| BUp
For use with
Lesson 6.8
The Shade feature of a graphing calculator can be used to graph an inequality.
Sampl*
Graph the inequality x — 2y < — 6.
Solution
© Rewrite the inequality to isolate y on the left side of the equation.
x — 2y < — 6 Write original inequality.
— 2y < —x — 6 Subtract x from each side.
x
y > — + 3 Divide each side by -2 and reverse the inequality.
Student MeCp
► Keystroke Help
See keystrokes for
several models of
calculators at
www.mcdougallittell.com
©Use your calculator’s procedure for graphing and shading an inequality to
x
graph y > — + 3. It may not be clear on the screen whether the graph of the
corresponding equation is part of the graph. In that case, you must decide.
The inequality is >. So the
region abovethe graph is
shaded and the graph of
y= | + 3 is part of the
solution.
© In the revised inequality, the inequality is >. So the graph of the
corresponding equation should be indicated by a solid line.
TVyTtas*
Use a graphing calculator to graph the inequality. Use an appropriate
viewing window.
1 _ y < — 2x — 3 2. y>2x + 2 3. x + 2y < — 1 4. x — 3y > 3
5- y > 0.5x + 2 6- y < 3x — 3.2 7. + y > 1 8- ^ — 2y < 2
9. y<x + 25 10. y>— x + 25 11.y<0.1x 12. y > lOOx
13. Write an inequality that represents all points that lie above the line y = x.
Use a graphing calculator to check your answer.
14. Write an inequality that represents all points that lie below the line
y = x + 2.Usea graphing calculator to check your answer.
■ —
SMipteir
X Chapter Summary
w and Review
• graph of an inequality, p. 323
• multiplication property of
• absolute-value equation,
• equivalent inequalities, p. 324
inequality, pp. 330, 331
p. 355
• addition property of
• division property of
• absolute-value inequality,
inequality, p. 324
inequality, pp. 330 , 331
p. 361
• subtraction property of
• compound inequality, p. 342
• linear inequality in two
inequality, p. 324
variables, p. 367
_>
Solving Inequalities Using Addition or Subtraction
Examples on
pp. 323-325
Solve n - 5 < -10. Then graph the solution.
n — 5 < —10 Write original inequality.
n — 5 + 5< —10 + 5 Add 5 to each side.
n<— 5 Simplify.
ANSWER ► The solution is all real numbers less than —5.
« — i — i — i — i—
-7 -6 -5 -4
Solve the inequality. Then graph the solution.
1.x-5<-3 2. a + 6 > 28 3. -8<-10 + x
4. 7 + z > 20
Solving Inequalities Using Multiplication or Division
Examples on
pp. 330-332
Solve — 14x < 56. Then graph the solution.
14x < 56 Write original inequality.
WUx 56
>
Divide each side by -14 and reverse the inequality.
-14 -14
x > -4 Simplify.
ANSWER ► The solution is all real numbers greater than —4.
H-1-I-h
-5 -4 -3 -2
Solve the inequality. Then graph the solution.
5. 64 < 8x 6 - —6k > —30 7. -81 >-3 p 8 . -81>9r
9. ~n > 9
10 . 3 <
"■i4 £4
l
12. > 3
Chapter Summary and Review
Chapter Summary and Review continued
Solving Multi-Step Inequalities
Examples on
pp. 336-338
Solve 7 + 2x >
1 + 2x> -3
7-7 + 2x >-3-7
2x> -10
Write original inequality.
Subtract 7 from each side.
Simplify.
2x .10 , ......
— > —— Divide each side by 2.
x>—5 Simplify.
ANSWER ^ The solution is all real numbers greater than or equal to —5.
Solve the inequality.
13 . 6x- 8 >4
16 . 5(x — 2) < 10
19.5 — 8x < — 3x
14 . 10 - 3x< -5
17 . -3(jc - 1)>4
20 . 5x > 12 + x
15 . 4jc — 9 > 11
18 . j(x + 8 )< 1
21 . 3x - 9<2x + 4
Solving Compound Inequalities Involving “And’
Examples on
pp. 342-344
Solve —I<3x + 2<11. Then graph the solution.
— I<3x + 2<11
2<3x + 2 — 2<11
— 3 < 3x < 9
_3 3x 9
3 < 3 _ 3
1 < x < 3
Write original inequality.
Subtract 2 from each expression.
Simplify.
Divide each expression by 3.
Simplify.
ANSWER ^ The solution is all real numbers greater than — 1 and less than or
equal to 3. The graph of the solution is shown below.
| - (,
-2 -1
H-b
Solve the inequality. Then graph the solution.
22 . 9 < x + 1 < 13 23 . -3<3x<15 24 . - l<x-2<3
25 . 1 < 2x - 3 < 5 26 . 0 < 4 - x < 5 27 . -7 < 3 - ^x < 1
Chapter 6 Solving and Graphing Linear Inequalities
Chapter^ Summary and Review continued
Solving Compound Inequalities Involving “Or’
Examples on
pp. 348-350
Solve the compound inequality x + 3 < 7 or 4x > 20. Then graph the solution.
Solve each of the parts separately.
x + 3 < 7 or 4x > 20 Write original inequality.
x + 3 — 3 < 7 — 3 or
4x 20
4 > 4
Isolate x.
x < 4 or x > 5 Simplify.
ANSWER ► The solution is all real numbers less than or equal to 4 or greater than 5.
The graph of the solution is shown below.
+ -1-1-1-1-1-1-1-1-►
0 1 2 3 4 5 6 7
Solve the inequality. Then graph the solution.
28 - x > 4 or 3x < —9 29 - 2x < —10 or x + 3 > 1 30 - x — 7>0 or 3+x< —2
31 . 6x — 2 < 4 or 3x > 21
32 . 3x + 2< -1 or 2x + 1 >9
33 . —x < “ or 3x — 6 > 24
Solving Absolute-Value Equations
Examples on
pp. 355-357
Because | x — 4
X - 4 IS POSITIVE
x — 4 = 6
x — 4 + 4 = 6 + 4
x = 10
Solve | x — 4 | —6.
| = 6, the expression x — 4 is equal to 6 or —6.
or X - 4 IS NEGATIVE
x — 4 = —6
x — 4 + 4= -6 + 4
6>r x = —2
ANSWER ^ The equation has two solutions: 10 and —2.
CHECK / |l0-4| = I 6 I = 6
-2-4 = -6 =6
Solve the equation and check your solutions. If the equation has no
solution, write no solution.
34. | x | = 13 35. | x | = — 7 36. | x — 1 | =6
37. | 3x | = 27 38. | 2x - 3 | = 1 39. | 6x - 1 | + 5 = 2
40. Write an absolute-value equation that has 9 and 21 as its solutions.
Chapter Summary and Review
Chapter Summary and Review continued
Solving Absolute-Value Inequalities
Examples on
pp. 361-363
Solve | x + 1 | <2. Then graph the solution.
The inequality involves <, so the related inequalities are connected by and.
| x + 1 | <2 Write original inequality.
x + 1 < 2 and x + 1 > — 2 Write related inequalities.
x+1 — 1<2— 1 and x + 1 — 1 > —2 — 1 Subtract 1 from each side.
x<l and x>— 3 Simplify.
ANSWER ► The solution is all real numbers greater than —3 and less than 1. This can be written
— 3 < x < 1. The graph of the solution is shown below. Check the solution.
- 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 -►
-4 -3 -2 -1 0 1 2 3
Solve the inequality. Then graph and check the solution.
41.
<N
VI
K
42.
1 6x |
>24
43.
| x — 10 |
<8
44.
1 4x + 8
>20
45.
| 2x ■
00
V
<N
1
46.
| 5x + 3 |
>2
47.
U-4|
- 5< 1
48.
1 3x |
+ 2< 11
49.
| 2x + 1|
- 5>7
Graphing Linear Inequalities in Two Variables
Examples on
pp. 367-369
Graph y - x > 4.
Write the corresponding equation in slope-intercept form.
y — x = 4 Write corresponding equation.
y = x + 4 Add x to each side.
The graph of the line has a slope of 1 and a y-intercept of 4.
The inequality is >, so use a dashed line.
Test the origin. (0, 0) is not a solution. Since it lies below
the line, shade above the line.
ANSWER ^ The graph of y — x > 4 is all points above the line.
Check by testing any point above the line.
5
/
w _
/
V -
Lx.
>4
<
/
/
/
‘ 3
/
/
/
1
(0,
0)
_ l
v.
3
-
1
:
L x
/
-1
j 1
Graph the inequality.
50. y > —5 51. x < 2
53.x - 3y > 3 54. 2y - 6x > -2
Chapter 6 Solving and Graphing Linear Inequalities
52. — 2x + y >4
55. 3x + 6y < 12
Chapter Test
Solve the inequality. Then graph the solution.
1.x — 3 < 10 2.— 6>x + 5
3. -5 +x> 1
4.|x<2
5. —3x < 21
6. — jx < 3
7. 6 — x > 15
8. 3x + 2 < 35
9. |x + 1 > 7
10.2(x + 1) > 6
11. 3x + 5 <2x - 1
12. — 2(x + 4) > 3x +
Solve the compound inequality. Then graph the solution.
13. -15<5x<20 15. —5 < 3x — 4 < 17
16 . — 17 < 3x + 1 < 25 17 . x — 2>8<?rx+l<7 18 . — ^x < — 3or2x< —12
19. x < —2 or 3x — 5 > 1 20. 8x — 11 < 5 or 4x — 1 > 13 21. 6x + 9 > 21 or 9x — 5 < 4
22. PAPER MAKING A machine makes rolls of paper. The rolls can be as wide
as 33 feet or as narrow as 12 feet. Write a compound inequality that describes
the possible widths w of a roll of paper produced by this machine.
Solve the equation and check your solution. If the equation has no
solution, write no solution.
23. | jc + 7 | =11 24. | 3x + 4 | = 16 25. | jc — 8 | —3 = 10
Write an absolute-value equation that has the given solutions.
26. 1 and 5 27. —8 and —4 28. — 1 and 9
Solve the inequality. Then graph and check the solution.
29. | 2x | > 14 30. 1 4x + 5 | < 1 31. | 3x - 9 | + 6 < 18
Graph the inequality in a coordinate plane.
32. x > — 1 33. y > 5 34. y > 3x — 3
35. x + y <1 36. x + 2y > 6 37. 3x + 4y > 12
ALGEBRAIC MODELING In Exercises 38 and 39, use the following
information. Your club plans to buy sandwiches and juice drinks for a club
picnic. Each sandwich costs about $2 and each drink about $1. You want to
find out how many of each you can buy if you have to spend less than $100.
38. Write and graph an algebraic model that models the situation.
39. Name two solutions of the inequality you graphed in Exercise 38. Interpret
the solutions.
Chapter Test
Chapter Standardized Test
Tip
Ca^*£!DC^CjD
Work as fast as you can through the easier
problems, but not so fast that you are careless.
1. Which graph represents the solution of the
inequality x + 5 < 12?
(A) I-1-1-1- $ I I - Hi-
13 14 15 16 17 18 19 20
CD — I — I — I — $—I—I—I—*
13 14 15 16 17 18 19 20
CD —I-1-1-1-1-<NH - H*
2 3 4 5 6 7 8 9
CD — I — I — I — I — $—I—M*
2 3 4 5 6 7 8 9
2. Which phrase describes the solution of the
inequality 5x < 10?
(A) All real numbers greater than 5
CD All real numbers less than 5
CD All real numbers greater than 2
CD All real numbers less than 2
3. Which inequality is equivalent to
2 — 3x > —4?
(A) x > 2 CD * < 2
CD x > —2 CD x < —2
4. For which values of x is the inequality
— 3x + 4 < x — 2 true?
(A) x < — 3 CD x > —3
CD x<l CD x -\
5. Which number is not a solution of the
inequality — 4 < x — 1 < 5?
(A) -4 CD -3
CD 5 CD 6
6. What is the solution of the compound
inequality — 3x + 2 > 11 or 5x + 1 > 6?
(A) x < — 3 or x > 1
CD x < 3 or x > 1
CD x > — 3 or x < 1
CD x > 3 or x < 1
CD None of these
7. Which numbers are solutions of the
equation | x — 7 + 5 = 17?
(A) —19 and 15 CD — 15 and 19
CD — 15 and 29 CD — 5 and 19
8_ Which graph represents the solution of the
inequality | 2x — 10 | >6?
(A) I-I - 1 - 1 - 1 - 1 ♦ >
1 2 3 4 5 6 7 8
CD ! :—I—I—I—I—I—:
2 4 6 8 10 12 14 16
cd I :—i—i— \ —i—i—:
1 2 3 4 5 6 7 8
CD :—i—I—I—I—t —4 i
-8 -7 -6 -5 -4 -3 -2 -1
9. Choose the inequality whose solution is
shown in i
the graph.
(A)
2x +
y <4
CD
2x +
y> 4
CD
2x —
y <4
CD
2x —
y>4
Chapter 6 Solving and Graphing Linear Inequalities
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
1 1 Evaluate an Expression
Evaluate 2x + 4y when v = 3 and y = 10.
Solution
2x + 4y Write original expression.
2(3) + 4(10) Substitute 3 for x and 10 for y.
6 + 40 Simplify using the order of operations. First multiply.
46 Then add.
ANSWER ► When x = 3 and y = 10, 2x + 4y = 46.
Try These
Evaluate the expression when x = 5 and y = 9.
1- 5x + 5y 2. 9y + x 3. 2x + 3^
5- 12x + y 6- 4y + 4x 7. 3x + 4y
4. 7^ + 3x
8. 6y + 2x
■ 2 Add Integers
Find the sum.
a. -4 + (-7) b. 4 + (-7)
Student HcCp
► Extra Examples
M° re exam Pl es
and practice
exercises are available at
www.mcdougallittell.com
Solution
a. Since —4 and —7 have the same sign, add the absolute values. Give the
sum the same sign as the integers being added.
-4 + (-7) = — (| —41 + | —7 |) = — (4 + 7) = —11
ANSWER t -4 + (-7) = -11
b. Since 4 and —7 have opposite signs, subtract the smaller absolute value (4)
from the larger absolute value (7). Give the difference the same sign as the
integer with the larger absolute value.
4 +(-7)= -(|-7 | - |4 |) = -(7 - 4) = -3
ANSWER ► 4 + (-7) = -3
Try These
Find the sum.
9. -6 + (-11) 10. -4 + (-10) 11.8 + (— 1) 12. -9 + (-9)
13.-21 + 24 14.-11 + 9 15. 15 + (-8) 16. 30+ (-16)
Maintaining Skills
flap W:
Cumulative Practice
Evaluate the expression for the given value of the variable. (1.1,1.2,1.3)
1 _ x + 8 when x = — 1 2. 3x — 2 when x — 1 3- x(4 + x) when x = 5
4. 3(x — 5) when x = l 5- * + 8 when x = 4 6. x 3 — 3x + 1 when x = 2
7. PHOTO COSTS A photography studio charges $65 for a basic package of
graduation photos. Each additional wallet-sized photo costs $1. Use the
equation C = 65 + n, where C is the total cost and n is the number of
additional wallet-sized photos. Make an input-output table that shows the
cost of ordering 0 through 6 additional wallet-sized photos. ( 1 . 8 )
Evaluate the expression. (2.2, 2.3,2.4)
8. — | 3 | 9. | -2.5 | 10. -15 + 7
11.2 + (-6) + (-14) 12.-8-12 13.3.1 - (-3.3) - 1.8
14. VELOCITY AND SPEED A hot-air balloon descends at a rate of 800 feet per
minute. What are the hot-air balloon’s velocity and speed? (2.2)
15. TEMPERATURES On February 21, 1918, the temperature in Granville,
North Dakota, rose from —33°F to 50°F in 12 hours. By how many degrees
did the temperature rise? (2.4)
Simplify the expression. (2.6, 2.7)
16.40 - 4) 17. 3(6 + x) 18. (5 + n) 2 19. (3 - t)(- 5)
20. 20x - 17x 21. 4b + 7 + lb 22. 5x - 3(x - 9) 23. 4 (y + 1) + 2(y + 1)
Solve the equation. (3.1-
24.x + 4 = -1
27.J = (>
30. 6 + jx = 14
33.|(x - 15) = 20
25. —3 = n — (-15)
28. 3x + 4 = 13
31. 2x + lx - 15 = 75
34. x — 8 = 3(x — 4)
26. 6b — —36
29. 5x + 2 = -18
32. 5(x — 2) = 15
35. —(x — 6) = 4x + 1
In Exercises 36 and 37, use the formula for density, d = —, where
m represents mass and v represents volume. (3.7)
36. Find a formula for v in terms of d and m.
37. Use the formula you wrote in Exercise 36 to find the volume (in cubic
centimeters) of a piece of cork that has a density of 0.24 gram per cubic
centimeter and a mass of 3 grams.
Chapter 6 Solving and Graphing Linear Inequalities
Find the unit rate. (3.8)
38. $1 for two cans of dog food 39. 156 miles traveled in 3 hours
40. $480 for working 40 hours 41 . 125 feet in 5 seconds
Plot and label the ordered pairs in a coordinate plane. (4.1)
42. A(2, 3), B(2, -3), C(-1, 1) 43.A(0, -2),B(-3, -3), C(2, 0)
44. A(2, 4), 5(3, 0), C(—1, -4) 45.A(1, -4), B(-2, 4), C(0, -1)
CATFISH SALES In Exercises 46 and 47, use the following information.
The table below shows the number of catfish (in millions) sold in the United States
from 1990 through 1997. The numbers are rounded to the nearest million. (4.1)
Year
1990
1991
1992
1993
1994
1995
1996
1997
Number of catfish (millions)
273
333
374
379
348
322
375
387
► Source: U.S. Bureau of the Census
46. Draw a scatter plot of the data. Use the horizontal axis to represent the time.
47. Describe the relationship between the number of catfish sold and time.
Use a table of values to graph the equation. (4.2)
48. x + j = 0 49. 2x + y=l2 50. x - y = 8
51. x — y = 4 52. 2x — y = — 1 53. x + 2y = 4
Write the equation of the line in slope-intercept form. (5.1)
54. Slope = 1; y-intercept = —3. 55. Slope = —2; v-intercept = 5.
56. Slope = 0; v-intercept = 0. 57. Slope = 4; v-intercept = 1.
Write in slope-intercept form the equation of the line that passes through
the given point and has the given slope. (5.2)
58. (-1, 1), m = 2 59. (0, 1), m = 1 60. (3, 3), m = 0
61 . (3, — 1), m = -j 62. (—3, 6), m = —5 63. (—2, 2 ),m= —3
Write in point-slope form the equation of the line that passes through the
given points. (5.3)
64. (2, 0) and (0,-2) 65. (1, 4) and (3, 6) 66. (1, 10) and (3, 2)
67. (-1, -7) and (-2, 1) 68. (0, 3) and (2, 4) 69. (4, 7) and (8, 10)
Solve the inequality. (6.1
70. -6<x + 12
73.-4 - 5x<31
76. -3 <x + 1 <7
79. x + 3>7or2x + 3<
6.5, 6.7)
71. 6>3x
74. —Ax + 3 > -21
77. —4 < —2x < 10
-1 80. |x- 8 | >10
72. -f > 8
6
75. -x + 2 < 2(x - 5)
78. 2x > 10 or x + 1 < 3
81. | 2x + 5 | <7
Cumulative Practice
Materials
• hole punch
• paper cup
• string
• scissors
• rubber band
• masking tape
• paper clip
• metric ruler
• 100 pennies
OBJECTIVE Model the movement of a spring.
When a weight is attached to a spring, the spring stretches as shown.
Unweighted spring Spring with weight attached
- 1 —
Amount of
stretch
Let y be the weight attached to a spring and let x be the amount of stretch.
The variables x and y vary directly, that is y = kx. The constant of variation
k is called the spring constant for that particular spring.
In this project you will make a model of this direct variation using a rubber
band to represent the spring. The weights attached to the spring will be groups
of 10 pennies. You will estimate the “spring constant” k for your “spring” by
finding the ratio of the number of pennies to the amount of stretch for each
group of pennies.
Collecting The Data
Q Punch two holes on opposite
sides of the cup, about one
half inch down from the rim.
Thread the string through
both holes and knot the ends.
Trim any excess string. Attach
the rubber band to the string.
© Tape the paper clip to the
edge of a table or desk so
that one end hangs over the
edge. Attach the rubber band
to the paper clip. The cup
should be hanging over the
side of the table as shown in
the photo above.
© Tape the ruler to the table as
shown. The “0” on the ruler
should line up with the top of
the rubber band. Record the
distance from the top of the
rubber band to the bottom of
the cup. This is the initial
distance d.
Chapter 6 Solving and Graphing Linear Inequalities
0 Add pennies to the cup in groups of 10. Each time, measure the distance D
from the top of the rubber band to the bottom of the cup. The amount of
stretch x is given by x = D — d. Copy and complete the table below.
Number of pennies y
0
10
20
30
40
50
Distance D(mm)
d
?
?
?
?
?
Amount of stretch x(mm)
0
?
?
?
?
?
y
X
-
?
?
?
?
?
Investigating The Data
y
1. Use the values of — in the last row of your table to estimate the value of k ,
the spring constant for your spring. (The values may not all be the same
because of minor variations in the weights of individual pennies or
measuring errors.)
2 . Use your answer to Exercise 1 to write a direct variation model that relates
the number of pennies to the amount of stretch.
3- What do you the think the amount of stretch would be if you added a total
of 100 pennies to the cup? Test your conjecture.
Presenting Your Results
Write a report or make a poster to present your results.
• Include a table with your data and include your answers to Exercises 1-3.
• Describe any patterns you found when you discussed the results with others.
• Tell what advice you would give to someone who is going to do this project.
Extending The Project
• How does the length of the rubber band affect the total distance it stretches?
Tie a knot in the rubber band to shorten it and repeat the experiment.
• How does the thickness of the rubber band affect the distance it stretches?
Repeat the experiment with a thicker rubber band of the same length.
• A grocery store scale operates in a similar way. When you put fruits or
vegetables on the scale, the spring inside the scale stretches. The heavier
the item, the larger the stretch. Can you think of other items that work in
a similar way?
Project
Systems of Linear
Equations and
Inequalities
APPLICATION: Housing
To see how the need for low-income rental
housing changes over time, you can construct a model.
The graph below shows the number of households with
annual earnings of $12,000 or less that need to rent
housing and the number of rental units available that
they can afford.
In this chapter you will learn how to use pairs of linear
equations, as well as inequalities, to analyze problems.
Think & Discuss
Gr¬
ilse the graph to answer the following questions.
1. How many low-cost housing units were available
in 1995?
2 . In 1995 how much greater was the need for
low-income housing than the availability of
low-cost units?
Learn More About It
You will use a linear system to analyze the need
for low-income housing in Exercises 32 and 33 on
page 413.
APPLICATION LINK More about housing is available at
www.mcdougallittell.com
PREVIEW
What’s the chapter about?
• Graphing and solving systems of linear equations
• Determining the number of solutions of a linear system
• Graphing and solving systems of linear inequalities
Key Words
• system of linear equations, p. 389
• solution of a linear system, p. 389
• point of intersection, p. 389
• linear combination, p. 402
• system of linear inequalities, p. 424
• solution of a system of linear
inequalities, p. 424
PREPARE
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to pp. 134 , 153)
1 _ Which of the following is not a linear equation?
(A) 2x + y = 5
CD x = 3
CD y = 2x 2 - l
CD y = 3x
2_ Which equation is an identity?
(A) lx + 6 = 5(2x + 1)
CD 5(2x + 4) = 2(10 + 5x)
Cg) + 4 = —2(4x + 4)
CD -4(2 - 3jc) = -8 - I2x
Skill Check (refer to pp. 146, 367)
3. What is the solution of the equation 2x + 6(x + 1) = — 2?
CS> -1 CD -f CD j CD l
4. Which ordered pair is a solution of the inequality ly — 8x > 56?
(5) (0,8) CD (0,0) CD (-6,1) CD (-7,2)
STUDY TIP
List Kinds
of Problems
In your notebook keep
a list of different types
of problems and how
to solve them.
Mixture Problems (p. 4 jq)
*+-/ = <?() -
0.2x +■ Q.Sy =: 36
volume of mixture
~~ acid in mixture
Chapter 7 Systems of Linear Equations and Inequalities
Graphing Linear Systems
Goal
Estimate the solution
of a system of linear
equations by graphing.
Key Words
• system of linear
equations
• solution of a linear
system
• point of intersection
How many hits are you getting at your Web site?
^ flLfcienceclub <i
Welcome
to our
Website!
Introduction Experiments
Calendar Members
In this chapter you will study
systems of linear equations.
In Example 3 you will use two
equations to predict when two
Web sites will have the same
number of daily visits.
^^Introduction
Experiments
Calendar
of Events
Link to
Members
Two or more linear equations in the same variable form a system of linear
equations, or simply a linear system. Here is an example of a linear system.
x + 2y = 5 Equation 1
2x — 3y = 3 Equation 2
A solution of a linear system in two variables is a pair of numbers a and b for
which x = a and y = b make each equation a true statement.
Such a solution can be written as an ordered pair ( a , b) in which a and b are the
values of x and y that solve the linear system. The point ( a , b) that lies on the
graph of each equation is called the point of intersection of the graphs.
Student MeCp
► Morn Examples
More examples
are ayajiabie a t
www.mcdougallittell.com
Find the Point of Intersection
Use the graph at the right to estimate
the solution of the linear system. Then
check your solution algebraically.
3x + 2y = 4 Equation 1
—x + 3 y = —5 Equation 2
Solution
The lines appear to intersect at the point (2, — 1).
CHECK S Substitute 2 for x and — 1 for y in each equation.
EQUATION 1
EQUATION 2
3x + 2y = 4
-x + 3j =
-5
3(2) + 2(— 1) 1 4
-(2) + 3(— 1) 1
-5
6-214
—2 — 3 1
-5
s
II
-5 =
-5/
ANSWER ► Because the ordered pair (2, — 1) makes each equation true,
(2, — 1) is the solution of the system of linear equations.
7.1 Graphing Linear Systems
SOLVING A LINEAR SYSTEM USING GRAPH-AND-CHECK
step Q Write each equation in a form that is easy to graph.
step 0 Graph both equations in the same coordinate plane.
step © Estimate the coordinates of the point of intersection.
step 0 Check whether the coordinates give a solution by
substituting them into each equation of the original
linear system.
Student HeGp
► Look Back
For help with writing
equations in slope-
intercept form,see
p. 243.
| 2 Graph and Check a Linear System
Use the graph-and-check method to solve the linear system.
x + y = —2
2x — 3 y = —9
Equation 1
Equation 2
Solution
0 Write each equation in slope-intercept form.
EQUATION 1
x + y = —2
y — —x — 2
EQUATION 2
2x — 3 y = —9
— 3 y = —2x
2
y
x + 3
0 Graph both equations.
0 Estimate from the graph that the point of intersection is (—3, 1).
0 Check whether (—3, 1) is a solution by substituting — 3 for x and 1 for y in
each of the original equations.
EQUATION 1
EQUATION 2
x + y = — 2
II
1
X
<N
-3 + 11-2
2(— 3) - 3(1) 1
s
<N
1
II
<N
1
-6-31
-9 =
ANSWER ► Because the ordered pair (—3, 1) makes each equation true,
(—3, 1) is the solution of the linear system.
Graph and Check a Linear System
Use the graph-and-check method to solve the linear system.
1. x + y = 4 2. x — y — 5 3- x — y — — 2
2x + y = 5 2x + 3y = 0 x + y = —4
Chapter 7 Systems of Linear Equations and Inequalities
WEBMASTERS build Web
sites for clients. They design
Web pages and update
content.
More about
r Webmasters at
www.mcdougallittell.com
■afMUM 3 Write and Solve a Real-Life Linear System
WEBMASTER You are the Webmaster of the Web sites for the science club
and for the math club. Assuming that the number of visits at each site can be
represented by a linear function, use the information in the table to predict
when the number of daily visits to the two sites will be the same.
Club
Current daily visits
Increase (daily visits per month)
Science
400
25
Math
200
50
Solution
Verbal
Model
Labels
Algebraic
Model
Daily
visits
Daily
visits
Current visits
to science site
+
Increase for
science site
Current visits
to math site
+
Increase for
math site
Number of
months
Number of
months
Daily visits = V
Current visits (science) = 400
Increase (science) = 25
Number of months = t
Current visits (math) = 200
Increase (math) = 50
(daily visits)
(daily visits)
(daily visits per month)
(months)
(daily visits)
(daily visits per month)
V = 400 + 25 1 Equation 1 (science)
V = 200 + 50 t Equation 2 (math)
Student HeGp
->
► Reading Algebra
The graph at the
right tells you that in
8 months both sites
should have the
same number of daily
visits, 600.
A J
Graph both equations. The point of intersection appears to be (8, 600).
CHECK / Check this solution in each of
the original equations.
Equation 1 600 3= 400 + 25(8)
600 = 400 + 200 /
Equation 2 600 3= 200 + 50(8)
600 = 200 + 400 /
ANSWER ► According to the model, the
sites will have the same number
of visits in 8 months.
i
(A
i
V*
i
(A
• — f.
^Scienci
e
> c
_>«
)UU
're
~a
!
1
m
c
l
h.
re
.S
*00
E
=
j
2
°<
)
i
t
l
5
12
t
Number of months
W3M
Write and Solve a Real-Life Linear System
4. The Spanish club Web site currently receives 500 daily visits. If the number
of daily visits increases by 20 each month, when will the Spanish club site
have the same number of daily visits as the science club site?
7.1 Graphing Linear Systems
Exercises
Guided Practice
Vocabulary Check 1. Explain what it means to solve a linear system using the graph-and-check
method.
2. Use the graph at the right to find
the point of intersection for the
system of linear equations.
y = — x + 2
y = x + 2
Skill Check In Exercises 3-6, use the linear system below.
—x + y = —2
2x + y = 10
3. Write each equation in slope-intercept form.
4. Graph both equations in the same coordinate plane.
5. Estimate the coordinates of the point of intersection.
6 . Check the coordinates algebraically by substituting them into each equation
of the original linear system.
Practice and Applications
CHECKING SOLUTIONS Check whether the ordered pair is a solution of
the system of linear equations.
7. 3x — 2y = 11
—x + 6y = 7
(5, 2)
8. 6x — 3 y = —15
2x + y = — 3
(-2, 1)
9. x + 3y = 15
4x + y = 6
(3, -6)
10. -5x + y = 19
x — ly — 3
(-4,-1)
11. — 15x + ly = 1
3x — y = 1
(3, 5)
12. —2x + y = 11
—x —9y = -15
(6,1)
FINDING POINTS OF INTERSECTION Use the graph given to estimate the
solution of the linear system. Then check your solution algebraically.
13. — x + 2y = 6 14. 2x — y = —2 15. x + y = 3
x + 4y = 24 4x — y — —6 —2x + y — — 6
Student HeCp
► Homework Help
Example 1: Exs. 7-15
Example 2: Exs. 16-24
Example 3: Exs. 25-28
1 _ J 1
Chapter 7 Systems of Linear Equations and Inequalities
WOMEN S EMPLOYMENT
In 1870 only 5% of all office
workers were women. By
1910 that number had risen
to 40%.
GRAPH AND CHECK Estimate the solution of the linear system
graphically. Then check the solution algebraically.
16. y = —x + 3
17. y = -6
18.
1
II
y = x + l
x = 6
1
II
19. 2x-3y = 9
20. 5x + 4y= 16
21.
x — y — \
x = —3 y = —16 5x — 4y = 0
22. 3x + 6y = 15 23. y = — 2x + 6
—2x + 3y = —3 y — 2x + 2
25. CARS Car model A costs $22,000 to
purchase and $.12 per mile to maintain.
Car model B costs $24,500 to purchase
and $.10 per mile to maintain.
Use the graph to determine how many
miles each car must be driven for
the total costs of the two models to be
the same.
26. AEROBICS CLASSES A fitness club offers an aerobics class in the morning
and in the evening. Assuming that the number of people in each class can be
represented by a linear function, use the information in the table below to
predict when the number of people in each class will be the same.
Class
Current
attendance
Increase (people
per month)
Morning
40
2
Evening
22
8
27. History Link / The fast-changing world of the 1920s produced new roles
for women in the workplace. From 1910 to 1930 the percent of women
working in agriculture decreased, while the percent of women in professional
jobs increased, as shown in the table.
Job type
Percent holding that
job type in 1910
Average percent increase
per year from 1910 to 1930
Agriculture
22.4%
-0.7%
Professional
9.1%
0.25%
24. 5x + 6y = 54
-v + y = 9
(/>
h.
_cc
37
o
CO
~a
36
cc
(/)
B
3
o
-=
35
»]
A
GO
o
o
110
120
130
Miles (thousands)
Assuming that both percentages can be represented by a linear function, use
the information in the table above to estimate when the percent of women
working in agriculture equaled the percent of women working in professional
jobs between 1910 and 1930.
28. PERSONAL FINANCE You and your sister are saving money from your
allowances. You have $25 and save $3 each week. Your sister has $40 and
saves $2 each week. After how many weeks will you and your sister have the
same amount of money?
7.1 Graphing Linear Systems
Standardized Test
Practice
Mixed Review
Maintaining Skills
29. CHALLENGE You know how to solve the equation x + 2 = 3x — 4
algebraically. This equation can also be solved by graphing the following
system of linear equations.
y = x + 2
y = 3* - 4
a. Explain how the system of linear equations is related to the original
equation given.
b. Estimate the solution of the linear system graphically.
c. Check that the x-coordinate from part (b) satisfies the original equation by
substituting the x-coordinate for x in x + 2 = 3x — 4.
30. MULTIPLE CHOICE Which ordered pair is a solution of the following
system of linear equations?
x + y = 3
2x + y = 6
(A) (0, 3) CD (1, 2)
CD (2, 1) CD (3, 0)
31. MULTIPLE CHOICE Which system
of linear equations is graphed?
CD — x + 2y = 2
CD
II
<n‘
+
H
— 3x + 4y = 2
II
cn‘
1
CD — 2x + y = 1
GD
2x + y =
—4x + 3y = 2
4x + 3 y =
SOLVING EQUATIONS Solve the equation. (Lesson 3.3)
32. 3x + 7 = -2 33. 15 - 2a = 7 34. 2y + 3y = 5
35. 21 = 7(w - 2) 36. -2 (t - 5) = 26 37. 4(2x + 3) = -4
WRITING EQUATIONS Write in slope-intercept form the equation of the
line that passes through the given point and has the given slope.
(Lesson 5.2)
38. (3, 0), m = —4 39. (—4, 3), m = 1 40. (1, —5), m = 4
41. (-4, — l),m = -2 42. (2, 3), m = 2 43. (-l,5),m= -3
44. SUSPENSION BRIDGES The Verrazano-Narrows Bridge in New York is the
longest suspension bridge in North America, with a main span of 4260 feet.
Let x represent the length (in feet) of every other suspension bridge in North
America. Write an inequality that describes x. Then graph the inequality.
(Lesson 6.1)
DECIMAL OPERATIONS Perform the indicated operation.
(Skills Review pp. 759, 760)
45. 3.71 + 1.054
48. (2.1)(0.2)
46. 10.35 + 5.301
0.3
49.
0.03
47. 2.5 - 0.5
50.
5.175
1.15
Chapter 7 Systems of Linear Equations and Inequalities
You can use a graphing calculator to graph linear systems and to estimate
their solution.
Samplt
Use a graphing calculator to estimate the solution of the linear system.
y — — 0.3x + 1.8 Equation 1
y — 0.6x — 1.5 Equation 2
Solution
Q Enter the equations.
Student HeCp
► Keystroke Help
See keystrokes for
several models of
calculators at
www.mcdougallittell.com
© Graph both equations. You can
use the direction keys to move
the cursor to the approximate
intersection point.
© Set an appropriate viewing
window to graph both equations.
Q Use the Intersect feature
to estimate a point where the
graphs intersect. Follow your
calculator’s procedure to display
the coordinate values.
ANSWER ^ The solution of the linear system is approximately (3.7, 0.7).
TtyTtos*
Use a graphing calculator to estimate the solution of the linear system.
Check the result in each of the original equations.
1 . y = v + 6 2. y = — 3x — 2
y = —v — 1 y = x + 8
4. y = 1.33a: — 20
y = 0.83x — 8.5
3. y = — 0.25a: - 2.25
y = x — 1.25
Using a Graphing Calculator
Solving Linear Systems by
Substitution
Goal
s° |v e a linear system How many softballs were ordered?
by substitution. _ _ ^ 7
Key Words
• substitution method
In Exercise 29 you will solve
a linear system to analyze a
problem about ordering
softballs. You will use a method
called the substitution method.
There are several ways to solve a linear system without using graphs. In this
lesson you will study an algebraic method known as the substitution method.
J i Substitution Method: Solve for y First
Solve the linear system.
—x + y = 1 Equation 1
2x + y — —2 Equation 2
Solution
Q Solve for y in Equation 1.
—x + y = 1 Original Equation 1
y = x + 1 Revised Equation 1
0 Substitute x + 1 for y in Equation 2 and find the value of x.
2x + y = —2
2x + (x + 1) — ~2
3x + 1 = -2
3x = —3
x = — 1
Write Equation 2.
Substitute x + 1 for y.
Combine like terms.
Subtract 1 from each side.
Divide each side by 3.
© Substitute — 1 for x in the revised Equation 1 to find the value of y.
y — x + 1 — —1 + 1—0
0 Check that (— 1, 0) is a solution by substituting — 1 for x and 0 for y in each
of the original equations.
ANSWER ► The solution is (— 1, 0).
Chapter 7 Systems of Linear Equations and Inequalities
Student HeCp
► Study Tip
When using
substitution, you will
get the same solution
whether you solve for
/first or xfirst. You
should begin by solving
for the variable that is
easier to isolate.
v _ j
2 Substitution Method: Solve for x First
Solve the linear system.
2x + 2y = 3 Equation 1
x — 4y = — 1 Equation 2
Solution
Q Solve for x in Equation 2 because it is easy to isolate x.
x — 4 y = — 1 Original Equation 2
x = 4y — 1 Revised Equation 2
e Substitute 4y — I for x in Equation 1 and find the value of y.
2x + 2y = 3
2(4 y - 1) + 2y = 3
8y - 2 + 2y = 3
10y — 2 = 3
10y = 5
y =
i
Write Equation 1.
Substitute 4y - 1 for x.
Use the distributive property.
Combine like terms.
Add 2 to each side.
Divide each side by 10.
© Substitute ~ for y in the revised Equation 2 to find the value of x.
4y
1 = 4 k
1 = 2 - 1 = 1
© Check by substituting 1 for x and ^ for y in the original equations.
ANSWER ^ The solution is ( 1, —).
Substitution Method
Name the variable you would solve for first. Explain.
1- 3x — y = — 9 2. x + 3y = — 11 3. x — 3y = 0
2x + 4y — 8 2x — 5y = 33 x — 2 y = 10
_
_ -s
ITT'
TT
L_
Solving a Linear System by Substitution
step © Solve one of the equations for one of its variables.
step © Substitute the expression from Step 1 into the other
equation and solve for the other variable.
step © Substitute the value from Step 2 into the revised
equation from Step 1 and solve.
step© Check the solution in each of the original equations.
7.2 Solving Linear Systems by Substitution
NATIONAL CIVIL RIGHTS
MUSEUM The National Civil
Rights Museum educates
people about the history of
the civil rights movement
through its unique collections
and powerful exhibits.
More about
museums at
www.mcdougallittell.com
3 Write and Use a Linear System
MUSEUM ADMISSIONS In one day the National Civil Rights Museum in
Memphis, Tennessee, admitted 321 adults and children and collected $1590.
The price of admission is $6 for an adult and $4 for a child. How many adults
and how many children were admitted to the museum that day?
Solution
Verbal
Model
I
Labels
Number
Number
Total
of +
of =
number
adults
children
admitted
Price of
Number
Price of
Number
Total
adult
•
of +
child
• of =
amount
admission
adults
admission
children
collected
Number of adults = x
Number of children = y
Total number admitted = 321
Price of adult admission = 6
Price of child admission = 4
Total amount collected = 1590
(people)
(people)
(people)
(dollars per person)
(dollars per person)
(dollars)
Algebraic x + y = 321
Model
6 x +4 j = 1590
Use the substitution method to solve
x = —y + 321
6(-y + 321) + 4 y= 1590
-6 y + 1926 + 4 y= 1590
—2 y + 1926 = 1590
—2 y = -336
y = 168
x = -(168) + 321 = 153
Equation 1 (Number admitted)
Equation 2 (Amount collected)
the linear system.
Solve Equation 1 for*. (Revised Equation 1)
Substitute -y + 321 for x in Equation 2.
Use the distributive property.
Combine like terms.
Subtract 1926 from each side.
Divide each side by -2.
Substitute 168 for yin revised Equation 1.
ANSWER^ 153 adults and 168 children were admitted to the National Civil
Rights Museum that day.
Write and Use a Linear System
4. In one day a movie theater collected $4275 from 675 people. The price of
admission is $7 for an adult and $5 for a child. How many adults and how
many children were admitted to the movie theater that day?
Chapter 7 Systems of Linear Equations and Inequalities
Exercises
Guided Practice
Vocabulary Check
1. What four steps do you use to solve a system of linear equations by the
substitution method?
2. When solving a system of linear equations, how do you decide which
variable to isolate in Step 1 of the substitution method?
Skill Check
In Exercises 3-6, use the following system of equations.
3x + 2y = 7 Equation 1
5x — y = 3 Equation 2
3. Which equation would you use to solve for y? Explain why.
4. Solve for y in the equation you chose in Exercise 3.
5. Substitute the expression for y into the other equation and solve for x.
6 - Substitute the value of x into your equation from Exercise 4. What is the
solution of the linear system? Check your solution.
Use substitution to solve the linear system. Justify each step.
7. 3x + y = 3 8- 2x + y = 4 9. 3x — y = 0
lx + 2y = 1 — x + y = 1 5y = 15
Practice and Applications
CRITICAL THINKING Tell which equation you would use to isolate a
variable. Explain.
10 . 2x + y = -10
3x — y = 0
11. m + An = 30
m — 2/i = 0
12 . 5c + 3d = 11
5c — d = 5
13. 3x — 2 y
x + y
19
8
14. 4 a + 3b
a — b
-5
-3
15. 3x + 5 y = 25
x — 2 y = —10
SOLVING LINEAR SYSTEMS Use the substitution method to solve the
linear system.
17. s = t + 4
2t + s = 19
16. y = x — 4
4x + y = 26
18. 2c — d = —2
4c + d = 20
00
II
<N
O)
r-
20. 2x + 3y = 31
21.
P + 9 = 4
Student HeCp
a + b = 2
y — x + 7
+ q — 1
1^ - N,
► Homework Help
22. x — 2 y = —25
23. u — v = 0
24.
o
II
1
H
Example 1: Exs. 10-27
1
II
o
lu + v = 0
12x — 5 y = —21
Example 2: Exs. 10-27
Example 3: Exs. 28-34
25. m + 2/7 = 1
26. x — y = —5
27.
— 3 w + z — 4
1 j
5m + 3 n = —23
x + 4= 16
—9vv + 5z = — 1
7.2 Solving Linear Systems by Substitution
jT7
28. TICKET SALES You are selling tickets for a high school play. Student
tickets cost $4 and general admission tickets cost $6. You sell 525 tickets and
collect $2876. Use the following verbal model to find how many of each type
of ticket you sold.
Number of
Number of
student
+
general
=
admissions
admissions
Total number
of tickets
Price of
Number of
Price of
Number of
r 1 1 Q 1
student
•
student
+
general
•
general
=
ioiai
admission
admissions
admission
admissions
price
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 29-31 is available at
www.mcdougallittell.com
29. SOFTBALL You are ordering softballs for two softball leagues. The size
of a softball is measured by its circumference. The Pony League uses an
11 inch softball priced at $3.50. The Junior League uses a 12 inch softball
priced at $4.00. The bill smeared in the rain, but you know the total was
80 softballs for $305. How many of each size did you order?
30. Geomotry Link / The rectangle at the right has
a perimeter of 40 centimeters. The length of the
rectangle is 4 times as long as the width. Find the
dimensions of the rectangle.
31. INVESTING One share of ABC stock is worth three times as much as XYZ
stock. An investor has 100 shares of each. If the total value of the stocks is
$4500, how much money is invested in each stock?
RUNNING at a rate of
200 meters per minute for one
hour, a 140 pound person will
burn 795 Calories. At a rate
of 250 meters per minute,
the same person will burn
953 Calories.
RUNNING In Exercises 32 and 33, use the following information.
You can run 200 meters per minute uphill and 250 meters per minute downhill.
One day you run a total of 2200 meters in 10 minutes.
32. Assign labels to the verbal model below. Then write an algebraic model.
Meters uphill
+
Meters downhill
=
Total meters
33. Find the number of meters you ran uphill and the number of meters you
ran downhill.
34. ERROR ANALYSIS Find and correct the error shown below.
Chapter 7 Systems of Linear Equations and Inequalities
Standardized Test
Practice
Mixed Review
Maintaining Skills
35- MULTIPLE CHOICE Which linear system has the solution (6, 6)?
(A) 4x — 3y = — 1
— 2x + y = — 3
Cb) x + y = 12
3x — 2y = 6
CD 3x + y
4x — 3y
4
1
CS) 4x + 3y
2x — y
0
0
36- MULTIPLE CHOICE Which linear system has been correctly solved for one
of the variables from the following system?
2x — y = — 1
2x + y = —1
CD 2x — y = — 1
y — 2x — 1
CED y = 2x + 1
2x + y = — 7
Cep 2x — y = — 1
y = —2x + 7
GD y — ~ 2x — l
2x + y = — 7
37- MULTIPLE CHOICE Your math test is worth 100 points and has
38 problems. Each problem is worth either 5 points or 2 points. How
many problems of each point value are on the test?
(A) 5 points: 54
2 points: 46
CD 5 points: 46
2 points: 54
CD 5 points: 30
2 points: 8
CD 5 points: 8
2 points: 30
SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 2.7)
38. 4 g + 3 + 2g — 3 39. 3x + 2 — (5x + 2)
40. 6(2 — m) — 3m — 12 41. 4(3 a + 5) + 3(—4 a + 2)
GRAPHING LINES Write the equation in slope-intercept form. Then graph
the equation. (Lesson 4.7)
42. 6x + y = 0 43. 8x - 4y = 16 44. 3x + y = -5
45. 5x + 3y = 3 46. x + y = 0 47. y = —4
SOLVING AND GRAPHING Solve the inequality. Then graph the solution.
(Lessons 6.4, 6.5)
48. -5 < -x< 1 49. — 14 <jc + 5 < 14 50. -2< -3jc + 1 < 10
51. x + 6 < 7 or 4x > 12 52. 3x — 2>4or5 — x>9
COMMON FACTORS List all the common factors of the pair of numbers.
(Skills Review p. 761)
53.3,21 54.4,28 55.21,27 56.10,50
57. 12, 30 58. 18, 96 59. 78, 105 60. 84, 154
7.2 Solving Linear Systems by Substitution
Solving Linear Systems by
Linear Combinations
Goal
Solve a system of linear
equations by linear
combinations.
Key Words
• linear combination
How can a farmer find the location of a beehive?
In Exercise 44 you will solve a
linear system to find the location
of a beehive. You will use a method
called linear combinations.
Sometimes it is not easy to isolate one of the variables in a linear system. In that
case it may be easier to solve the system by linear combinations. A linear
combination of two equations is an equation obtained by (1) multiplying one or
both equations by a constant if necessary and (2) adding the resulting equations.
1 Add the Equations
Solve the linear system. 4x + 3y = 16 Equation 1
2x — 3y = 8 Equation 2
Solution
Q Add the equations to get an equation in one variable.
4x + 3y = 16 Write Equation 1.
2x — 3y — 8 Write Equation 2.
Add equations.
Solve for x.
6x
= 24
x — 4
0 Substitute 4 for x into either equation and solve for y.
4(4) + 3y = 16 Substitute 4 for x.
y = 0 Solve for y.
© Check by substituting 4 for x and 0 for y in each of the original equations.
ANSWER ^ The solution is (4, 0).
Add the Equations
Solve the linear system. Then check your solution.
1. 3x + 2y = 7 2. 4x — 2y = 2 3- 5x + 2 y = —4
— 3x + 4y = 5 3x + 2y = 12 —5x + 3y = 19
Chapter 7 Systems of Linear Equations and Inequalities
Sometimes you can solve by adding the original equations because the
coefficients of a variable are already opposites, as in Example 1. In Example 2
you need to multiply both equations by an appropriate number first.
2 Multiply Then Add
Solve the linear system. 3x + 5y = 6 Equation 1
—Ax + 2y = 5 Equation 2
Solution
0 Multiply Equation 1 by 4 and Equation 2 by 3 to get coefficients of x that
are opposites.
3x + 5y = 6 Multiply by 4.
12x + 20y = 24
—4x + 2y — 5 Multiply by 3.
— 12x + 6y = 15
0 Add the equations and solve for y.
26 y = 39
Add equations.
y = 1.5 Solve for y.
© Substitute 1.5 for y into either equation and solve for x.
—Ax + 2(1.5) = 5 Substitute 1.5 for y.
—Ax + 3 = 5 Multiply.
—Ax — 2 Subtract 3 from each side.
x — —0.5 Solve for x.
0 Check by substituting —0.5 for x and 1.5 for y in the original equations.
ANSWER ► The solution is (—0.5, 1.5).
Multiply Then Add
Solve the linear system. Then check your solution.
4. 2x — 3 y = 4 5. 3x + Ay = 6 6. 6x + 2y = 2
—4x + 5y = — 8 2x — 5y = —19 — 3x + 3 y = —9
Solving a Linear System by Linear Combinations
step Q Arrange the equations with like terms in columns.
step © Multiply, if necessary, the equations by numbers to obtain
coefficients that are opposites for one of the variables.
step © Add the equations from Step 2. Combining like terms
with opposite coefficients will eliminate one variable.
Solve for the remaining variable.
step 0 Substitute the value obtained in Step 3 into either of the
original equations and solve for the other variable.
step © Check the solution in each of the original equations.
7.3 Solving Linear Systems by Linear Combinations
Student HaCp
► More Examples
More examples
are ava j| a bie at
www.mcdougallittell.com
3 Solve by Linear Combinations
Solve the linear system.
3x + 2y = 8
2y = 12 — 5x
Equation 1
Equation 2
Solution
© Arrange the equations with like terms in columns.
3x + 2y = 8 Write Equation 1.
5x + 2y = 12 Rearrange Equation 2.
0 Multiply Equation 2 by — 1 to get the coefficients of y to
be opposites.
3x + 2y = 8 3x + 2y = 8
5x + 2y = 12 Multiply by -tl — 5x — 2y = —12
© Add the equations. — 2x = — 4
x — 2
© Substitute 2 for x into either equation and solve for y.
3x + 2y = 8 Write equation 1.
3(2) + 2y = 8 Substitute 2 for x.
6 + 2y = 8 Multiply.
2y = 2 Subtract 6 from each side.
y = 1 Solve for y.
ANSWER ► The solution is (2, 1).
0 Check the solution in each of the original equations.
First check the solution in Equation 1.
3x + 2y = 8 Write Equation 1.
3(2) + 2(1) d= 8 Substitute 2 for x and 1 for y.
6 + 228 Multiply.
8 = 8/ Add.
Then check the solution in Equation 2.
2 y = 12 — 5x Write Equation 2.
2(1) =2 12 — 5(2) Substitute 2 for x and 1 for y.
2 2= 12 — 10 Multiply.
2 = 2 y Subtract.
Add equations.
Solve for x.
Solve by Linear Combinations
Solve the linear system. Then check your solution.
7- 2x + 5 y = —11
5y = 3x- 21
8. -13 = 4x - 3y
5x + 2y = 1
9- 4x + 7y = —9
3x = 3y + 18
Chapter 7 Systems of Linear Equations and Inequalities
Guided Practice
Vocabulary Check
1. When you use linear combinations to solve a linear system, what is the
purpose of using multiplication as a first step?
Skill Ch&ck ERROR ANALYSIS In Exercises 2 and 3, find and correct the error.
Describe the steps you would use to solve the system of equations using
linear combinations. Then solve the system. Justify each step.
4. x + 3y = 6 5. 3x — 4y = 7 6- 2y = 2x — 2
x — 3y = 12 2x — y = 3 2x + 3y = 12
Practice and Applications
USING ADDITION Use linear combinations to solve the linear system.
Then check your solution.
7.x + y = 4
x — y = —10
11 . p + 4q = 23
~P + <7 = 2
8. a — b = 8
a + b — 20
12. 3v — 2w = 1
2v + 2w = 4
9. 2x + y = 4
x — y — 2
13. g + 2h = 4
g h — 2
10. m + 3/7 = 2
—m + 2/? = 3
14. 13x — 5 y = 8
3x + 5y = 8
Student ftedp
► Homework Help
Example 1: Exs. 7-14,
31-42
Example 2: Exs. 15-22,
31-42
Example 3: Exs. 23-42
h j 1
USING MULTIPLICATION AND ADDITION Use linear combinations to
solve the linear system. Then check your solution.
15. x + 3y = 3
x + 6y = 3
19. 2a + 6z — 4
3a — lz — 6
16. v - w = -5
v + 2w = 4
20. 5e + 4/= 9
4e + 5/= 9
17. 2g — 3h = 0
3g - 2h = 5
21. 2p — q — 2
2p + 3q = 22
18. x — y = 0
-3x - y = 2
22. 9m-3n = 20
3/77 + 6/7 = 2
ARRANGING LIKE TERMS Use linear combinations to solve the linear
system. Then check your solution.
23. x - 3y
3y + v
30
12
24. 3b + 2c
5 c + b
46 25. y — x — 9
11 x + 8y = 0
26. m = 3/7
/77 + 10 ^
13
27. 2q = 1 — 5p 28. 2v = 150 — u 29. g — 10// = 43 30. 5s + 8t = 70
4/7 — 16 = 7/ 2// = 150 — v 18 = —g + 5/? 60 = 5s — 8f
7.3 Solving Linear Systems by Linear Combinations
LINEAR COMBINATIONS Use linear combinations to solve the linear
system. Then check your solution.
Link-
Science
VOLUME AND MASS
Legend has it that
Archimedes (above) was
asked to prove that a
crown was not pure gold.
Archimedes compared the
volume of water displaced by
the crown with the volume
displaced by an equal mass
of gold. The volume of water
displaced was not the same,
proving that the crown was
not pure gold.
31. v + 2y = 5
32. —3 p + 2 — q
33. t + r = 1
34. 3g — 24 = —4h
5x — y = 3
—q + 2p = 3
2r — t = 2
-2 + 2 h = g
35. x + 1 = 3y
36. 4 a = -b
37. 2/77 — 4 = 4/7
38. 3y = —5x + 15
CO
1
II
a — b = 5
m — 2 = n
-y = — 3jc + 9
39. 3 j + 5k=\9
40. 6x + 2y = 5
41. 3jc + ly = 6
42. 5_y — 20 = -4x
j ~ 2k = — 1
8v + 2y = 3
2x + 9y — 4
4 y = —20x + 16
Write and Use a Linear System
VOLUME AND MASS A gold crown, suspected of containing some silver,
was found to have a mass of 714 grams and a volume of 46 cubic centimeters.
The density of gold is about 19 grams per cubic centimeter. The density of
silver is about 10.5 grams per cubic centimeter. What percent of the crown
is silver?
Solution
Verbal
Model
Gold
volume
+
Silver
volume
Total
volume
Labels
Gold
density
Gold
volume
Silver
density
Silver
volume
Total
mass
Volume of gold = G
(cubic centimeters)
Volume of silver = S
(cubic centimeters)
Total volume = 46
(cubic centimeters)
Density of gold = 19
Density of silver = 10.5
Total mass = 714
(grams per cubic centimeter)
(grams per cubic centimeter)
(grams)
Algebraic
Model
G + S = 46
19 G + 10.5 S = 714
Equation 1
Equation 2
Use linear combinations to solve for 5.
-19 G- 195 = -874
19G + 10.55 = 714
-8.55 = -160
5 ~ 18.8
Multiply Equation 1 by -19.
Write Equation 2.
Add equations.
Solve for 5.
ANSWER ^ The volume of silver is about 19 cm 3 . The crown has a volume of
19
46 cm 3 , so the crown is oy ~ 41% silver by volume.
4o
Chapter 7 Systems of Linear Equations and Inequalities
MODELING Use the example on the previous page as a model for
Exercise 43.
43, VOLUME AND MASS A bracelet made of gold and copper has a mass of
46 grams. The volume of the bracelet is 4 cubic centimeters. Gold has a
density of about 19 grams per cubic centimeter. Copper has a density of
about 9 grams per cubic centimeter. How many cubic centimeters of copper
are mixed with the gold?
44. BEEHIVE A farmer is tracking
two wild honey bees in his
field. He maps the first bee’s path
to the hive on the line ly = 9x. The
second bee’s path follows the line
y = — 3x + 12. Their paths cross at
the hive. At what coordinates will
the farmer find the hive?
45. Hist ory Link / The first known system of linear equations appeared in
Chinese literature about 2000 years ago. Solve this problem from the book
Shu-shu Chiu-chang which appeared in 1247.
A slrxGbousG Ppfe tt joa Kinds cfl stuff cotton, floss
silK onp rckk! si1k~TbQU) taka iqcioptonj) of1t)G
MqfGrals of )d ffsb tocuf off an d pate ganqGnts
flor ti)G ar.Mu\ As flor tt)G cotton, fl gjg .usg % rolls flor
(o MGD; pg fffiG a sbortago ofl &0 rolls,- ifl so usg ,v
t rolls flor 7 Mpn, tf)GTG is a surplus ofl rffo rdls....\ff .
GJIsb to Knopj tt)G nupbGT ofl PGO UkG CUT) dotbG
tuff):...
-x = - Y - 2(00
U - M: - Ol . 0
Q> 7
Standardized Test
Practice
46. CHALLENGE Solve for x, y, and z in the system of equations. Explain each
step of your solution.
3x + 2y + z — 42
2y + z + 12 = 3x
x — 3y = 0
47. MULTIPLE CHOICE Solve the system and choose the true statement.
x + y = 4
x — 2 y = 10
(A) The value of x is greater than y. Cb) The value of y is greater than x.
Cep The values of x and y are equal. (Tp None of these
48. MULTIPLE CHOICE Solve the system and choose the true statement.
3x + 5 y = —8
x - 2y = 1
(T) The value of x is greater than y. Cg) The value of y is greater than x.
(H) The values of x and y are equal. C p None of these
7.3 Solving Linear Systems by Linear Combinations
Mixed Review
Maintaining Skills
Quiz 7
WRITING EQUATIONS Write in slope-intercept form the equation of the
line that passes through the given point and has the given slope, or that
passes through the given points. (Lessons 5.2 , 5.3)
49. (-2, 4), m = 3 50. (5, 1), m = 5 51. (9, 3), m = -3
52. (-2, -1) and (4, 2) 53. (6, 5) and (2, 1) 54. (4, -5) and (-1, -3)
CHECKING SOLUTIONS Check whether each ordered pair is a solution of
the inequality. (Lesson 6.8)
55. 3x - 2y < 2; (1, 3), (2, 0) 56. 5x + 4y > 6; (-2, 4), (5, 5)
SOLVING LINEAR SYSTEMS Use the substitution method to solve the
linear system. (Lesson 7.2)
57. — 6x — 5y = 28 58. m + 2n = 1 59. g — 5h = 20
x — 2y = 1 5/77 — 4/7 = —23 4g + 3h = 34
SIMPLIFYING FRACTIONS Decide whether the statement is true or false.
Explain. (Skills Review p. 763)
60.
12
63.
11
10
55
61.
25
35
64.
28
15
6 , 1-3
62 ‘16 " 7
65.
250
350
2
3
Estimate the solution of the linear system graphically. Then check the
solution algebraically. (Lesson 7.1)
1 . 3x + y = 5 2. x — 2y = 0 3. 2x + 3y = 36
-x + y=-l 3x - y = 0 — 2x + y = -4
Use substitution to solve the linear system. (Lesson 7.2)
4. 4x + 3y = 31 5.—12x + y = 15 6. x + 2y = 14
= 2x + 7 3x + 2y = 3 2x + 3y = 18
Use linear combinations to solve the linear system. (Lesson 7.3)
7. 2x + 3y = 36 8. jc + 7}/ = 12 9. 3x — 5y = —4
2x-y = 4 3x — 5y = 10 ~9x + 7y = 8
Choose a method to solve the linear system. (Lessons 7.1-7.3)
10. 2x + 3y= 1 11- jc + 18 y= 18 12. 5x - 3y = 7
4x — 2 v = 10 x — 3 y = —3 jc + 3.y = 5
13. COMPACT DISCS A store is selling compact discs for $10.50 and $8.50.
You buy 10 discs for $93. Write and solve a linear system to find how many
compact discs you bought at each price. (Lessons 7.1-7.3)
EEE
Chapter 7 Systems of Linear Equations and Inequalities
Linear Systems and Problem
Solving
Goal
Use linear systems to
solve real-life problems.
How many violins were sold?
Key Words
• substitution method
• linear combinations
method
In Example 1 you will use a system
of linear equations to find the number
of violins a store sold. Once you have
written a linear system that models
a real-life problem, you need to
decide which solution method is
most efficient.
Student HeCp
I ►Study Tip
Examples 1 and 2
are called mixture
problems. Mixture
problems often have
one equation of the
form
x + y = amount
and another equation in
which the coefficients
of x and yare not 1.
■'x
J
1 Choosing a Solution Met hod
VIOLINS In one week a music store sold 7 violins for a total of $1600. Two
different types of violins were sold. One type cost $200 and the other type cost
$300. How many of each type of violin did the store sell?
Solution
Verbal Number of , Number of
MoDEL type A type B
Labels
Number of type A = x
(violins)
Number of type B = y
(violins)
Total number sold = 7
(violins)
Price of type A = 200
(dollars per violin)
Price of type B = 300
(dollars per violin)
Total sales = 1600
(dollars)
Algebraic
Model
■ + y =7
200 x + 300 .y = 1600
Equation 1
Equation 2
The coefficients of x and y are 1 in Equation 1, so use the substitution method.
You can solve Equation 1 for x and substitute the result into Equation 2. After
simplifying, you will obtain y — 2. Then substitute this y -value into the revised
Equation 1 and simplify to obtain x = 5.
ANSWER ► The store sold 5 type A violins and 2 type B violins.
Price of
type A
Number of
type A
+
Price of
type B
Number of
type B
Total
sales
Total number
sold
7.4 Linear Systems and Problem Solving
CHEMISTRY To test the
acidity of a substance,
scientists use litmus paper.
When the paper comes in
contact with acid, it turns red.
2 Solve a Mixture Problem
CHEMISTRY You combine 2 solutions to form a mixture that is 40% acid. One
solution is 20% acid and the other is 50% acid. If you have 90 milliliters of the
mixture, how much of each solution was used to create the mixture?
Solution
Verbal
Model
Volume of
solution A
+
Acid in
solution A
+
Volume of
solution B
Volume of
mixture
Acid in
solution B
Acid in
Mixture
Labels Volume of solution A = x (milliliters)
Volume of solution B — y (milliliters)
Volume of mixture = 90 (milliliters)
Acid in solution A = 0.2 x (milliliters)
Acid in solution B = 0.5 y (milliliters)
Acid in mixture = 0.4(90) = 36 (milliliters)
Algebraic
Model
x y — 90
0.2 x + 0.5 y = 36
Equation 1
Equation 2
Solve Equation 1 for x and multiply each side of Equation 2 by 10 so that it
contains only integers. Then use substitution to solve the system.
II
o
1
Revised Equation 1
2x + 5 y = 360
Revised Equation 2
2(90 -y) + 5y = 360
Substitute 90 - y for x in Revised Equation 2.
180 - 2y + 5y = 360
Use the distributive property.
o
00
II
CO
Combine like terms.
o
vo
II
Solve for y.
X = 90 - 60 = 30
Substitute 60 for yin Revised Equation 1.
ANSWER ^ 30 mL of solution A and 60 mL of solution B were used.
Solve Mixture Problems
1. A store sold 32 pairs of jeans for a total of $1050. Brand A sold for $30 per
pair and Brand B sold for $35 per pair. How many of each brand were sold?
2 . A 10-pound mixture of peanuts and cashews sells for $5.32 per pound. The
price of peanuts is $3.60 per pound and the price of cashews is $7.90 per
pound. How many pounds of each type are in the mixture?
Chapter 7 Systems of Linear Equations and Inequalities
Student MeCp
► Mom Examples
More examples
are available at
www.mcdougallittell.com
| 3 Compare Two Salary Plans
SALES JOBS Job A offers an annual salary of $30,000 plus a bonus of 1% of
sales. Job B offers an annual salary of $24,000 plus a bonus of 2% of sales.
How much would you have to sell to earn the same amount in each job?
Solution
Verbal
Model
Total earnings = Job A salary + 1 % • Total sales
Total earnings = Job B salary + 2% • Total sales
Labels Total earnings = y (dollars)
Total sales = x (dollars)
Job A salary = 30,000 (dollars)
Job B salary = 24,000 (dollars)
Algebraic y = 30,000 + 0.01 x Equation 1 (Job A)
Model
y — 24,000 + 0.02 x Equation 2 (Job B)
It is convenient to use the linear combinations method.
—y = —30,000 — O.Olx Multiply Equation 1 by -1.
y = 24,000 + 0.02x Write Equation 2.
0 = —6000 + O.Olx Add Equations.
x = 600,000 Solve for x.
Substitute x = 600,000 into Equation 1 and simplify to obtain y = 36,000.
ANSWER ► You would have to sell $600,000 of merchandise to earn $36,000
in each job.
When a linear system has a solution ( a , b ), this solution can be found by
substitution or by linear combinations.
EEEEH3
Ways to Solve a System of Linear Equations
SUBSTITUTION requires that one of the variables be isolated on
one side of the equation. It is especially convenient when one of
the variables has a coefficient of 1 or -1.
(Examples 1-3, pp. 396-398)
LINEAR COMBINATIONS can be applied to any system, but it
is especially convenient when a variable appears in different
equations with coefficients that are opposites.
(Examples 1-3, pp. 402-404)
GRAPHING can provide a useful method for estimating a solution.
(Examples 1-3, pp. 389-391)
H
7.4 Linear Systems and Problem Solving
/■ j Exercises
Guided Practice
Vocabulary Check 1 . Describe a system that you would use linear combinations to solve.
Skill Check Choose a method to solve the linear system. Explain your choice.
2. x + y = 300 3, 3x + 5y = 25 4. 2x + y = 0
x + 3y = 18 2x — 6y = 12 x + y = 5
5. Solve Example 3 on page 411 using the substitution method.
POCKET CHANGE In Exercises 6-8, use the following information.
You have $2.65 in your pocket. You have a total of 16 coins, with only quarters
and dimes. Let q equal the number of quarters and d equal the number of dimes.
6. Complete: ? + ? = 16
7. Complete: 25 q + ? = 265
8. Use the equations you wrote in Exercises 6 and 7 to find how many of each
coin you have.
Practice and Applications
COMPARING METHODS Solve the linear system using both methods
described on page 411. Then represent the solution graphically.
9- x + y = 2 10. x — y = 1 11. 3x - y = 3
6x + y = 2 x + y = 5 —x + y = 3
CHOOSING A SOLUTION METHOD Choose a solution method to solve
the linear system. Explain your choice, but do not solve the system.
12 . 6x + y = 2
9x - y = 5
13. 2x + 3y = 3
5x + 5y = 10
14. — 3x = 36
— 6x + y = 1
15. 2x — 5y = 0
x - y = 3
16. 3x + 2y = 10
2x + 5y = 3
17.x + 2y = 2
x H - 4y — 2
Student HeCp
► Homework Help
Example 1: Exs. 9-26
Example 2: Exs. 27-31
Example 3: Exs. 32-34
^ ^
SOLVING LINEAR SYSTEMS Choose a solution method to solve the
linear system. Explain your choice, and then solve the system.
18. 2x + y = 5
II
1
i
3
20. x 2y — 4
x — y = 1
4x + 3y =
21
6x + 2y = 10
21.
3x + 6y = 8
22. x + y =
0
23. 2x — 3y = —7
—6x + 3y = 2
3x + 2y =
1
3x + y = — 5
24.
8x + 4y = 8
25. x + 2y =
1
26. 6x — y = 18
—2x + 3y = 12
5x - 4y =
-23
8x + y = 24
Chapter 7 Systems of Linear Equations and Inequalities
CRITICAL THINKING In Exercises 27-29, match the situation with the
corresponding linear system.
27. You have 7 packages of paper towels. Some packages have 3 rolls, but some
have only 1 roll. There are 19 rolls altogether.
28. You buy 5 pairs of socks for $19. The wool socks cost $5 per pair and the
cotton socks cost $3 per pair.
29. You have only $1 bills and $5 bills in your wallet. There are 7 bills worth a
total of $19.
A. x + y = 7
x T 3y — 19
B. x + y = 7
x T 5y — 19
C. x + y = 5
3x + 5y = 19
COMMUNITY GARDENS
allow people without yards to
plant their own gardens. A
25 foot by 35 foot garden can
produce enough vegetables
for a family of four.
More about
community gardens at
www.mcdougallittell.com
30. TREADMILLS You exercised on a treadmill for 1.5 hours. You jogged at
4 miles per hour and then sprinted at 6 miles per hour. The treadmill monitor
says that you ran for a total of 7 miles. Using the verbal model below,
calculate how long you ran at each speed.
Time spent
+
Time spent
jogging
sprinting
Jogging
•
Time spent
+
speed
jogging
Total time
on treadmill
Sprinting
speed
Time spent
sprinting
Total
distance
31. COMMUNITY GARDENS You designate one row in your garden to broccoli
and pea plants. Each broccoli plant needs 12 inches of space and each pea
plant needs 6 inches of space. The row is 10 feet (120 inches) long. If you
want a total of 13 plants, how many of each plant can you have?
HOUSING In Exercises 32 and 33, use the following information.
The graph below represents the need for low-income rental housing in the United
States and the number of affordable rental units available.
QATA UPDATE of Center on Budget and Policy
t ** 1 Priorities data at www.mcdougallittell.com
32. Use the points (0, 6200) and (25, 10,500) to write an equation for the number
of housing units needed. Then use the points (0, 6500) and (25, 6100) to
write an equation for the number of affordable units available.
33. Solve the system you wrote in Exercise 32. Use the graph to check the
reasonableness of your solution.
34. TREE GROWTH You plant a 14-inch spruce tree that grows 4 inches per year
and an 8-inch hemlock tree that grows 6 inches per year. After how many
years will the trees be the same height? How tall will each be?
7.4 Linear Systems and Problem Solving
Standardized Test
Practice
Mixed Review
Maintaining Skills
35, CHALLENGE It takes you 3 hours to drive to a concert 135 miles away. You
drive 55 miles per hour on highways and 40 miles per hour the rest of the
time. How much time did you spend driving at each speed?
36. 4ftjZzl9 Let the variables a , b , g, and p represent the weights of an
apple, a banana, a bunch of grapes, and a pineapple, respectively. Use these
variables to write three equations that model the first three diagrams below.
Then use substitution to determine how many apples will balance the
pineapple and two bananas in the fourth diagram.
37. MULTIPLE CHOICE You and your friend go to a Mexican restaurant.
You order 2 tacos and 2 enchiladas and your friend orders 3 tacos and
1 enchilada. Your bill was $4.80 and your friend’s bill was $4.00. Which
system of linear equations represents the situation?
CE)2t + 2e = 4.00 CD 2t + 2<? = 4.00
3t + e = 4.80 t + 3e = 4.80
CD 2t + 2e = 4.80
3t + e = 4.00
CD 2t + 2e = 4.80
t + 3e — 4.00
38. MULTIPLE CHOICE Solve the system of equations you chose in Exercise 37.
CD t = $1.60 CD t = $.80 CH) t = $1.40 CD t = $.60
e — $.80 e — $1.60 e — $.60 e — $1.40
PARALLEL LINES Determine whether the graphs of the two equations
are parallel lines. Explain. (Lesson 4.7)
39. line a\y — 4x + 3 40. line a\ 4y + 5x = 1
line b\ 2y — 8x = —3 line b\ lOx + 2y = 2
41. line a\ 3x + 9y + 2 = 0
line b\ 2y = ~6x + 3
42. line a\ 4y — 1 = 5
line b\ 6y + 2 = 8
GRAPHING FUNCTIONS Graph the function. (Lesson 4.8)
43. f\x) = 2x + 3 44. h(x) = x + 5 45. g(x) = 5x — 4
46. g(x) = — x + 2
47./(x) = —4x + 1 48. h(x) = — 3x - 1
ADDING FRACTIONS Add. Write the answer as a fraction or a mixed
number in simplest form. (Skills Review p. 764)
„ 9 , 3
49 -T5 + 5
„ 1 , 1
5 °-l2 + 2
53 -L+2
10 + 3
55 IZ + i
55 ‘ 32 + 4
56. — + -
20 8
Chapter 7 Systems of Linear Equations and Inequalities
K /,^) O
DEVELOPING CONCEPTS
For use with
Lesson 7.5
Goal
Use reasoning to discover
graphical and algebraic
rules for finding the
number of solutions of a
Question
How can you identify the number of solutions of a linear system by
graphing or by using an algebraic method?
linear system.
Materials
• graph paper
Explore
Q Graph each linear system.
a. x + y = 0 b. 2x — 4y = 6 c. x — y =
3x — 2y = 5 x — 2y = 3 — 3x + 3 y =
Q How are the three graphs different?
© Write both equations of each system in the form y = mx + b.
© How are the equations within each system alike or how are they
different?
Student HeCp
Think About It
be -^
p Look Back
For help with graphing
linear systems, see
p. 390.
Il _ *
1 _ Repeat Steps 1 through 4 for the following systems.
a. x — 3 y = 9 b. 4x — y = 20 c. x + 2y = 3
2x + 6 y = — 18 20x + y = 28 x + 2y = 6
Write a linear system for the graphical model. If only one line is shown,
write two different equations for the line.
LOGICAL REASONING The graph of a linear system is described.
Determine whether the system has no solution , exactly one solution ,
or infinitely many solutions. Explain.
5. The lines have the same slope and the same ^-intercept.
6. The lines have the same slope but different ^-intercepts.
7. The lines have different slopes.
Developing Concepts
Developing Concepts: continued
Question
How can you solve systems that have many solutions or recognize
systems that have no solution?
Explore
O Try to solve each linear system.
a. x + y = 0 b. 2x — 4y = 6 c. x — y = l
3x — 2y = 5 x — 2y = 3 — 3x + 3y = 3
Q Refer to your graph of part (a) from Step 1 on page 415. What does the
algebra of part (a) tell you about the graphs of the equation?
© Refer to your graph of part (b) from Step 1 on page 415. What does the
algebra of part (b) tell you about the graphs of the equation?
Q Refer to your graph of part (c) from Step 1 on page 415. What does the
algebra of part (c) tell you about the graphs of the equation?
Think About It
Describe the algebraic solution of the system. Then check your answer by
solving the appropriate equation you wrote for Exercises 2-4 on page 415.
Solve the linear system using linear combinations. Then describe the
graphical solution of the system.
4. 2x — y = 3 5- 2x + y = 5 6- x + 3y = 2
—4x + 2y = 0 x — 3y — — 1 2x + 6 y = 4
7. LOGICAL REASONING Summarize your results from Exercises 1-6
by writing a rule for determining algebraically whether a system of
linear equations has exactly one solution, no solution, or infinitely
many solutions.
Chapter 7 Systems of Linear Equations and Inequalities
Special Types of Linear
Systems
Goal
Identify how many
solutions a linear
system has.
Key Words
• linear system
What is the weight of a bead in a necklace?
Some linear systems have no
solution or infinitely many
solutions. In Exercise 31 you
will see why this can be a
problem as you try to find the
weight of a jewelry bead.
Student HeCp
.
p Look Back
For help with
equations in one
variable that have no
solution, see p. 153.
v_ -J
B222233I 1 A Linear System with No Solution
Show that the linear system has no solution. 2x + y = 5 Equation 1
2x + y = 1 Equation 2
Solution
Method 1 GRAPHING Rewrite each
equation in slope-intercept form.
Then graph the linear system.
y = — 2x + 5 Revised Equation 1
y = — 2x + 1 Revised Equation 2
Because the lines have the same slope but
different y-intercepts, they are parallel. Parallel
lines never intersect, so the system has no solution.
Method 2 SUBSTITUTION Because revised Equation 2 is y = — 2x + 1,
you can substitute — 2x + 1 for y in Equation 1.
2x + y = 5 Write Equation 1.
2x + (— 2x + 1) = 5 Substitute -2x + 1 fory.
1^5 Combine like terms.
The variables are eliminated and you are left with a statement that is false. This
tells you that the system has no solution.
I
A Linear System with No Solution
1 _ Show that the linear system has no solution. v + 3y = 4 Equation 1
2x + 6y = 4 Equation 2
H
7.5 Special Types of Linear Systems
2 A Linear System with Infinitely Many Solutions
Show that the linear system has infinitely many solutions.
— 2x + y = 3 Equation 1
—Ax + 2y = 6 Equation 2
Solution
Method 1 GRAPHING Rewrite each
equation in slope-intercept form.
Then graph the linear system.
y = 2x + 3 Revised Equation 1
y = 2x + 3 Revised Equation 2
You can see that the equations represent
the same line. Every point on the line is
a solution of the system.
Method 2 LINEAR COMBINATIONS You can multiply Equation 1 by 2
to obtain an equation that is identical to Equation 2.
—Ax + 2y = 6 Revised Equation 1
—Ax + 2y = 6 Equation 2
The two equations are identical. Any solution of — 4x + 2y = 6 is also a solution
of the system. This tells you that the linear system has infinitely many solutions.
A Linear System with infinitely Many Solutions
2 . Show that the linear system has infinitely many solutions.
x — 2y = A Equation 1
—x + 2 y = —A Equation 2
EEEE
Number of Solutions of a Linear System
If the two equations have
If the two equations
have different slopes,
then the system has
one solution.
Lines intersect
Exactly one solution
the same slope but different
/-intercepts, then the system
has no solution.
N
X
\ i
\
k
Lines are parallel
No solution
If the two equations have
the same slope and the same
/-intercept, then the system
has infinitely many solutions
« /
/
X
Lines coincide
Infinitely many solutions
Chapter 7 Systems of Linear Equations and Inequalities
3 Identify the Number of Solutions
a. 3x + y = — 1
— 9x — 3 y = 3
b. x — 2y = 5
—2x + 4y = 2
c. 2x + y = 4
4x - 2y = 0
Solution
a. Use linear combinations.
You can multiply Equation 1 by —3 to obtain Equation 2.
— 9x — 3y = 3 Revised Equation 1
— 9x — 3y = 3 Equation 2
ANSWER ^ The two equations are identical. Any solution of — 9x — 3y = 3
is also a solution of the system. Therefore the linear system has
infinitely many solutions.
b. Use linear combinations.
x — 2y = 5 Multiply by 2. 2x — 4y — 10
— 2x + 4y — 2
— 2x + 4y — 2
0 A 12
Add equations.
ANSWER ► The resulting statement is false. The linear system has
no solution.
c. Use the substitution method.
2x + y = 4
y = —2x + 4
4x — 2y — 0
4x — 2(— 2x + 4) = 0
4x + 4x — 8 = 0
8x — 8 = 0
8x = 8
x = 1
j = -2(1) + 4
y=~ 2+4
J = 2
Write Equation 1.
Solve Equation 1 for y. (Revised Equation 1)
Write Equation 2.
Substitute -2x + 4 for y.
Use the distributive property.
Combine like terms.
Add 8 to each side.
Solve for x.
Substitute 1 for x in Revised Equation 1.
Multiply.
Solve for y.
ANSWER ► The linear system has exactly one solution, which is the ordered
pair (1, 2).
Identify the Number of Solutions
Solve the linear system and tell how many solutions the system has.
3. x + y = 3 4. x + y = 3 5. x + y = 3
2x + 2y = 4 2x + 2y = 6 x + 2y = 4
7.5 Special Types of Linear Systems
Exercises
Guided Practice
Vocabulary Check Describe the graph of a linear system that has the given number of
solutions. Sketch an example.
1. No solution 2. Infinitely many solutions 3- Exactly one solution
Skill Check Graph the system of linear equations. Does the system have exactly one
solution, no solution, or infinitely many solutions ? Explain.
4. 2x + y = 5 5- — 6x + 2y = 4 6- 2x + y = 7
—6x — 3 y = —15 — 9x + 3y = 12 3x — y = —2
Use the substitution method or linear combinations to solve the linear
system and tell how many solutions the system has.
7. —x + y — 7 8. —4x + y = — 8 9. ~4x + y = -8
2x — 2 y — —18 — 1 2x + 3y = —24 2x — 2y — —14
Practice and Applications
Student He dp
^
►Homework Help
Example 1: Exs. 10-33
Example 2: Exs. 10-33
Example 3: Exs. 10-33
LINEAR SYSTEMS Match the linear system with its graph and tell how
many solutions the system has.
10. —2x + 4y = 1
3x — 6y = 9
13. -x + y = 1
x-y= 1
y*
A
Z
3 ^
-1 ,
, 1
3 \
>y
1
-
1
-1
5 X
/
z
11. 2x - 2y = 4
-x + y = -2
14. 5x + = 17
x - 3j = -2
12. 2x + y = 4
—4x - 2j = -8
15. x-y = 0
5x — 2j = 6
\
D
\
1
v
\
\
-
1 ,
, 1
k
4 x
16. ERROR ANALYSIS Patrick says that the
graph of the linear system shown at the
right has no solution. Why is he wrong?
y j
/7
2
f
/
2
i
1
3 x
T
Chapter 7 Systems of Linear Equations and Inequalities
Student HeCp
► Homework Help
Extra help with
problem solving
in Ex. 23 is available at
www.mcdougallittell.com
v _/
Link to
Careers
CARPENTERS must be
familiar with codes that
specify what types
of materials can be used.
Carpenters also must be
able to estimate how much
material will be needed and
what the total cost will be.
More about
' carpenters at
www.mcdougallittell.com
INTERPRETING GRAPHICAL RESULTS Use the graphing method to tell
how many solutions the system has.
17.x + y = 8
x + y=-l
18. 3x — 2 v = 3 19. x — y = 2
— 6x + 4 y = —6 — 2x + 2y = 2
20 . -x + 4 y= -20
3x - 12y = 48
21 . 6x — 2y = 4
12x — 6 v = 8
22 . 3x + 2y = 40
— 3x — 2 v = 8
23. CRITICAL THINKING Explain how you can tell from the equations how
many solutions the linear system has. Then solve the system.
x — y — 2 Equation 1
4x — 4y = 8 Equation 2
INTERPRETING ALGEBRAIC RESULTS Use the substitution method or
linear combinations to solve the linear system and tell how many
solutions the system has. Then describe the graph of the system.
24. -lx + ly = 1 25. 4x + 4 y= -8 26. 2x + y = -4
2x — 2 y = — 18 2x + 2 y = —4 4x — 2y = 8
27. 15x — 5y = —20 28. —6x + 2 y = —2 29. 2x + y = —1
— 3x + y = 4 —4x — y = 8 —6x — 3 y = —15
30. BUSINESS A contracting company rents a generator for 6 hours and a
heavy-duty saw for 6 hours at a total cost of $48. For another job the
company rents the generator for 4 hours and the saw for 8 hours for a total
cost of $40. Find the hourly rates g (for the generator) and s (for the saw)
by solving the system of equations 6g + 6s = 48 and 4g + 8s = 40.
31. JEWELRY You have a necklace and matching bracelet with 2 types of beads.
There are 40 small beads and 6 large beads on the necklace. The bracelet has
20 small beads and 3 large beads. The necklace weighs 9.6 grams and the
bracelet weighs 4.8 grams. If the threads holding the beads have no
significant weight, can you find the weight of one large bead? Explain.
CARPENTRY In Exercises 32 and 33, use the following information.
A carpenter is buying supplies for the next job. The job requires 4 sheets of oak
paneling and 2 sheets of shower tileboard. The carpenter pays $99.62 for these
supplies. For the following job the carpenter buys 12 sheets of oak paneling and
6 sheets of shower tileboard and pays $298.86.
32. Can you find how much the carpenter is spending on 1 sheet of oak
paneling? Explain.
33. If the carpenter later spends a total of $139.69 for 8 sheets of oak paneling
and 1 sheet of shower tileboard, can you find how much 1 sheet of oak
paneling costs? Explain.
CHALLENGE In Exercises 34 and 35, use the following system.
6x — 9 y — n Equation 1
— 2x + 3y = 3 Equation 2
34. Find a value of n so that the linear system has infinitely many solutions.
35. Find a value of n so that the linear system has no solution.
Student HeCp
► Homework Help
Extra help with
problem solving
in Ex. 23 is available at
www.mcdougallittell.com
k _/
Link to
Careers
CARPENTERS must be
familiar with codes that
specify what types
of materials can be used.
Carpenters also must be
able to estimate how much
material will be needed and
what the total cost will be.
More about
' carpenters at
www.mcdougallittell.com
7.5 Special Types of Linear Systems
Standardized Test 36. multiple choice
Practice Which graph corresponds
to a linear system that has
no solution?
(A) I CD II
CD III (D) IV
37. MULTIPLE CHOICE
Which graph corresponds
to a linear system that has
infinitely many solutions?
CD I CD II
CED ill CD IV
ROCK CLIMBING In Exercises 38 and 39, use the following information.
You are climbing a 300 foot cliff. By 1:00 P.M. you have climbed 110 feet up
the cliff. By 3:00 P.M. you have reached a height of 220 feet. (Lesson 4.5)
38. Find the slope of the line that passes through the points (1, 110) and (3, 220).
What does it represent?
39. If you continue climbing the cliff at the same rate, at what time will you
reach the top of the cliff?
GRAPHING INEQUALITIES Graph the inequality. (Lesson 6.8)
40. x < 2 41. y > 5 42. y < 3x + 1
43. y > x + 4 44. 4x + y < 4 45. 2x — 3y < 6
Maintaining Skids ESTIMATING AREA Estimate the area of the figure to the nearest square
unit. Then find the exact area, if possible. (Skills Review p. 775)
Chapter 7 Systems of Linear Equations and Inequalities
For use with
Lesson 7.6
r
Goal
Question
Use graphing to describe
the solution of a system of
linear inequalities.
How can you graph a system of linear inequalities?
Materials
• graph paper
• red and blue pencils
Explore
"".
Consider the following system of linear inequalities.
x + y < 5 Inequality 1
x — y > 1 Inequality 2
© Graph the boundary lines x + y = 5 and x — y = 1 in the same coordinate
plane.
© Test several points with integer coordinates in the first inequality. If a point is
a solution, circle the point in blue.
© Test several points with integer coordinates in the second inequality. If a point
is a solution, circle the point in red.
© Describe the points that are solutions of both inequalities (the points that are
circled with both colors).
Think About It
Follow Steps 1 through
describe the solution.
1, x + y > 4
x — 2y < —2
4. x > 3
x < 5
to graph the system
2 - x — y < 0
x + y < 6
5- y < 4
y — i
linear inequalities. Then
3- 3x + 2y > 8
— 3x + y < 1
6- 4x + y > 2
4x + y < 8
LOGICAL REASONING Use your results from Exercises 1-6 to answer the
following questions.
7. When would the solution of a system of two linear inequalities be a
horizontal strip? When would the solution of a system of two linear
inequalities be a vertical strip?
8- When would a system of two linear inequalities have no solution?
9. When would a half-plane be the solution of a system of two linear
inequalities?
10, What are the possible graphs of a general system of two linear inequalities?
Developing Concepts
Systems of Linear
Inequalities
Goal
Graph a system of linear Hqw m spotlights CO 11 VOU afford?
inequalities. 7 r 27 7
Key Words
• system of linear
inequalities
• solution of a system
of linear inequalities
In Exercises 34-36 you will
graph a system of linear
inequalities to analyze the
number of spotlights that
can be ordered for a theater.
From Lesson 6.8 remember that the graph of a linear inequality in two variables
is a half-plane. The boundary line of the half-plane is dashed if the inequality is
< or > and solid if the inequality is < or >, as shown in the graphs below.
iVJ
\
\
l
k
\
X
X
-l
]
i
i
-l
x + y<3
\
\
is ~
Two or more linear inequalities in the same variables form a system of linear
inequalities, or a system of inequalities. A solution of a system of linear
inequalities in two variables is an ordered pair that is a solution of each
inequality in the system.
Student HeCp
► Study Tip
Notice how the two
half-planes above can
be used to find the
solution in Example 1.
L j
Graph a System of Two Linear Inequalities
Graph the system of linear inequalities. x + y <3 Inequality 1
x + 4y > 0 Inequality 2
Solution
Graph both inequalities in the same
coordinate plane. The graph of the
system is the overlap, or intersection ,
of the two half-planes shown at the
right as the darker shade of blue.
X j
1
\
\ x + 4y>0
-1
5 ]
^ X
-1
x + y < 3
\
Chapter 7 Systems of Linear Equations and Inequalities
2 Graph a System of Three Linear Inequalities
Graph the system of linear inequalities.
y<2
x>-l
y>x - 2
Inequality 1
Inequality 2
Inequality 3
Solution
The graph of y < 2 is the
half-plane below the dashed
line y — 2.
The graph of x > — 1 is the half-plane
on and to the right of the solid line
x = — 1.
The graph of y > x — 2 is the
half-plane above the dashed
line y — x — 2.
Finally, the graph of the system
is the intersection of the three
half-planes shown.
Graph a System of Linear Inequalities
Graph the system of linear inequalities.
1. jc + 2y<6 2. y < 3 3. x>0
—jc + y < 0 y > 1 y > 0
2x + 3y < 12
EEEEH3
Graphing a System of Linear Inequalities
step O Graph the boundary lines of each inequality. Use a
dashed line if the inequality is < or > and a solid line if
the inequality is < or >.
step 0 Shade the appropriate half-plane for each inequality.
step © Identify the solution of the system of inequalities as the
intersection of the half-planes from Step 2.
7.6 Systems of Linear Inequalities
3 Write a System of Linear Inequalities
Write a system of linear inequalities
that defines the shaded region shown.
Solution
Since the shaded region is bounded by
two lines, you know that the system
must have two linear inequalities.
Student MeCp
\
> p Look Back
For help with
writing equations
in slope-intercept
form, see p. 269.
I ^
INEQUALITY 1 The first inequality is bounded by the line that passes through
the points (0, 1) and (3, 4). The slope of this line can be found using the
formula for slope.
y2 ~ y 1
m = - Write formula for slope.
v — y r
x 2 x \
4-1
m — ^ _ q Substitute coordinates into formula.
m — 1 Simplify.
Since (0, 1) is the point where the line crosses the y-axis, an equation for this
line can be found using the slope-intercept form.
y — mx + b Write slope-intercept form.
y — lx + 1 Substitute 1 for m and 1 for b.
y — x + 1 Simplify.
Since the shaded region is below this solid boundary line, the inequality is
y<*+ 1.
INEQUALITY 2 The second inequality is bounded by the vertical line that
passes through the point (3, 0). An equation of this line is v = 3.
Since the shaded region is to the left of this dashed boundary line, the
inequality is v < 3.
ANSWER ► The system of inequalities that defines the shaded region is:
y < x + 1 Inequality 1
x<3 Inequality 2
l_
Write a System of Linear inequalities
Write a system of linear inequalities that defines the shaded
region shown.
t 1
i
i
- )
r
■
i
-
3
i
:
i :
5 X
1
i
1
4 \
Chapter 7 Systems of Linear Equations and Inequalities
1 Exercises
Guided Practice
Vocabulary Check 1 . Determine whether the following statement is true or false. Explain.
A solution of a system of linear inequalities is an ordered pair
that is a solution of any one of the inequalities in the system.
Skill Check
Graph the system of linear inequalities.
2. y>—2x + 2 3. y>x
y < — 1 x < 1
4. x 4- 1 > y
y > 0
ERROR ANALYSIS Use both the student
graph shown at the right and the system
of linear inequalities given below.
y > —i
x>2
y>x - 4
5. Find and correct the errors the student
made while graphing the system.
6 . Graph the system correctly.
Write a system of linear inequalities that defines the shaded region.
Practice and Applications
LINEAR INEQUALITIES Match the graph with the system of linear
inequalities that defines it.
Student HeCp
► Homework Help
Example 1: Exs. 9-17,
37-39
Example 2: Exs. 18-23,
31-36
Example 3: Exs. 24-30,
40,41
A. 2x + y < 4
—2x + y < 4
V
T
/
/
\
/
/
\
/
1
y
\
\
1
i
\ 3 ?
-1
\
jf
B. 2x + y > -4
x - 2y< 4
C. 2x + y < 4
2x + y> —4
7.6 Systems of Linear Inequalities
GRAPHING SYSTEMS Graph the system of linear inequalities.
12 . y >0
x> -2
15. y < 2x - 1
y> -x + 2
18. x + y <6
x> 1
j>0
21 . x > 0
y>0
x < 3
13 .y> -2
y < 4 - 2x
16. 2x — 2y < 6
x-y<9
19. x<3
2y< 1
2x + y > 2
22 . x > —2
j >-2
_y < 4
14. 2x + 3y < 5
3x + 2y > 5
17.x - 3y> 12
x — 6y < 12
20. 3x — 2 y> —6
x T 4y > 2
4x + y < 2
23. x — 2y < 3
3x + 2y > 9
x + y < 6
WRITING SYSTEMS Write a system of linear inequalities that defines the
shaded region.
Geometry Link, - Plot the points and draw line segments connecting the
points to create the polygon. Then write a system of linear inequalities
that defines the polygonal region.
27. Triangle: (-2, 0), (2, 0), (0, 2) 28. Rectangle: (1, 1), (7, 1), (7, 6), (1, 6)
Link to
Careers
CHEFS prepare meals that
appeal to both the taste buds
and the eye. They develop
menus, direct kitchen
workers, and estimate food
needs.
More about chefs
4*^ is available at
www.mcdougallittell.com
29. Triangle: (0, 0), (-7, 0), (-3, 5) 30. Trapezoid: (-1, 1), (1, 3), (4, 3), (6, 1)
FOOD BUDGET In Exercises 31-33, use the following information.
You are planning the menu for your restaurant. For Saturday night you plan to
serve roast beef and teriyaki chicken. You expect to serve at least 240 pounds of
meat that evening and that less beef will be ordered than chicken. The roast beef
costs $5 per pound and the chicken costs $3 per pound. You have a budget of at
most $1200 for meat for Saturday night.
31. Copy and complete the following system of linear inequalities that shows the
pounds b of roast beef meals and the pounds c of teriyaki chicken meals that
you could prepare for Saturday night.
b + c> ?
b ? c
?|*fc+? • c< 1200
32. Graph the system of linear inequalities.
33. CRITICAL THINKING What quadrant should the graph in Exercise 32 be
restricted to for the solutions of the system to make sense in the real-world
situation described? Explain.
Chapter 7 Systems of Linear Equations and Inequalities
Student HeCp
► Homework Help
Extra help with
^ problem solving in
Ex. 34-36 is available at
www.mcdougallittell.com
LIGHTING In Exercises 34-36, use the following information.
You have $10,000 to buy spotlights for your theater. A medium-throw spotlight
costs $1000 and a long-throw spotlight costs $3500. The current play needs at
least 3 medium-throw spotlights and at least 1 long-throw spotlight.
34, Write a system of linear inequalities for the number m of medium-throw
spotlights and the number / of long-throw spotlights that models both your
budget and the needs of the current play.
35. For 0 < m < 7 and 0 < / < 7, plot the pairs of integers (ra, /) that satisfy the
inequalities you wrote in Exercise 34.
36. Which of the options plotted in Exercise 35 correspond to a cost that is less
than $8000?
EARNING MONEY In Exercises 37-39, use the following information.
You can work a total of no more than 20 hours per week at your two jobs.
Baby-sitting pays $5 per hour, and your job as a cashier pays $6 per hour. You
need to earn at least $90 per week to cover your expenses.
37. Write a system of inequalities that shows the various numbers of hours you
can work at each job.
38. Graph the system of linear inequalities.
39. Give two possible ways you could divide your hours between the two jobs.
TREE FARMING In Exercises 40-42, use the tree farm graph shown.
40. Write a system of inequalities that
defines the region containing
maple trees.
41. Write a system of inequalities that
defines the region containing
sycamore trees.
42. CHALLENGE Find the area of the
oak tree region. Explain the method
you used.
Standardized Test
Practice
43. MULTIPLE CHOICE Which system of
inequalities is graphed?
(A) y <3x — 1
2x + y > 4
(IT) y < 3x + 1
2x + y > 4
Cg) y <3x — 1 (D) y < 3x + 1
2x — y > —4 2 x — y> —4
44. MULTIPLE CHOICE Which ordered pair is a solution of the following
system of linear inequalities?
y < x + 2
y + x > 4
CD (1, 3) <3D (2, 1) CE) (2, 6) Q) (4, 2)
7.6 Systems of Linear Inequalities
Mixed Review
EVALUATING NUMERICAL EXPRESSIONS Evaluate the expression.
(Lessons 1.2, 1.3)
45. 3 5 46. 8 2 - 17 47. 5 3 + 12
48. 2(3 3 - 20)
49. 2 6
17
3 + 1
50. 5 • 2 + 4 2
EVALUATING EXPONENTIAL EXPRESSIONS Evaluate the expression for
the given values of the variables. (Lesson 1.2)
51. (x + y) 2 when x = 5 and y — 2 52, (b — c) 2 when b — 2 and c — 1
53. g ~ h 2 when g = 4 and h = 8 54. x 2 + z when x = 8 and z = 12
55. TEST QUESTIONS Your teacher is giving a test worth 250 points. There are
68 questions. Some questions are worth 5 points and the rest are worth
2 points. How many of each question are on the test? (Lesson 7.4)
Maintaining Skills
FRACTIONS, MIXED NUMBERS, AND DECIMALS Write the fraction or
mixed number as a decimal. (Skills Review pp. 763, 767)
22
56. T
37
57 —
0/ - 4
60. 1
1
61. 3i
58 51
58 ‘ 12
__ 56
59 ‘ 20
62.4
1
63. 6
8
Quiz 2
1. Ge ometry Link / The perimeter of the
rectangle is 22 feet and the perimeter of the
triangle is 12 feet. Find the dimensions of
the rectangle. (Lesson 7.4)
2. GASOLINE The cost of 12 gallons of regular gasoline and 18 gallons of
premium gasoline is $44.46. Premium costs $.22 more per gallon than
regular. What is the cost per gallon of each type of gasoline? (Lesson 7.4)
Use any method to solve the linear system and tell how many solutions
the system has. (Lesson 7.5)
3. 3x + 2 y = 12 4. 4x + 8y = 8 5. —Ax + lly = 44
9x + 6y = 18 x + y = 1 4x - 11 y= -44
Graph the system of linear inequalities. (Lesson 7.6)
6. y<—x + 3 7. x —2y<—6 8. x + y<l
y > 1 5x — 3y < —9 — x + y < 1
y >0
9. Write a system of linear inequalities that
defines the shaded region. (Lesson 7.6)
T
Chapter 7 Systems of Linear Equations and Inequalities
Chapter Summary
and Review
• system of linear equations,
• point of intersection, p. 389
• solution of a system of
>
p. 389
• linear combination, p. 402
linear inequalities, p. 424
• solution of a linear system,
p. 389
• system of linear inequalities,
p. 424
Graphing Linear Systems
Examples on
pp. 389-391
Estimate the solution of the linear system graphically. Then check the
solution algebraically.
—x + y = 3 Equation 1
x + y = 7 Equation 2
First write each equation in slope-intercept form so that they are easy to graph.
EQUATION 1 EQUATION 2
—x + y = 3 x + y = 7
y = x + 3 y = —x + 7
Then graph both equations.
Estimate from the graph that the point of intersection is (2, 5).
Check whether (2, 5) is a solution by substituting 2 for x and
5 for y in each of the original equations.
EQUATION 1 EQUATION 2
— x + y = 3 x + y = 7
-(2) + 5 2= 3 2 + 527
3=3/ 7=7/
ANSWER ► Because the ordered pair (2, 5) makes each equation true, (2, 5) is the solution
of the linear system.
Estimate the solution of the linear system graphically. Then check the
solution algebraically.
1- x + y = 6
x — y = 12
2 . 4x — y = 3
3x + y = 4
3. x + 9y = 9
3x + 6y = 6
4. 5x — y = —5
3x + 6y = —3
5, lx + 8y = 24
x - 8y = 8
6 . 2x — 3y = —3
x + 6y = -9
Chapter Summary and Review
Chapter Summary and Review continued
1.1
1.3
Solving Linear Systems by Substitution
Examples on
pp. 396-398
Solve the linear system by substitution.
o
II
1
Equation 1
£
1
II
00
Equation 2
Solve for y in Equation 1 because it is easy to isolate y.
o
II
1
Original Equation 1
II
Revised Equation 1
00
II
1
3
Write Equation 2.
00
II
g
1
$
Substitute 2x for y.
x = 4
Solve for x.
II
eg
II
£
II
8 Substitute 4 for x in Revised Equation 1 to solve for y.
ANSWER ^ The solution is (4, 8). Check the solution in the original equations.
Use the substitution method to solve the linear system.
7. x + 3y = 9
8. —2x — 5y = 7
9. 4x — 3 y
= -2
'sO
1
II
1
$
lx + y = —8
4x + y
= 4
10. -x + 3y = 24
11. 4x + 9y = 2
12. 9x + 6 y
= 3
5x + 8y = —5
2x + 6y = 1
3x - ly
= -26
Solving Linear Systems by Linear Combinations
Examples on
pp. 402-404
Solve the linear system by linear combinations.
2x — 15y = —10 Equation 1
— 4x + 5y = —30 Equation 2
You can get the coefficients of x to be opposites by multiplying Equation 1 by 2.
2x — 15y = —10 Multiply by 2. 4x — 30 y = —20
— 4x + 5 y = —30 —4x + 5 y = —30
— 25y = — 50 Add equations.
y = 2 Solve for y.
Substitute 2 for y in Equation 2 and solve for x.
—4x + 5y = —30 Write Equation 2.
—4x + 5(2) = -30 Substitute 2 for y.
x = 10 Solve for x.
ANSWER ^ The solution is (10, 2). Check the solution in the original equations.
Chapter Summary and Review continued
Use linear combinations to solve the linear system.
13. —4x - 6y = 7
x + 5y = 8
14. 2x + y = 0
5x — Ay — 26
16. 9x + 6y = 3
3y + 6x = 18
17. 2-lx = 9y
2y — Ax = 6
15. 3x + 5 y = —16
—2x + 6 y = —36
18. Ax - 9y = 1
25x + 6y = 4
Linear Systems and Problem Solving
Examples on
pp. 409-411
Your teacher is giving a test worth 150 points. There are
46 three-point and five-point questions. How many of each are on the test?
Write an algebraic model. Let x be the number of three-point questions and let
y be the number of five-point questions.
3x + 5 y = 150 Equation 1
x + y = 46 Equation 2
Since at least one variable has a coefficient of 1, use substitution to solve the system.
y = —x + 46
Solve Equation 2 for y. (Revised Equation 2)
3x + 5(— x + 46) =
150
Substitute -x + 46 for y in Equation 1.
3x — 5x + 230 =
150
Use the distributive property.
— 2x =
-80
Combine like terms.
x =
40
Divide each side by -2.
y = -(40) + 46 =
6
Substitute 40 for x in Revised Equation 2.
ANSWER ^ There are 40 three-point questions and 6 five-point questions.
19. RENTING MOVIES You spend $13 to rent five movies for the weekend. New
releases rent for $3 and regular movies rent for $2. How many regular movies
did you rent? How many new releases did you rent?
Special Types of Linear Systems
Examples on
pp. 417-419
Tell how many solutions the following linear system has.
3x + 5y = 1 Equation 1
— 3x — 5y = 8 Equation 2
Use linear combinations.
3x + 5y = 1 Write Equation 1.
— 3x — 5y = 8 Write Equation 2.
0^15 Add equations.
ANSWER ► The resulting statement is false. The linear system has no solution.
Chapter Summary and Review
Chapter Summary and Review continued
Tell how many solutions the following linear system has.
—x — 3 y = —5
2x + 6y = 10
Equation 1
Equation 2
You can multiply Equation 1 by —2 to obtain Equation 2.
2x + 6y = 10 Revised Equation 1
2x + 6y = 10 Equation 2
ANSWER ^ The two equations are identical. Any solution of 2x + 6 y = 10 is also a solution
of the system. This tells you that the linear system has infinitely many solutions.
Use the substitution method or linear combinations to solve the linear
system and tell how many solutions the system has.
20 . —2x — 6y = —12 21 . 2x — 3y = 1 22 . — 6x + 5y = 18
2x + 6y = 12 —2x + 3y = 1 lx + 2y = 26
7.6 Systems of Linear Inequalities
Examples on
pp. 424-426
Graph the system of linear inequalities.
x > 0 Inequality 1
y < — 2x + 2 Inequality 2
y > 3x — 7 Inequality 3
Graph all three inequalities in the same coordinate plane. Use
a dashed line if the inequality is < or > and a solid line if the
\
y
/
inequality is < or >.
y>3x—l i
\ /
The graph of x > 0 is the half-plane on and to the right of the
line x = 0.
\ /
-3 -1
-1
3 5 x
Y
The graph of the y < — 2x + 2 is the half-plane below the
line y = — 2x + 2.
-3
/ \ x> 0
/ \
v
The graph of y > 3x — 7 is the half-plane on and above the
line y — 3x — 7.
The graph of the system is the intersection of the
three half-planes shown.
-5
7
/ \
f \
y< -2x+2\
'
Graph the system of linear inequalities.
23.x > —5
24. 2x — lOy > 8
25. — x + 3y < 15
<N
1
V
x — 5y <12
9x > 27
26. x < 5
27. x + y < 8
28. ly > -49
y > —2
x — y < 0
—lx + y > — 14
x T 2y > —4
y >4
x + y < 10
■ ——
Chapter Test
Estimate the solution of the linear system graphically. Then check the
solution algebraically.
1 - y — 2x — 3 2 . 6x 4- 2y = 16 3 - 4x — y = 10
—y — 2x — \ —2x 4- y = —2 — 2x 4- 4y — 16
Use the substitution method to solve the linear system.
4. —4x 4- ly — —2 5- lx 4- 4y — 5 6 - — 3x + 6y = 24
x — —y — 5 x — 6y = —19 — 2x — y = 1
Use linear combinations to solve the linear system.
7- 6x + 7_y = 5 8. —lx + 2y = —5 9- —3x + 3j = 12
4x — 2y — —10 lOx — 2y = 6 4x + 2y = 20
10, WILD BIRD FOOD You buy 6 bags of wild bird food to fill the feeders in
your yard. Oyster shell grit, a natural calcium source, sells for $4.00 a bag.
Sunflower seeds sell for $5.00 a bag. If you spend $28.00, how many bags of
each type of bird food are you buying?
Use the substitution method or linear combinations to solve the linear
system and tell how many solutions the system has.
11.8* + 4y = -4
2x — y — —3
12 , — 6x + 3 y = —6
2x + 6y = 30
13- — 3x + y = —18
3x — y = —16
14- 3jc + y = 8
4x + 6y = 6
15- 3jc — 4y = 8
9x — 12 y = 24
16- 6x 4- y = 12
—4x — 2y = 0
Graph the system of linear inequalities.
17,jc<4
y>l
18- — 3x 4- 2y > 3
x 4- 4y < —2
20.x > -1
J<3
J>-3
21 . y<2
y>x-2
y>-x- 2
19. 2x -3y<12
-x- 3y >-6
22 . x < 5
y<6
y > —2x + 3
Write a system of linear inequalities that defines the shaded region.
Chapter Test
Chapter Standardized Test
Tip
Ca^c£!DC^Cj£>
Go back and check as much of your work as
you can.
1. Which point appears to be the solution of
the linear system graphed below?
(A) (-4, 0)
CD (-3,-1)
CD (-1, -3)
CD (0, -2)
2 . The ordered pair (3, 4) is a solution of
which linear system?
(A) x + y = 7
x + 2y = 11
CD x — y = 1
2x — y = 9
Cp x — y = 1
2x + y = 10
CD x + y = 7
2x — 2y — 14
6 - How many solutions does the following
linear system have?
4x — 2y = 6
2x - y = 3
(A) One CE) Two
CD Infinitely many Cp None
7. Which system of linear equations has
no solution?
(A) y = 2x + 4
y = 2
CD 3x + 4y = 10
3x + 2y = 8
CD 5x + 2y = 11 Cp
10x + 4y = 11
2x - 4y = -5
3x + 6y = 15
Cp None of these
3. What is the solution of the following
linear system?
— 2x + 7y = —3
x-ly = -2
(5) 1 CD 5
CD (1, 5) CD (5, 1)
8. Which point is a solution of the following
system of linear inequalities?
y < -x
y <x
(5) (6, -2) CD (-2, 6)
CD (-1, -6) CD (-6, -1)
4. What is the solution of the following
linear system?
5x — 6y = —10
— 15x+ 14y = 10
(£>(-5,-8) CD (-2,0)
CD (4, 5) CD (10, 10)
5. You have 50 ride tickets. You need 3 tickets
to ride the Ferris wheel and 5 tickets to ride
the roller coaster. You ride 12 times. How
many times did you ride the roller coaster?
(£> 5 CD 7
CD 10 CD 18
9. Which system of inequalities is
represented by the graph below?
(A) y < 2x + 1
2y < — 3x
Cp y > 2x + 1
2y > — 3x
Cp y < 2x + 1
2y < 3x
Cp y > 2x + 1
2y > 3x
Chapter 7 Systems of Linear Equations and Inequalities
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
Maintaining Skills
i Volume of a Solid
Find the volume of the figure shown.
Solution
Volume = Area of base X Height
= tit 2 X h
= tt(6) 2 5
= 1 80tt
Try These
Find the volume of the geometric figure shown.
2 Decimals and Percents
a, Write 30% as a decimal. b_ Write 0.705 as a percent.
Solution
a. 30% =
30
100
= 0.3
b. 0.705 = 0.705 X 100%
= 70.5%
Student HeCp
► Extra Examples
M° re examples
7^0 l "' and practice
exercises are available at
www.mcdougallittell.com
Try These
Write the percent as a decimal.
5. 47% 6. 4% 7. 3.5%
Write the decimal as a percent.
9.0.61 10.0.07 11.2
8 . 120 %
12 . 0.025
Maintaining Skills
Exponents and
Exponential Functions
our bike and
APPLICATION: Bicycle Racing
Shifting into a higher gear helps racers increase
speed but makes pedaling more difficult. When racers
use more energy, their air intake increases.
The relationship between air intake and bicycle speed
can be represented by a type of mathematical model
that you will study in Chapter 8.
Think & Discuss
1. Construct a scatter plot of the data below. Draw a
smooth curve through the points.
Bicycle speed, x
(miles per hour)
Air intake, y
(liters per minute)
0
6.4
5
10.7
10
18.1
15
30.5
20
51.4
2 . Describe the change in the air intake after each
increase of 5 miles per hour in bike speed. Does
air intake increase by the same amount? Does it
increase by the same percent?
Learn More About It
You will use an exponential model that relates air
intake and bicycle speed in Exercises 35 and 36
on page 480.
APPLICATION LINK More about bicycle racing is available
www.mcdougallittell.com
nipTtri
Study Guide
PREVIEW
What’s the chapter about?
Multiplying and dividing expressions with exponents
Using scientific notation to solve problems
Using exponential growth and decay models
Key Words
-
• exponential function,
• exponential growth,
• exponential decay,
p. 455
p . 476
p. 482
• scientific notation, p. 469
< _
• growth factor, p. 476
• decay factor, p. 482
_ >
PREPARE
STUDY TIP
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to p. 9)
1. Complete: In the expression 7 6 , 7 is the ? .
(A) base Cb) factor Cc) exponent
2 _ Complete: In the expression 7 6 , 6 is the ? .
(a) base Cb) factor Cc) exponent
Skill Check (refer to pp. 11,16,177)
3- Evaluate (3x) 2 when x = 2.
(A) 12
CD 18
(© 24
CD) power
CD power
CD 36
4. Evaluate — when x = 4 and y = 2.
(A) 6 CD 8 CD 21.5 CD 32
5. How much do you earn per hour if you earn $123.75 for working 15 hours?
(A) $8.25 CD $12.12 CD $108.75 CD $1856.25
Plan Your Time
A schedule or weekly
planner can be a useful tool
that allows you to coordinate
your study time with time for
other activities.
Chapter 8 Exponents and Exponential Functions
For use with
Lesson 8.1
DEVEL
Goal
Find a pattern for
multiplying exponential
expressions.
Materials
• paper
• pencil
Question
i i
How do you multiply powers with the same base?
Explore
0 One way to multiply powers with the same base is to write the product in
expanded form. Then count the number of factors and use this number as the
exponent of the product of the powers.
Number of
factors
7 3 • 7 2 = (7 • 7 • 7)(7 • 7) = (7)(7)(7)(7)(7) 5
-— -- ■—.—' v-^-'
3 factors 2 factors 5 factors
0 Notice that the exponent for the product of powers with the same base is the
sum of the exponents of the powers: 3 + 2 = 5. See if the same pattern
applies to the following products.
7 3 • 7 3 2 3 • 2 2 x 3 • x 4
© What can you conclude?
Product as
a power
7 5
Think About It
Find the product. Write your answer as a single power.
1 . 6 3 • 6 2 2. 2 • 2 4 3. a 4 * a 6 4. x 2 • x 7
5_ Complete: For any nonzero number a and any positive integers m and n ,
Question
How do you find the power of a power?
Student fteCp
-\
► Reading Algebra
When you read a
power of a power,
start with the power
within the parentheses.
For example, (7 3 ) 2 is
read "seven cubed,
squared."
V_ )
Explore
0 To find the power of a power, you can write the product in expanded form.
Then count the number of factors and use this number as the exponent of the
product of the powers.
2 times 3 factors 3 factors
(7 3 ) 2 = (^XT 3 ) = (7X7X7) (7X7X7)
6 factors
Number of
factors
6
Product as
a power
7 6
Developing Concepts
Developing Concepts: continued
© Notice that the exponent for the power of a power is the product of the
exponents: 2*3 = 6. See if the same pattern applies to the following powers
of powers.
(5 2 ) 3 (3 2 ) 2 (* 5 ) 3
© What can you conclude?
Think About It
Find the power of a power. Write your answer as a single power.
1.(4 2 ) 3 2. (5 4 ) 2 3.(J 3 ) 3 4. (n 3 ) 4
5. Complete: For any nonzero number a and any positive integers m and n,
0 a m ) n = ? .
Question
& ■ i»
How do you find the power of a product?
Explore
1 1
© One way to find the power of a product is to write the product in expanded
form and group like factors. Then count the number of each factor and write
the answer as a power of each factor.
Number of
Product as
each factor
a power
(5 • 4) 2 = (5 • 4)(5 • 4) = (5 • 5)(4 • 4)
2 and 2
5 2 • 4 2
2 times 2 factors 2 factors
© Notice that the exponent for a product of factors is distributed to each of the
factors: (5 • 4) 2 = 5 2 • 4 2 . See if the same pattern applies to the following
powers of products.
(3 • 2) 3 (3 • 6) 4 (a • b ) 5
© What can you conclude?
Think About It
Find the power of the product.
1. (2 • 6) 3 2. (3 • 4) 5 3 .(a-b) 2 4. (x • y) 4
5. Complete: For any nonzero numbers a and b and any positive integer m,
(a • b) m = ? .
Chapter 8 Exponents and Exponential Functions
Multiplication Properties of
Exponents
Goal
use multiplication How do the areas of two irrigation circles compare?
properties of exponents. _ M
Key Words
• power
• base
• exponent
What does it mean to say that one
circle is twice as big as another? Does
it mean that the radius r is twice as big
or that the area is twice as big? In
Example 5 you will see that these two
interpretations are not the same.
PRODUCT OF POWERS As you saw in Developing Concepts 8.1, page 441, to
multiply powers that have the same base, you add the exponents. This property is
called the product of powers property. Here is an example.
5 factors
f * \
a 2 • c? — a • a • a • a • a = a 2 + 3 = a 5
2 factors 3 factors
Student HeCp
► Look Back
For help with
exponential
expressions, see p. 9.
L j
03Z!mZ 219 1 Use the Product of Powers Property
Write the expression as a single power of the base,
a. 5 3 • 5 6 b. — 2(—2) 4 c. x 2 • x 3 ■
Solution
a. 5 3 • 5 6 = 5 3 + 6
= 5 9
b. — 2 (- 2) 4 = (— 2 )‘(— 2) 4
= (— 2) 1 + 4
= (- 2) 5
Use product of powers property.
Add the exponents.
Rewrite -2 as (-2) 1 .
Use product of powers property.
Add the exponents.
Use product of powers property.
Add the exponents.
Use the Product of Powers Property
Write the expression as a single power of the base.
1 .4 2 • 4 3 2 . (—3) 2 (—3) 3. a-a 1 4 . n 5 • n 2 • n 3
8.1 Multiplication Properties of Exponents
POWER OF A POWER To find a power of a power, you multiply the exponents.
This property is called the power of a power property. Here is an example.
(a 2 y = a 2 -a 2 -a 2 = a 2 + 2 + 2 = a 6
Student Hedp
-\
► Look Back
For help with
exponents and
grouping symbols,
see p. 11.
v__ J
Use the Power of a Power Property
Write the expression as a single power of the base.
a. (3 3 ) 2
b. (/) 4
Solution
a. (3 3 ) 2 = 3 3 ’ 2
Use power of a power property.
= 3 6
Multiply exponents.
b .(P 4 ) 4 =P 4 ' 4
Use power of a power property.
= p 16
Multiply exponents.
Use the Power of a Power Property
Write the expression as a single power of the base.
5.(4 4 ) 3 6. [(—3) 5 ] 2 7. (n 4 ) 5 8. (x 3 ) 3
POWER OF A PRODUCT To find a power of a product, find the power of each
factor and multiply. This property is called the power of a product property. Here
is an example.
(<a • bf — (a • b)(a • b)(a • b) = (a • a • a)(b • b • b) = a 3 b 3
Student tfeCp
^
► Study Tip
Notice that (—6) 2 • 5 2
is equivalent to:
(-6 • 5) 2 = (-30) 2 .
= 900
l J
J 3 Use the Power of a Product Property
Simplify the expression.
a. (-6 • 5) 2
Solution
^ a. (-6 • 5) 2 = (—6) 2 • 5
= 36-25
= 900
b. (4 yz) 3 = 4 3 • v’ 3 • z 3
= 64v 3 z 3
b. (4jz) 3
Use power of a product property.
Evaluate each power.
Multiply.
Use power of a product property.
Evaluate power.
Use the Power of a Product Property
Simplify the expression.
9. (2 • 4) 3 10. (-3- 5) 2 11. (2w) 6
12 . (7a) 2
Chapter 8 Exponents and Exponential Functions
4 Use All Three Properties
Simplify the expression (4x 2 ) 3 • x 5 .
Solution
(■4x 2 ) 3 • x 5 = 4 3 • (x 2 ) 3 • x 5
= 64 • x 6 • x 5
= 64X 11
Use power of a product property.
Use power of a power property.
Use product of powers property.
Use All Three Properties
Simplify the expression.
13. (4x 3 ) 4 14. (-3a 4 ) 2
15. 9 • (9z 5 ) 2 16. (n 2 ) 3 • n 1
Student HeCp
>
► Study Tip
In the formula for
the area of a circle,
A = ttt 2 , r is the
radius of the circle,
and 7T is a constant,
approximately 3.14.
5 Use Multiplication Properties of Exponents
FARMING Find the ratio
of the area of the larger
irrigation circle to the
area of the smaller
irrigation circle.
Solution
The area of a circle can be found using the formula A = irr 2 .
7r(2r) 2 • 2 2 • r 2 77 • 4 • r 2 4
Ratio =-— =-~— =-«— = —
Trr 77 • r 77 • r f
ANSWER ► The ratio of the areas is 4 to 1.
Use Multiplication Properties of Exponents
17. Find the ratio of the area of the smaller irrigation circle in Example 5 to the
area of an irrigation circle with radius 3r.
- - N
Multiplication Properties of Exponents
Let a and b be real numbers and let m and n be positive integers.
PRODUCT OF POWERS PROPERTY POWER OF A POWER PROPERTY POWER OF A PRODUCT PROPERTY
To multiply powers that have To find a power of a power. To find a power of a product,
the same base, add the multiply the exponents. find the power of each factor
exponents. and multiply.
a m . a n = a m + n (a m ) n = 3 m * n (a • b) m = 3 m • b m
8.1 Multiplication Properties of Exponents
Exercises
Guided Practice
Vocabulary Check
Match the multiplication property of exponents with the example that
illustrates it.
1. Product of powers property
2. Power of a power property
3. Power of a product property
A. (3 • 6) 2 = 3 2 • 6 2
B. 4 3 • 4 5 = 4 3 + 5
C. (2 4 ) 4 = 2 4 ‘ 4
Skill Check Use the product of powers property to write the expression as a single
power of the base.
4. 2 2 • 2 3 5. (—5) 4 • (—5) 2 6. a 4 • a 6
Use the power of a power property to write the expression as a single
power of the base.
7. (2 4 ) 3 8. (4 3 ) 3 9. (y 4 ) 5
Use the power of a product property to simplify the expression.
10. (3 *4 ) 3 11.(2 n) 4 12.(3 pq) 3
Practice and Applications
COMPLETING EQUATIONS Copy and complete the statement.
13. 3 2 • 3 7 = 3 7 14. 5 ? * 5 8 = 5 9 15 . 410.48 = 4 ?
16. x 3 • x 2 = x 7 17. r 7 • r 1 = r 14 18. a 2 • a 7 = a 5
PRODUCT OF POWERS Write the expression as a single power of
the base.
19. 4 3 • 4 6 20. 8 9 • 8 5 21. (-2 ) 3 • (-2 ) 3
22. b • b 4 23. x 6 • x 3 24. t 3 • t 2
Student HeCp
► Homework Help
Example 1: Exs. 13-24
Example 2: Exs. 25-36
Example 3: Exs. 37-51
Example 4: Exs. 52-60
Example 5: Exs. 61-68
COMPLETING EQUATIONS Copy and complete the statement.
25. (5 ? ) 3 = 5 9 26. (2 2 ) ? = 2 8 27. [(— 9 ) 4 ] 3 = (-9) ?
28. (a 2 ) 7 =
JO
29. (x 3 ) 3 =
?
x •
30. {p 2 ) 6 = p
12
POWER OF A POWER Write the expression as a single power of
the base.
31. (2 3 ) 2 32. (7 4 ) 2
34. (? 5 ) 6
35. (c 8 )
10
33. [(—4) 5 ] 3
36. (x 3 ) 2
Chapter 8 Exponents and Exponential Functions
POWER OF A PRODUCT
37. (3 • 7) 2
40. (5x) 3
43. (2mn) 6
Simplify the expression.
38. (4 • 9) 3 39. (-4 • 6) 2
41. (-2 df 42. (ab) 2
44. (1 Oxv) 2 45. (— rst) 5
Student HeCp
► Homework Help
Extra help with
“^5"^ problem solving in
Exs. 46-51 is available at
www.mcdougallittell.com
WRITING INEQUALITIES
46. (5 • 6) 4 ? 5 • 6 4
49. 4 2 • 4 8 ? (4 • 4) 10
Copy and complete the statement using < or >.
47. 5 2 • 5 3 ? (5 • 5) 6 48. (3 • 2) 6 ? (3 2 ) 6
50. 7 3 - 7 4 ? (7 • 7) 4 51. (6 • 3) 3 ? 6-3-3
SIMPLIFYING EXPRESSIONS Simplify the expression.
52. (3b) 3 • b
55. (rV ) 4
58. 4x • (—x • x 3 ) 2
53. — 4x • (x 3 ) 2
56. (6z 4 ) 2 • 7 ?
59. ( abc 2 ) 3 • ab
54. (5a 4 ) 2
57. 2x 3 • (—3x) 2
60. (5 y 2 ) 3 - (y 3 ) 2
Link
Careers
ALTERNATIVE ENERGY
TECHNICIANS solve
technical problems in the
development maintenance,
and inspection of machinery,
such as windmills.
More about alternative
energy technicians is
at www.mcdougallittell.com
61 . Ge ometry M The volume V of a sphere
4 a
is given by the formula V = —tit J , where r is
the radius. What is the volume of the sphere in
terms of al
62. Geometry Link / The volume V of a circular
cone is given by the formula V = where
r is the radius of the base and h is the height.
What is the volume of the cone in terms of bl
ALTERNATIVE ENERGY The power generated by a windmill can be
modeled by w = 0.015s 3 , where w is the power measured in watts and
s is the wind speed in miles per hour.
63. Find the ratio of the power generated when the wind speed is 20 miles per
hour to the power generated when the wind speed is 10 miles per hour.
64. Find the ratio of the power generated when the wind speed is 5 miles per
hour to the power generated when the wind speed is 10 miles per hour.
PENNIES Someone offers to double the amount of money you have
every day for 1 month (30 days). You have 1 penny.
65. At the end of the first day, you will have 2*1=2 pennies. On the second
day, you will have 2*2 = 4 pennies. On the third day, you will have
2*4 = 8 pennies. Write each of these equations using only powers of 2.
66 . Using the pattern you find in Exercise 65, write an expression for the number
of pennies you will have on the nth day.
67. How many pennies will you have on the 30th day?
68 . How much money (in dollars) will you have after 30 days?
8.1 Multiplication Properties of Exponents
Standardized Test
Practice
69. CHALLENGE Fill in the
blanks and give a reason for
each step to complete a
convincing argument that the
power of a power property is
true for this case.
(b 3 ) 2 = b 2 -_l_
= ?
70- 1W1ULTIPLE CHOICE Simplify the expression 5 2 • 5 4 .
(a) 5 6 CD 5 8 CD 10 6 CD 25 8
71. IVIULTIPLE CHOICE Evaluate the expression (2 3 ) 2 .
CD 18 CD 32 CD 36 CD 64
72. IVIULTIPLE CHOICE Evaluate the expression (4 • 6 ) 2 .
(a) 48 CD 96 CD 144 CD 576
73. IVIULTIPLE CHOICE Simplify the expression (3x 2 y) 3 .
CD 3x 2 y 3 CD 9x 5 y 3 CD 9x 6 y 3 CD 21x 6 y 3
VARIABLE EXPRESSIONS Evaluate the expression for the given value of
the variable. (Lesson 1.3)
74. b 2 when b = 8
77. when y = 5
/
GRAPHING EQUATIONS Use a table of values to graph the equation.
(Lessons 4.2, 4.3)
80. y = x + 2 81. y = — (x — 4) 82. y = — 5
83. y = |x + 2 84. _y = 2 85. x = -3
SOLVING INEQUALITIES Solve the inequality. (Lesson 6.3)
86. —jc — 2 < —5 87. 3 -x>-4 88. 7 + 3x>-2
89. 6 x - 10 < -4 90. 2 < 2x + 7 91. 9 - 4x< 2
75. (5y ) 4 when y — 2
24
78. — when x = 2
1 q
76. —when n — —2
_ 45 t
79. ~r when a = 2
Maintaining Skills LCM AND GCF Decide whether the statement is true or false. If it is
false, correct the statement to make it true. (Skills Review p. 761)
92. The least common multiple of 6 and 10 is 60.
93. The greatest common factor of 6 and 10 is 2.
94. The least common multiple of 10 and 30 is 30.
95. The greatest common factor of 10 and 30 is 5.
96. The least common multiple of 45 and 82 is 105.
97. The greatest common factor of 45 and 82 is 3.
Chapter 8 Exponents and Exponential Functions
Zero and Negative
Exponents
Goal
Evaluate powers that
have zero or negative
exponents.
Key Words
• zero exponent
• negative exponent
• reciprocal
What was the population of the U.S. in 17761
Many real-life quantities can be
modeled by functions that contain
exponents. In Exercise 64 you will
use such a model to estimate the
population of the United States
in 1776.
The definition of a° is determined by the product of powers property:
a°a n = a 0 + n = a n
In order to have a°a n = a n , a° must equal 1.
The definition of a~ n is similarly determined:
a n a~ n = a n ~ n = a° = 1
In order to have a n a~ n = 1, a~ n must be the reciprocal of a n .
Student HeGp
->
►Writing Algebra
The definition of a
negative exponent can
also be written as:
v __ j
ZERO AND NEGATIVE EXPONENTS
Let a be a nonzero number and let n be an integer.
• A nonzero number to the zero power is 1:
a 0 = 1, a ¥= 0
• a~ n is the reciprocal of a n :
a ~ n = i*’ a * 0
1 Powers with Zero Exponents
Evaluate the expression.
a. 5° = 1
b. (Undefined)
c. (- 2 )° = 1
0
d -1 = 1
o° is equal to 1.
o° is defined only for a nonzero number o.
o° is equal to 1.
o° is equal to 1.
8.2 Zero and Negative Exponents
■TO 2 Powers with Negative Exponents
Evaluate the expression.
a. 2 -2 = |p 2 -2 is the reciprocal of 2 2 .
= Evaluate power.
b .
1
(- 3)“ 4
= (- 3) 4
= 81
(-3) 4 is the reciprocal of (-3) 4 .
Evaluate power.
Powers with Zero or Negative Exponents
Evaluate the expression.
1 .
2 - (- 9 )
-2
3 .
>-3
4 .
(- 5 )"
Student Hadp
p More Examples
More examples
are available at
www.mcdougallittell.com
3 Evaluate Exponential Expressions
Evaluate the expression.
3 . 6 4 • 6 4
Solution
a. 6 “ 4 . 6 4 = 6“ 4 + 4
= 6 °
= 1
b . ( 2- 3 )- 2 = 2 - 3 *(- 2 >
= 2 6
= 64
b . ( 2- 3 )" 2
c. (-3 • 2 )
-2
c. (-3 • 2)“ 2 =
1
(-3 • 2) 2
1
( — 3) 2 • 2 2
1
9 • 4
J_
36
Use product of powers property.
Add exponents.
o° is equal to 1.
Use power of a power property.
Multiply exponents.
Evaluate power.
Use definition of negative exponent.
Use power of a product property.
Evaluate powers.
Multiply.
Evaluate Exponential Expressions
Evaluate the expression without using a calculator.
5 . 4 2 • 4“ 3 6 . ( 3 -1 ) -2 7 . (2 • 5)“ 2
Chapter 8 Exponents and Exponential Functions
Student HeCp
► Keystroke Help
Use £g| or to
input the exponent.
Evaluate Expressions with a Calculator
. Use a calculator to evaluate (2 -2 ) 4 .
Solution You can simplify the expression first.
( 2 - 2)4 = 2 -8
KEYSTROKES
21AHIB
ANSWER l (2 -2 ) 4 ~ 0.0039
Use power of a power property.
DISPLAY
0.00390625
Evaluate Expressions with a Calculator
Use a calculator to evaluate the expression.
8 . 7~ 3 a a -2 . a -1
9 . 6 “
10 . (3 3 )
-2
5 Simplify Exponential Expressions
Rewrite the expression with positive exponents.
a. 2x 2 y 3
r -2
b. —v c. (5 a)~ 2
a 3
Solution
a. 2x“ 2 y“ 3 = 2 • 4r*
Use definition of negative exponents.
2
x 2 y 3
Multiply.
. C -2 _2 1
b. -7 = C • -7
<T 3 <T 3
Multiply by reciprocal.
= -y • d 3
c
Use definition of negative exponents.
II
Multiply.
c. (5a) 2 = ,
(5a ) 2
Use definition of negative exponents.
1
5 2 • a 2
Use power of a product property.
1
25a 2
Evaluate power.
Simplify Exponential Expressions
Rewrite the expression with positive exponents.
11 . 2x _ 3 y 3 12 . -37 13 . (5 &) -3
8.2 Zero and Negative Exponents
BP — MP ^7* 1B£^ I £
Exercises
Guided Practice
Vocabulary Check Tell whether the statement is true or false. Explain your answer.
1 . A nonzero number to the zero power is zero.
2 . Let a be a nonzero number and let n be an integer. Then a~ n =
Skill Check Evaluate the expression.
3 - 6 ° 4 ' 3 "
Evaluate the expression without using a calculator.
7 . 2 -4 • 2 5 8 . (3 4 ) -1 9 . (4 • l) -2 10 . (9 -1 ) 2
B Use a calculator to evaluate the expression. Round your answer to
the nearest ten-thousandth.
11 . 5" 4 12 . 7" 1 • 7“ 3 13 . (8 2 )" 1 14 . (3 • 4)" 3
Rewrite the expression with positive exponents.
15 . m ~ 2 16 . a 5 b~ 8 17 . A 18 . (2x)“ 3
c D
Practice and Applications
RECIPROCALS Copy and complete the table.
ZERO AND NEGATIVE EXPONENTS Evaluate the expression.
21 . 3 ° 22 . (-5)° 23 . 4“ 2 24 . 9“ 1
Student HeCp
► Homework Help
Examples 1 and 2:
Exs. 19-28
I Example 3: Exs. 29-40
Example 4: Exs. 41-48
Example 5: Exs. 49-62
. _J
25 . (-7)" 3 26 .
EVALUATING EXPRESSIONS
a calculator.
29 . 2“ 3 • 2° 30 . 10" 5 •
33 . (4- 1 )- 3 34 . (5“ 2 ) 2
37 . (10 • 2)“ 2 38 . (1 • 7)
28 .
_ 1 _
(-§r 2
Evaluate the expression without using
10 7
-3
31 . 6 2 • 6~ 4
35 . (3 2 ) -1
39 . (-2 • 2)“ 2
32 . 4" 1 • 4" 1
36 . [(— 8) -2 ] -1
40 . [4 • (—3)] _1
Chapter 8 Exponents and Exponential Functions
Link to
History
STATEHOOD After 1790,
when the last of the original
13 colonies became a state,
a population of at least
60,000 people was required
for statehood.
0 EVALUATING EXPRESSIONS Use a calculator to evaluate the
expression. Round your answer to the nearest ten-thousandth.
41 . 2~ 5 42 . 11~ 2 43 . 5" 1 • 5“ 3 44 . 9“ 4 • 9 2
45 . (4 2 ) -1
46 . (3“ 3 ) 2
47 . (2 • 7)
-1
48 . (8 • 3)
-2
ERROR ANALYSIS In Exercises 49 and 50, find and correct the error.
SIMPLIFYING EXPRESSIONS Rewrite the expression with positive
exponents.
1 ,,— 6
51 . x 5
52 . 3 x “ 4
53 . x " V
54 .
55 . —^
V z
56 .
X 3
/
5 V.„
58 .
59 . ( 4 x )“ 3
60 . (3xy)~ 2
61 . ( 6 x “ 3 ) 3
62 .
9x
-3
(4*r
Using Zero and Negative Exponents
STATEHOOD The population P (in millions) of the United States from
the late 1700s to the mid-1800s can be modeled by P = 5.31(1.03)^, where
y represents the number of years since 1800. Estimate the population of the
United States in 1790 when the first census was taken.
Solution Since 1790 is 10 years before 1800, you want to know the value
of P wheny = —10.
P = 5.31(1.03)* y Write model.
= 5.31(1.03) — 10 Substitute -10 for y.
~ 3.95 Use a calculator to evaluate.
ANSWER^ The population in 1790 was about 3.95 million people.
63 . Estimate the population of the United States in 1800.
64 . Estimate the population of the United States in 1776.
Puzzler Refer to the squares shown.
65 . What fraction of each figure is shaded?
66 . Rewrite each fraction from Exercise 65
in the form 2 X .
67 . Look for a pattern in your answers to
Exercise 66. If this pattern continues,
what fraction of Figure 10 will be shaded?
Figure 3
Figure 2
Figure 4
8.2 Zero and Negative Exponents
Standardized Test
Practice
Mixed Review
Maintaining Skills
68. MULTIPLE CHOICE Which expression equals ' ?
O
(A) -8 CD 4“ 2 CD 2~ 3 CD l" 8
69 . MULTIPLE CHOICE Evaluate the expression (4 -1 ) -2 .
CD ^ CD CD 16 CD 64
70 . MULTIPLE CHOICE Evaluate the expression 3 • 3 -5 .
CD ^j- CD | CD 45 CD 81
3x~ 2
71 . IVIULTIPLE CHOICE Rewrite the expression - — with positive exponents.
yz
7 37 3 y 2 z 0 q
CD 7Y7 CD CD -p- CD 3 x 2 y 3 z
3x^ xy J x z
EVALUATING EXPRESSIONS Evaluate the expression. Then simplify the
answer. (Lesson 1.3)
72 .
75 .
6 ♦ 5
1+7*2
9 + 3 3 - 4
73 .
76 .
8 • 8
10 + 3 • 2
(5 ~ 3) 2
2 • (6 - 2 )
74 .
77 .
2 • 4 2
1 + 3 2 - 2
2 ♦ 3 4
20 - 4 2 + 8
SOLVING EQUATIONS Solve the equation. (Lesson 3.1)
78 . x + 1 = 6 79 . -2 = 7 + x 80 . 15 = x - (-4)
81 . 10 = x — 5 82 . -3 + x =-8 83 . x - (-6) =-9
SOLVING INEQUALITIES Solve the inequality. Then graph and check the
solution. (Lesson 6.7)
84. U - 3 | >4
85. x + 9 <4
86. 13x + 2 | >10
87. 15 + 2x | < 7
88. |x + 2 | + 6< 15
89. 13x + 7 | - 5 > 8
SOLVING SYSTEMS
Use substitution to solve the system. (Lesson 7.2)
90. 2x — y — —2
91. —3x + y = 4
92. x + 4y = 30
4x + y = 5
—9x + 5+ = 10
o
II
1
93. 2x — 3y = 10
94. x + 15j = 6
95. 4x - y = 5
x + j = 5
x 5y = 84
2x + 4 y = 16
EQUIVALENT FRACTIONS Write three equivalent fractions for the given
fraction. (Skills Review p. 764)
98-1
97 1
3/ - 5
“I
99 .
100 . f
15
101 16
ioz 4
103 .
Chapter 8 Exponents and Exponential Functions
Graphs of Exponential
Functions
Goal
Graph an exponential How many shipwrecks occurred from 1680 to 1980?
f 11 nrtinn # ■
Key Words
• exponential function
Many real-life relationships can be
modeled by exponential functions. In
Example 5 the number of shipwrecks that
occurred in the northern part of the Gulf
of Mexico from 1680 to 1980 is modeled
by an exponential function.
In Lesson 8.2 the definition of b n was extended to allow for zero and negative
integer values of n. This lesson makes use of the expression b x , where b > 0 and
v is allowed to be any real number.
A function of the form y = a • b x or simply y = ab x , where b > 0 and b ^ 1, is
an exponential function.
■a&maa J i| Evaluate an Exponential Function
Make a table of values for the exponential function y = 2 X . Use x- values of
—2, —1, 0, 1, 2, and 3.
Solution To evaluate an exponential function, use the definitions you
learned in Lesson 8.2. Lor example, when x = —2 you find y as follows:
X
-2
-1
0
1
2
3
X
C\l
II
1
4
1
2
1
2
4
8
Evaluate an Exponential Function
1 . Make a table of values for the exponential function y
—2, —1, 0, 1, 2, and 3.
2 . Make a table of values for the exponential function y
of —2, — 1, 0, 1, 2, and 3.
3 X . Use x-values of
(i\ x
21 — I .Usex-values
8.3 Graphs of Exponential Functions
Student HeCp
--n
► Study Tip
In later courses you
will learn to give 2* a
precise mathematical
definition for any real
value x. In this course
you will use a smooth
curve to represent
these values.
^ _ >
®22322SB 2 Graph an Exponential Function when b > 1
a. Use the table of values in Example 1 to graph the function y = 2 X .
b. j|=r : Use a calculator to evaluate y = 2 X when x = 1.5.
Solution Begin by writing the six points given by the table on page 455:
4
(0, 1), (1, 2), (2, 4), (3,8)
a. Draw a coordinate plane and plot the
six points listed above. Then draw a
smooth curve through the points.
Notice that the graph has a y-intercept
of 1, and that it gets closer to the
negative side of the x-axis as the
x-values decrease.
b. KEYSTROKES DISPLAY
2 |5 11 _) 5 I2.828M271251
ANSWER ^ 2 1,5 ~ 2.83
B22E2EB 3 Graph an Exponential Function when 0 < b < I
Graph the function y = 3
Solution Make a table of values that includes both positive and negative
x-values. Be sure to follow the order of operations when evaluating the
function. For example, when x = — 2 you find y as follows:
j = 2 = 3(2) 2 = 3(4) = 12
X
-2
-1
0
1
2
3
'-S*
12
6
3
3
2
3
4
3
8
Draw a coordinate plane and plot the
six points given by the table. Then draw
a smooth curve through the points.
Notice that the graph has a y-intercept
of 3, and that it gets closer to the
positive side of the x-axis as the
x-values increase.
Chapter 8 Exponents and Exponential Functions
Student HeCp
► More Examples
More exam Pl es
are available at
www.mcdougallittell.com
J 4 Find Domain and Range
a. Describe the domain and range of the function y = 2 X , which is graphed in
Example 2.
b_ Describe the domain and range of the function y
graphed in Example 3.
*
, which is
Solution
a. You can see from the graph of the function that y = 2 X is defined for
all x-values, but only has y -values that are greater than 0. So the
domain of y = 2 X is all real numbers and the range is all positive
real numbers.
(i\ x
b. You can see from the graph of the function that y = 31 — I is defined for
all x-values, but only has y -values that are greater than 0. So the domain
(iY
of y = 31 2 J is all real numbers and the range is all positive real numbers.
Graph an Exponential Function and Find its Domain and Range
3. Graph the function y = 3 X . Then describe its domain and range.
(\ Y
4. Graph the function y = 21 — I . Then describe its domain and range.
Link to
History
SHIPWRECKS In 1685
La Salle claimed part of what
would become the United
States for France. In 1686 his
ship the Belle sank near
Texas. This shipwreck wasn't
discovered until 1995.
More about
shipwrecks at
www.mcdougallittell.com
5 Use an Exponential Model
SHIPWRECKS From 1680 to 1980 the number of shipwrecks per 10-year
period t that occurred in the northern part of the Gulf of Mexico can be
modeled by S = 180(1.2)*, where 5 is the number of shipwrecks and t = 0
represents the 10-year period from 1900 to 1909. Graph the function.
Solution Make a table of values that includes positive and negative ^-values.
t
-4
-2
0
2
4
6
S= 180(1.2)'
87
125
180
259
373
537
Draw a coordinate plane and plot the
six points given by the table. Then
draw a smooth curve through the points.
Notice that the graph has a 5-intercept of
180, and that it gets closer to the negative
side of the £-axis as the t- values decrease.
8.3 Graphs of Exponential Functions
Exercises
Guided Practice
Vocabulary Check 1 _ Define exponential function.
Skill Check 2 . Copy and complete the table of values for the exponential function.
X
-2
-1
0
1
2
3
y — 4 X
?
?
?
?
?
?
3- Graph y = 4 X . Use the points found in Exercise 2.
4. Graph the function y = 3
Using the graph shown, describe the domain and range of the function.
Practice and Applications
CHECKING POINTS Tell whether the graph of the function contains the
point (0, 1). Explain your answer.
7 . y = 2 X 8.y = 5 x
9. y = 2(3 y
13. y = 7
10 . y = 5(7) x
i 4 -y = 4 (f) X
f Student HeCp
p Homework Help
Example 1: Exs. 7-22
Example 2: Exs. 23-41
Example 3: Exs. 31-41
Example 4: Exs. 42-49
Example 5: Exs. 50, 51
>_/
MAKING TABLES Make a table of values for the exponential function.
Use x-values of -2, -1, 0, 1, 2, and 3.
15. y = 3
19. y =
16. y = 8
20 . y =
17. y = 5(4)
21.y
21 i
18.? = 3(5)
22 . y
5 ' 1
B EVALUATING FUNCTIONS Use a calculator to evaluate the
exponential function when x = 2.5. Round your answer to the
nearest hundredth.
23. y = 5* 24. y = 9 X 25. y = 8(2) x 26. y = 3(4) x
27. y =
28. y =
29. y
Chapter 8 Exponents and Exponential Functions
Link
Computers
WORLD WIDE WEB
The phrase "World Wide
Web" was introduced by
Tim Berners-Lee. It's the
name he gave to the very
first web browser, which
he created.
EXPONENTIAL FUNCTIONS Match the equation with its graph.
31.y = 3* 32, y = 2 X 33.y = 9 x
GRAPHING FUNCTIONS
34. y = 4 X
35. y
39 . y
Graph the exponential function.
= —l x 36. y = 4(2)* 37. y = -3(8)*
DOMAIN AND RANGE Using your graphs from Exercises 34-41, describe
the domain and the range of the function.
42. y = 4* 43. y = -7* 44. y = 4(2)* 45. y = -3(8)*
49. y
50. SALARY INCREASE The company you work for has been giving a
5% increase in salary every year. Your salary S can be modeled by
S = 38,000(1.05)* where t = 0 represents the year 2000. Make a table
showing your salary in 1995, 2000, 2005, and 2010. Then graph the points
given by this table and draw a smooth curve through these points.
Standardized Test
Practice
51. WORLD WIDE WEB The number of users U (in millions) of the World Wide
Web can be modeled by U = 135(1.5)* where t = 0 represents the year 2000.
Make a table showing the number of users (in millions) in 1995, 2000, 2005,
and 2010. Then graph the points given by this table and draw a smooth curve
through these points. ►Source: WinOpportunity
52. CHALLENGE If a 0 = 1 (a 0), what point do all graphs of the form y = a x
have in common? Is there a point that all graphs of the form y = 2 (a) x have
in common? If so, name the point.
53. MULTIPLE CHOICE What is the
equation of the graph?
(5) y = 2* CD y = 2(2)*
® > = $ ® = 2 (i)'
54. MULTIPLE CHOICE What is the
equation of the graph?
CD y = 2 x (g) y = 2(2 y
® > = ® > = 2 (i)'
8.3
Graphs of Exponential Functions
Mixed Review
Maintaining Skills
Quiz 7
SOLVING AND CHECKING Solve the equation. Round the result to the
nearest hundredth. Check the rounded solution. (Lesson 3.6)
55. 8x + 9 = 12 56. 3y — 5 = 11 57. 13f +8 = 2
58. 14 — 6r = -17 59. Ilk + 12 = -9 60. -7x - 7 = -6
STANDARD FORM Write the equation in standard form with integer
coefficients. (Lesson 5.4)
61. y = — 8x + 4
„ 2
64. y = — —x
70. x + y = 0
x + 2y = 6
62. y = 5x — 2
- ii.-' • r«
71. 4x-y=-2
— 12x + 3y = 6
63. >’ =
„ 1 9
66 ^ = To x “To
to tell how many
69. 6x + 2y = 3
3x + y = —2
72. — x + 3y = 3
2x — y — — 8
GRAPHING SYSTEMS Use the graphing method
solutions the system has. (Lesson 7.5)
67. 2x — 2y = 4 68. — x + y = — 1
x + 3y = 9 2x + 3y = 12
ORDERING NUMBERS Write the numbers in increasing order.
(Skills Review pp. 770 , 7771
3 5 4 3 4 1
73. -4, -5, 6 74. y, | 75. ~2j, -3j, -2j
76. -6.57, -6.9, -6.56 77. 3.001, 3.25, 3.01 78. 7.99, 7.09, 7.9
Evaluate the expression. (Lessons 8.1 , 8.21
1. 3 4 • 3 6
2. (2 3 ) 2
3. (8 • 5) 2
4. 6“ 7 • 6 9
5.(5 2 )-‘
6. (4 • 9)°
Simplify the expression. Use only positive exponents.
(Lessons 8.1, 8.2)
7. r 5 • r ^
8. {k 4 ) 2
9. (3d) 2
10. 2x~ 3 y~ 9
11 1
5a -10 fe -12
12. ( mri )- 1
13. SAVINGS ACCOUNT
You started a savings account in 1994. The balance A
is given by A = 1600(1.08)* where 7 = 0 represents the year 2000. What is
the balance in the account in 1994? in 2004? (Lesson 8.2)
Graph the exponential function. (Lesson 8.3)
14. y = 10 x 15. y = 3(2) x
16 - y = 41 3
Chapter 8 Exponents and Exponential Functions
USING A GRAPHING CALCULATOR
j^cpvnanhul sunshvnz
For use with
Lesson 8.3
You can use a graphing calculator to graph exponential functions.
£ampl*
Student HeCp
► Keystroke Help
See keystrokes for
several models of
calculators at
www.mcdougallittell.com
Solution
Q To enter the function in your
graphing calculator, press .
Enter the function as
X. T, 0
Q Adjust the viewing window to get
the best scale for your graph.
e
Now you are ready to graph the
function. Press i|3332SLI t0 see
the graph.
TVyTlns*
Use a graphing calculator to graph the exponential function.
1. y = 2 X 2. y = 10 x 3.y=-3 x
LOGICAL REASONING Use your results from Exercises 1-6 to answer the
following questions.
7. If a > 1, what does the graph of y — a x look like? the graph of — a x l
8 . IfO < a < 1, what does the graph of y = a x look like? the graph of — a x 7
Using a Graphing Calculator
Division Properties of
Exponents
Goal
Use division properties
of exponents.
How much does a baseball player earn?
Key Words
• power
• base
• exponent
• quotient
One way to compare numerical
values is to look at their ratio. In
Exercise 59 you will use division
properties of exponents to compare
the average salary of a baseball
player in 1985 to the average salary
of a baseball player in 1990.
QUOTIENT OF POWERS To divide powers that have the same base, you subtract
the exponents. This is called the quotient of powers property. Here is an example.
45
43
5 factors
4 . 4 . 4 . 4 . 4
4.4.4
4.4 = 4 5 - 3 = 4 2
3 factors
2 factors
J 1 Use the Quotient of Powers Property
Simplify the quotient.
Student tfeCp
■ ^ -—V
► Study Tip
In Example 1(b) note
that the same answer
would have been
reached by cancelling
common factors:
y 3 j-j-s
y 5 X'Y'Y' V V .
X
■ y 1
K _ J
6 5 ,5 — 4
a -^ = 6
Use quotient of powers property.
Subtract exponents.
Evaluate power.
Use quotient of powers property.
Subtract exponents.
Use definition of negative exponent.
Using the Quotient of Powers Property
Simplify the quotient.
1 .
8 ^
8 6
(~3) 3
(— 3) 2
Chapter 8 Exponents and Exponential Functions
a
POWER OF A QUOTIENT Recall that • y
b b
b 1 '
To find a power of a quotient,
first find the power of the numerator and the power of the denominator, and then
divide. This is called the power of a quotient property. Here is an example.
(lY = 2 2 2 2 2 » 2 » 2 • 2 2 4
V 3 / 3 * 3 * 3 * 3 3 . 3 . 3 . 3 3 4
mi2 Use the Power of a Quotient Property
Simplify the quotient.
Use power of a quotient property.
4
9
b.
Student HeGp
,
► Study Tip
One step in simplifying
a quotient is to make
sure only positive
exponents are used. .
V _V
►
-27
7~ 3
4 -3
4f
7 3
64
343
Evaluate powers.
Use power of a quotient property.
Evaluate power.
Use power of a quotient property.
Use definition of negative exponents.
Evaluate powers.
Use the Power of a Quotient Property
Simplify the quotient.
•■if
6 . 1 ^
8 .
-5
Division Properties of Exponents
Let a and b be real numbers and let m and n be integers.
QUOTIENT OF POWERS PROPERTY
To divide powers that have the
same base, subtract the
exponents.
= a m n , a ± 0
POWER OF A QUOTIENT PROPERTY
To find a power of a quotient,
find the power of the
numerator and the power of
the denominator and divide.
a \m _ af_
4 b m ' b *°
8.4 Division Properties of Exponents
Student HeCp
p More Examples
M°r e examples
are available at
www.mcdougallittell.com
3 Simplify Expressions using Multiple Properties
Simplify the expression. Use only positive exponents.
2 x 2 y 9 xy 1
a -^ 7"
Solution
2 x 2 y 9 xy 2
a.
3x
y
18x 3 y 3
3 xy 4
6 x 2 y~ l
6x 2 _
y
(2xf_
(;y 2 ) 4
2 4 *x 4
2 • 4
r
I6x 4
Z /
2x
y 2
Use product of powers property.
Use quotient of powers property.
Use definition of negative exponents.
Use power of a quotient property.
Use power of a product property.
Use power of a power property.
Evaluate power.
Multiply exponents.
rjirr
4 Simplify Expressions with Negative Exponents
X (x 2 \
Simplify the expression —— • j
-3
. Use only positive exponents.
Solution
Mt)
“ 3 _ x . (x 2 )- 3
y 1 * y 3
Use power of a quotient property.
— x • y •
(x 2 ) 3
Use definition of negative exponents.
V
Use product of powers property.
X 6
Use power of a power property.
= x-y
A
Use quotient of powers property.
II
Use definition of negative exponents.
Simplify Expressions
Simplify the expression. Use only positive exponents.
3xy 4
y
/ 5 x\ 3
„ >~ 2 (A
X 3 '
• —
3
xy
(7)
Chapter 8 Exponents and Exponential Functions
Exercises
Guided Practice
Vocabulary Check
Match the division property of exponents with the example that
illustrates it.
1 . Quotient of powers property
2. Power of a quotient property
A.
3^
6 2
B . 7 = 4’-5
Skill Check Use the quotient of powers property to simplify the expression.
3 —
3 ‘ 5 1
J2
7.
4 —
■ 7 9
8 .
5.
(~ 2) 8
(— 2) 3
9.
m
li
6 .
10 .
5 3 • 5 5
7
X 7 • X
Use the power of a quotient property to simplify the expression.
11 .
15.
2
-5
m
12 .
16.
13.
14.
-3
,8 -'£r
Practice and Applications
COMPLETING EQUATIONS Copy and complete the statement.
19 — = 3 H
1S - 3 5
22. ^ = x 2
7 ?
20. ^ = 7 4
23
.,10 _
_ ?
- = //
9 5 ,
21 — = Q 6
■ ^
9
24.
w
w
Student HeCp
► Homework Help
Example 1: Exs. 19-32
Example 2: Exs. 33-46
Example 3: Exs. 47-54
Example 4: Exs. 55-57
QUOTIENT OF POWERS Simplify the quotient.
*6 «2 ( — 3) 6
25.
26.
27.
29.
x 5
30. x 3 •
x 2
(- 3) 6
31. • x 5
x 8
28.
32.
6 3 ♦ 6 2
6 5
x 3 • x 5
COMPLETING EQUATIONS Copy and complete the statement.
33.
36.
1
6
x \ ?
>’
?
34.
37.
-3
5
2\5
a~'
b
■
25
a
b 5
■>
35.
38.
2 \ 1
8
343
m
3 \4
i •
m
n'
12
8.4 Division Properties of Exponents
POWER OF A QUOTIENT Simplify the quotient.
SIMPLIFYING EXPRESSIONS Simplify the expression. Use only positive
exponents.
4x 3 y 3 5xy 2
49.-•-
2 xy 2y
52.
55.
f 2mV?
\ 3 mn )
x 2 _ 2x~y
xy - 4 3xy _1
50.
53.
56.
16x 3 y
— 2xy
51.
36 a 8 b 2
ab 2
—4xy 3
X
ab
6
6x 2 y 2
(4x 2 y) 2
54.
00
VO
/ x 3 y 2 \ 4
•
xy 3
xy 2
•
7 4
x'y
[ *xy)
sx-y
(V )- 2
57.
4xy
( 2xy 2 N
xV 1
xy
2x~ l y~ 3
\*xy,
STOCKBROKERS who work
on the floor of a stock
exchange use hand signals
that date back to the 1880s to
relay information about stock
trades.
More about
' stockbrokers at
www.mcdougallittell.com
Use Division Properties of Exponents
STOCK EXCHANGE The number of shares n (in billions) listed on the New
York Stock Exchange (NYSE) from 1977 through 1997 can be modeled by
n = 93.4 • (1.11)'
where t — 0 represents 1990. Find the ratio of shares listed in 1997 to the
shares listed in 1977. ►Source: New York Stock Exchange
Solution
Since 1997 is 7 years after 1990, use t = 1 for 1997. Since 1977 is 13 years
before 1990, use t = — 13 for 1977. Because 93.4 is a common factor to the
number of shares for both years, you may omit it from the ratio below.
Number listed in 1997 _ (1.11) 7
Number listed in 1977 (111) -13
= (i.ii) 7 -<~ 13 >
= (i.ii ) 20
~ 8.06 * - Use a calculator.
ANSWER ^ The ratio of shares listed in 1997 to the shares listed in 1977 is 8.06
to 1. There were about 8 times as many shares listed in 1997 as there
were in 1977.
Chapter 8 Exponents and Exponential Functions
Use the example on the previous page as a
model for Exercises 58-61.
58. RETAIL SALES From 1994 to 1998
the sales for a clothing store increased
by about the same percent each year.
The sales S (in millions of dollars) for
year t can be modeled by S = 3723
where t = 0 corresponds to 1994. Find
the ratio of 1998 sales to 1995 sales.
59. BASEBALL SALARIES The average salary s (in thousands) for a
professional baseball player in the United States can be modeled by
s = 136(1.18)'
where t = 0 represents the year 1980. Find the ratio of the average salary in
1985 to the average salary in 1990. ►Source: National Baseball Library and Archive
60. ATLANTIC COD The average weight w (in pounds) of an Atlantic cod can
be modeled by
w = 1.21(1.42)'
where t is the age of the fish (in years). Find the ratio of the weight of a
5-year-old cod to the weight of a 2-year-old cod.
► Source: National Marine Fisheries Service
61. LEARNING SPANISH You memorized a list of 200 Spanish vocabulary
words. Unfortunately, each week you forget one fifth of the words you knew
the previous week. The number of Spanish words S you remember after n
weeks can be modeled by:
5 =
Copy and complete the table showing the number of words you remember
after n weeks.
Weeks n
0
1
2
3
4
5
6
Words S
?
?
?
?
?
?
?
LOGICAL REASONING Give a reason for each step to show that the
definitions of zero and negative exponents hold true for the properties
of exponents.
62. a 0 = a n “ " 63. a~ n = a n ~ 2n
a n a n
a
n
= l
„2 n
a' 1
a n • a n
J_
r n
a
64. CHALLENGE A piece of notebook paper is about 0.0032 inch thick. If you
begin with a stack consisting of a single sheet and double the stack 25 times,
how thick will the stack be? HINT: You will need to write and solve an
exponential equation.
8.4 Division Properties of Exponents
Standardized Test
Practice
Mixed Review
65. MULTIPLE CHOICE Simplify the expression
C s-h:
(© X
X
-3
-3 ‘
66. MULTIPLE CHOICE Simplify the expression
-3
CH) 729
(D) X 3
Q)^r
67. MULTIPLE CHOICE Simplify the expression
CD I CDv
4x 3 y
1 8a- 2 1 6x 2 y'
3 I
68. MULTIPLE CHOICE Simplify the expression — - • j |
<E> —
X
/
CD —
X
CH) "t
r
(3)^
r
POWERS OF TEN Evaluate the expression. (Lessons 1.2, 8.2)
69.10 5 70.10 1 71.10° 72.10 “ 4
SLOPE-INTERCEPT FORM Write in slope-intercept form the equation of
the line that passes through the given points. (Lesson 5.3)
73. (-4, 2) and (4, 6 ) 74. (-4, -5) and (0, 3) 75. (-1, -7) and (3,-11)
76. (3, 9) and (1, -3) 77. (5, -2) and (-4, 7) 78. (1, 8 ) and (-4, -2)
CHECKING FOR SOLUTIONS Decide whether the ordered pair is a
solution of the system of linear equations. (Lesson 7. 1)
79. 2x + 4y = 2
—x + 5y = 13
(-3, 2)
80. 3x — 4y = 5
x + 6 y = 8
(3,1)
81. 8 x + 4y = 6
4x + y = 3
( 1 ,- 1 )
82. x - 5y = 9
3x + 5y = 11
(4,-1)
SOLVING LINEAR SYSTEMS Use linear combinations to solve the
system. Then check your solution. (Lesson 7.3)
83. x — y = 4
84. — p + 2q = 12
85. 2a + 3b= 17
x + y = 12
p + 6q = 20
3a + 4b = 24
86. 2/77 + 3/7 = 7
87.x + lOy = -1
88. Sr-3t = 2
m + n = 1
2x + 9y = 9
2r-2t = 3
Maintaining Skills ESTIMATION Use front-end estimation to estimate the sum or difference.
(Skills Review p. 774)
89. 287 + 165
90. 4672 + 1807
91.46.18 + 34.42
92. 172 - 112
93. 4882 - 3117
94. 3.84 - 1.68
Chapter 8 Exponents and Exponential Functions
Scientific Notation
Goal
Read and write numbers
in scientific notation.
Key Words
• scientific notation
What was the price of Alaska per square mile?
In 1867 the United States
purchased Alaska by writing a
check for $7.2 million. In
Example 5 you will use
scientific notation to find the
price per square mile of
that purchase.
Numbers such as 100, 14.2, and 0.07 are written in decimal form. Scientific
notation uses powers of ten to express decimal numbers. A number is written
in scientific notation if it is of the form c X 10” where 1 < c < 10 and n is an
integer. Here are three examples.
1.2 X 10 3 = 1.2 X 1000 = 1200
5.6 X 10° = 5.6 X 1 = 5.6
3.5 X 10" 1 = 3.5 X 0.1 = 0.35
Student UeCp
► Study Tip
When multiplying by
10 n and n > 0 , move
the decimal point n
places to the right.
When n < 0 move the
decimal point n places
to the left.
V J
1 Write Numbers in Decimal Form
Write the number in decimal form.
a. 2.83 X 10 1 b. 4.9 X 10 5 c. 8 X 10 _1 d. 1.23 X 10“ 3
Solution
a. 2.83 X 10 1 = 28.3
b. 4.9 X 10 5 = 490 000
\AAA>U
c. 8 X 10” 1 = 0.8
d. 1.23 X 10“ 3 = 0.00123
Move decimal point 1 place to the right.
Move decimal point 5 places to the right.
Move decimal point 1 place to the left.
Move decimal point 3 places to the left.
Write the number in decimal form.
1. 2.39 X 10 4
2. 1.045 X 10 7
3. 3.7 X 10 8
4. 8.4 X 10“ 6
5. 1.0 X 10“ 2
6. 9.2 X 10“
8.5 Scientific Notation
2 Write Numbers in Scientific Notation
Write the number in scientific notation,
a. 34,000 b. 1.78 c. 0.0007
Solution
a. 34,000 = 3.4 X 10 4
ItAAA/
b. 1.78 = 1.78 X 10°
c. 0.0007 = 7 X 10" 4
Move decimal point 4 places to the left.
Move decimal place 0 places.
Move decimal point 4 places to the right.
Write Numbers in Scientific Notation
Write the number in scientific notation.
7.423 8. 2,000,000 9.0.0001
10 . 0.0098
Student HeCp
p More Examples
More examples
are available at
www.mcdougallittell.com
3 Operations with Scientific Notation
Perform the indicated operation. Write the result in scientific notation.
a. (1.4 X 10 4 )(7.6 X 10 3 )
= (1.4 • 7.6) X (10 4 • 10 3 )
= 10.64 X 10 7
= ( 1.064 x 10 1 ) X 10 7
= 1.064 X10 8
. 1.2 X IQ' 1 = L2 IQ -1
4.8 X 10 -4 4.8 lo- 4
= 0.25 X 10 3
= (2.5 X 10 " 1 ) X 10 3
Use properties of multiplication.
Use product of powers property.
Write in scientific notation.
Use product of powers property.
Write as a product.
Use quotient of powers property.
Write in scientific notation.
= 2.5 X 10 2
Use product of powers property.
c. (4 X 10" 2 ) 3 = 4 3 X (10“ 2 ) 3
= 64 X 10“ 6
= ( 6.4 x 10 1 ) X 10" 6
= 6.4 X 10“ 5
Use power of a product property.
Use power of a power property.
Write in scientific notation.
Use product of powers property.
Operations with Scientific Notation
Perform the indicated operation. Write the result in scientific notation.
11. (2.3 X 10 3 )(l.8 X 10“ 5 ) 12. 5 ' 2 X 10 ^ 13. (5 X 10“ 4 ) 2
v A ' 1.3 X 10 1
T
Chapter 8 Exponents and Exponential Functions
Many calculators automatically use scientific notation to display large or small
numbers. Try multiplying 98,900,000 by 500 on a calculator. If the calculator
follows standard procedures, it will display the product using scientific notation.
( M.9M5 10 1 + -- Calculator display for 4.945 x 10 10
Student HeCp
► Keystroke Help
If your calculator does
not have an key,
you can enter a
number in scientific
notation as a product:
7.48 Efl 10 EM
7
i_/
Use a Calculator
| Use a calculator to multiply 7.48 X 10 -7 by 2.4 X 10 9 .
Solution
KEYSTROKES DISPLAY
7.48 0 2 . 4^90 I 1.7955 I
ANSWER ► The product is 1.7952 X 10 3 , or 1795.2.
Use a Calculator
Use a calculator to perform the indicated operation.
14. (5.1 X 10 2 )(0.8 X 1(T 4 ) 15. 8,9 X 10 ° 16. (1.5 X 10 6 )" 1
A ' 6.4 X 1(T 5 v '
Student HeCp
^
► Look Back
For help with unit
rates, see p. 177.
^ _ /
ms Scientific Notation in Real Life
ALASKA PURCHASE In 1867 the United States purchased Alaska from
Russia for $7.2 million. The total area of Alaska is about 5.9 X 10 5 square
miles. What was the price per square mile?
Solution
The price per square mile is a unit rate.
Total price
Price per square mile = —— ----—
1 Number ol square miles
= 7.2 X 10 6 - 7.2 million = 7.2 x 10 6
5.9 X 10 5
- 1.22 X 10 1
= 12.2
ANSWER ^ The price was about $12.20 per square mile.
Scientific Notation in Real Life
17. In 1994 the population of California was about 3.1 X 10 7 . In that year about
5.6 X 10 10 local calls were made in California. Estimate the number of local
calls made per person in California in 1994.
8.5 Scientific Notation
Exercises
Guided Practice
Vocabulary Check 1. Is the number 12.38 X 10 2 in scientific notation? Explain.
Skill Check Write the number in decimal form.
2. 9 X 10 4 3. 4.3 X 10 2
5. 5 X 10" 2 6. 9.4 X 10" 5
Write the number in scientific notation.
8. 15 9. 6,900,000
11.0.99 12.0.0003
4. 8.11 X 10 3
7. 2.45 X 10" 1
10. 39.6
13. 0.0205
Perform the indicated operation. Write the result in scientific notation.
14. (5 X 10 6 )(6 X 10“ 2 ) 15. — - 10 7 3 16. (9 X 10“ 9 ) 2
7 / 7 x to 7 v '
Practice and Applications
MOVING DECIMALS Tell whether you would move the decimal left or
right and how many places to write the number in decimal form.
17. 1.5 X 10 2 18. 6.89 X 10 5 19. 9.04 X 10“ 7
DECIMAL FORM
20. 5 X 10 5
23. 2.1 X 10 4
26. 3 X 10“ 4
29. 9.8 X 10" 2
Write the number in decimal form.
21. 8 X 10 3
24. 7.75 X 10°
27. 9 X 10“ 3
30. 6.02 X 10" 6
22. 1 X 10 6
25. 4.33 X 10 8
28. 4 X 10“ 5
31. 1.1 X 10" 10
LOGICAL REASONING Decide whether the number is in scientific
notation. If not, write the number in scientific notation.
32. 0.7 X 10 2 33. 2.9 X 10 5 34. 10 X 10 -3
Student HeCp
p Homework Help
Example 1: Exs. 17-31
Example 2: Exs. 32-46
Example 3: Exs. 47-55
Example 4: Exs. 56-61
Example 5: Exs. 62-69
SCIENTIFIC NOTATION
35.900
38. 1012
41. 0.1
44. 0.0422
Write the number in scientific notation.
36. 700,000,000
39. 95.2
42. 0.05
45. 0.0085
37. 88,000,000
40. 370.2
43. 0.000006
46. 0.000459
Chapter 8 Exponents and Exponential Functions
Link to
History
ANTONIO LOPEZ DE
SANTA ANNA, the
President of Mexico, sold the
Gadsden Purchase to the
United States. The purchase
was negotiated by James
Gadsden, the United States
Minister to Mexico.
EVALUATING EXPRESSIONS Perform the indicated operation without
using a calculator. Write the result in scientific notation.
47. (4.1 X 10 2 )(3 X 10 6 ) 48. (9 X 10“ 6 )(2 X 10 4 ) 49. (6 X 10 5 )(2.5 X 10“ 1 )
50.
8 X 10~ 3
4 X 10" 5
51.
3.5 X 10~ 4
5 X 10 -1
52.
6.6 X IQ" 1
1.1 X 10 _1
53. (3 X 10 2 ) 3
54. (2 X 10“ 3 ) 4
55. (0.5 X lO)" 2
i::: CALCULATOR Use a calculator to perform the indicated operation.
Write the result in scientific notation and in decimal form.
56. 6,000,000 • 324,000 57. (2.79 X 10" 4 )(3.94 X 10 9 )
58.
3,940,000
0.0002
-6
59.
6.45 X 10
4.3 X 10 5
60. (0.000094) 3 61. (2.4 X 10“ 4 ) 5
DECIMAL FORM Write the number in decimal form.
62. The distance that light travels in one year is 9.46 X 10 12 kilometers.
63. The length of a dust mite is 9.8 X 10 -4 foot.
SCIENTIFIC NOTATION Write the number in scientific notation.
64. At the end of 1999 the population of the world was estimated at
6,035,000,000. DATA UPDATE of U.S. Census Bureau data at www.mcdougallittell.conn
65. The mass of a carbon atom is 0.00000000000000000000002 gram.
66 . Science Link Light travels at a speed of about 3 X 10 5 kilometers per
second. It takes about 1.5 X 10 4 seconds for light to travel from the sun to
Neptune. What is the approximate distance (in kilometers) between Neptune
and the sun?
67. Find the price per square
mile of the Louisiana
Purchase.
68 . Find the price per square
mile of the Gadsden Purchase.
69. Science Unky Jupiter, the largest planet in our solar system, has a radius
of about 7.15 X 10 4 kilometers. Use the formula for the volume of a sphere,
4 q
V = , to estimate Jupiter’s volume.
Hist ory Unify In Exercises 67 and 68, use the following information.
In 1803 the Louisiana Purchase
added 8.28 X 10 5 square miles
to the United States. The price
of the land was $15 million. In
1853 the Gadsden Purchase
added 2.96 X 10 4 square miles.
The price was $10 million.
8.5 Scientific Notation
Standardized Test
Practice
Mixed Review
Maintaining Skills
Quiz 2
70. MULTIPLE CHOICE Which number is not in scientific notation?
(A) 1 X 10 4 CD 3.4 X 10“ 3 CD 9.02 X 10 2 CD 12.25 X 10“
11X1o -1
71. MULTIPLE CHOICE Evaluate —- 7 .
5.5 X 10“ 5
CD 0.2 X 10“ 6 CD 0.2 X 10“ 4 CD 2 X 10 3 CD 2 x 10 4
GRAPHING Use the graphing method to tell how many solutions the
system has. (Lesson 7.5)
72. 4x + 2y= 12 73. 3x - 2y = 0 74. x - 5j = 8
—6x + 3j = 6 3x — 2y = —4 — x + 5y = —8
GRAPHING Graph the system of linear inequalities. (Lesson 7.6)
75. 2x + y < 1 76. x + 2y < 3 77. 2x + y > 2
— 2x + y<l x — 3j > 1 x<2
FRACTIONS, DECIMALS, AND PERCENTS Write the given fraction,
decimal, or percent in the indicated form. (Skills Review pp. 767-769)
1 53
78. Write y as a decimal. 79. Write as a percent.
80. Write 1.45 as a fraction. 81 . Write 0.674 as a percent.
82. Write 15% as a fraction. 83. Write 756.7% as a decimal.
Simplify the quotient. (Lesson 8.4)
~T_
2
-5
Simplify the expression. Use only positive exponents. (Lesson 8.4)
5gfr 3 # 10a~ 3 fc
—2 a~ l b 2 * a 2 b~ 4
9 wz~ 2
3xv 5 4*4
20x 3 y
—6xy
5. — - 7-7 • —r
9x 4 y 6 xy 8
6. 2 *
4xv
—x
7 .
i-2m 2 n\ 4
O
X
C
(2 x 2 y) 4
10 .
\ 3 mn 2 j
^ 0 •
5 x 3 y 6
4x 3 y
-3 3
W Z
w 2 z 3 3
3 z
-1
Write the number in decimal form. (Lesson 8.5)
11.5 X 10 9 12. 4.8X10 3
14. 7 X 10“ 6 15. 1.1 X 10“ 2
13. 3.35 X 10 4
16. 2.08 X 10“ 5
Write the number in scientific notation. (Lesson 8.5)
17. 105
20 . 0.25
18. 99,000
21 . 0.0004
19. 30,700,000
22. 0.0000067
Chapter 8 Exponents and Exponential Functions
Goal
Use reasoning to compare
exponential and linear
functions.
Question
i i ^
How are linear and exponential functions different?
Materials
• graph paper
Explore
© The equation y = 5 X is an exponential
function. Copy and complete the
table using this equation.
©Use the table in Step 1 to graph y = 5 X .
© The equation y — 5x + 20 is a linear
function. Copy and complete the table
using this equation.
©Use the table in Step 3 to graph y = 5x + 20.
© Which of the graphs below shows a linear function! Which shows an
exponential function! Explain how you know.
Think About It
Graph the function.
1. y = x + 5
2.y = 3 x
3. y = 10 + 2x
4 .y= -3(2 y
5 .y = 5(4x - 7)
6 .y= 10(1.2)*
LOGICAL REASONING In Exercises 7-9, use the results from
Exercises 1-6.
7. Complete: A linear function increases the ? amount for each unit on
the x-axis.
8. Describe the rate of increase in an exponential growth model.
9- Explain one way that an equation for a linear function differs from an
equation for an exponential function.
Developing Concepts
Exponential Growth Functions
Goal
Write and graph
exponential growth
functions.
Key Words
• exponential growth
• growth rate
• growth factor
How does a catfish's weight change as it grows?
In Lesson 8.3 you learned about
exponential functions. One use of
exponential functions is to model
exponential growth. In Example 1
you will analyze the weight of a
newly hatched catfish when that
weight is increasing by 10% each day.
A quantity is growing exponentially if it increases by the same percent r in each
unit of time t. This is called exponential growth. Exponential growth can be
modeled by the equation
y = C(1 + rY
where C is the initial amount (the amount before any growth occurs), r is the
growth rate (as a decimal), t represents time, and both C and r are positive.
The expression (1 + r) is called the growth factor.
,
Student HeCp
► Study Tip
To write a percent as a
decimal, remove the
percent sign from the
number and divide the
number by 100.
.0% = JjL - 0.10
i Write an Exponential Growth Model
CATFISH GROWTH A newly hatched channel catfish typically weighs about
0.06 gram. During the first six weeks of life, its weight increases by about 10%
each day. Write a model for the weight of the catfish during the first six weeks.
Solution
Let y be the weight of the catfish during the first six weeks and let t be the
number of days. The initial weight of the catfish C is 0.06. The growth
rate is r is 10%, or 0.10.
y = C(1 + r)* Write exponential growth model.
= 0.06(1 + 0.10)* Substitute 0.06 for C and 0.10 for r.
= 0.06(1.1)* Add.
1- A TV station’s local news program has 50,000 viewers. The managers of the
station hope to increase the number of viewers by 2% per month. Write an
exponential growth model to represent the number of viewers v in t months.
Chapter 8 Exponents and Exponential Functions
COMPOUND INTEREST Compound interest is interest paid on the principal P,
the original amount deposited, and on the interest that has already been earned.
Compound interest is a type of exponential growth, so you can use the
exponential growth model to find the account balance A.
Find the Balance in an Account
COMPOUND INTEREST You deposit $500 in an account that pays 8% interest
compounded yearly. What will the account balance be after 6 years?
Student HeCp
►Writing Algebra
The model for
compound interest is
generally written using
A (for the account
balance) instead of y,
and P (for the
principal) instead of C.
V_ /
Solution
The initial amount P is $500, the growth rate is 8%, and the time is 6 years.
A = P( 1 + r/ Write yearly compound interest model.
= 500(1 + 0.08) 6 Substitute 500 for P, 0.08 for r, and 6 for t.
= 500(1.08) 6 Add.
~ 793 Use a calculator.
ANSWER ^ The balance after 6 years will be about $793.
\ _
Find the Balance in an Account
2 . You deposit $750 in an account that pays 6% interest compounded yearly.
What is the balance in the account after 10 years?
Student HeCp
i — >
► Study Tip
Growth factors are
usually given as whole
numbers and growth
rates as percents or
decimals.
v_ J
3 Use an Exponential Growth Model
POPULATION GROWTH An initial population of 20 mice triples each year
for 5 years. What is the mice population after 5 years?
Solution
You know that the population triples each year. This tells you the factor by
which the population is growing, not the percent change in the population.
Therefore the growth factor (not the growth rate) is 3. The initial population
is 20 and the time is 5 years.
y = C( 1 + rf Write exponential growth model.
= 20(3) 5 Substitute for 20 for C, 3 for 1 + r, and 5 for t.
= 4860 Evaluate.
ANSWER ► There will be 4860 mice after 5 years.
Use an Exponential Growth Model
3. An initial population of 30 rabbits doubles each year for 6 years. What is the
rabbit population after 6 years?
8.6 Exponential Growth Functions
J 4 A Model with a Large Growth Rate
Graph the exponential growth model from Example 3.
Solution
Make a table of values, plot
the points in a coordinate
plane, and draw a smooth
curve through the points.
0
1
2
3
4
20
60
180
540
1620
Student HeCp
► Study Tip
A large growth rate
corresponds to a rapid
increase in the
/-values.
v_/
BZES9 5 A Model with a Small Growth Rate
In 1980 there were only 73 peregrine falcons along the Colville River in
Alaska. From 1980 to 1987 the population grew by about 9% per year.
Therefore the population P of peregrine falcons can be modeled by
P = 73(1.09) r where t — 0 represents 1980. Graph the function.
Solution
Make a table of values, plot
the points in a coordinate
plane, and draw a smooth
curve through the points.
0
1
2
3
4
73
80
87
95
103
Student Hedp
► Study Tip
A small growth rate
corresponds to a
slow increase in the
/-values.
h J
Graph an Exponential Growth Model
4. Graph the exponential growth model you found in Checkpoint 3.
Chapter 8 Exponents and Exponential Functions
M Exercises
Guided Practice
Vocabulary Check 1. Complete: In the exponential growth model, y = C(1 + /■)'. C is the ?
and (1 + r) is the ? .
Skill Check COMPOUND INTEREST You deposit $500 in an account that pays
4% interest compounded yearly.
2. What is the initial amount P?
3. What is the growth rate r?
4. Complete this equation to write an exponential growth model for the balance
after t years: A = ? (1 + ? V.
5. Use the equation from Exercise 4 to find the balance after 5 years.
6. CHOOSE A MODEL Which model
best represents the growth curve
shown in the graph at the right?
A. y = 100(2)' B. y = 100(1.2/
C. y = 200(2)' D. y = 200(1.2)'
Practice and Applications
EXPONENTIAL GROWTH Identify the initial amount and the growth rate
in the exponential function.
7. y = 100(1 + 0.5)' 8. y = 12(1 + 2)' 9. y = 7.5(1.75)'
WRITING EXPONENTIAL FUNCTIONS Write an exponential function to
model the situation. Tell what each variable represents.
10. Your salary of $25,000 increases 7% each year.
11. A population of 310,000 increases by 15% each year.
12 . An annual benefit concert attendance of 10,000 increases by 5% each year.
! Student HeCp
► Homework Help
Example 1: Exs. 7-15
Example 2: Exs. 16-27
Example 3: Exs. 28-35
Examples 4 and 5:
Exs. 36-40
v j
BUSINESS Write an exponential growth model for the profit.
13. A business had a $10,000 profit in 1990. Then the profit increased by 25%
per year for the next 10 years.
14. A business had a $20,000 profit in 1990. Then the profit increased by 20%
per year for the next 10 years.
15. A business had a $15,000 profit in 1990. Then the profit increased by 30%
per year for the next 15 years.
8.6 Exponential Growth Functions
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COMPOUND INTEREST You deposit $1400 in an account that pays
6% interest compounded yearly. Find the balance at the end of the
given time period.
16-5 years 17. 8 years 18. 12 years 19. 20 years
COMPOUND INTEREST You deposit money in an account that pays
5% interest compounded yearly. Find the balance after 5 years for the
given initial amount.
20. $250 21. $300 22. $350 23. $400
COMPOUND INTEREST You deposit $900 in an account that compounds
interest yearly. Find the balance after 10 years for the given interest rate.
24. 4% 25. 5% 26. 6% 27. 7%
GROWTH RATES AND FACTORS Identify the growth rate and the growth
factor in the exponential function.
28. y = 50(1 + 1 Y 29. y = 31(4/ 30. y = 5.6(2.3/
POPULATION GROWTH An initial population of 1000 starfish doubles
each year for 4 years.
31. What is the growth factor for the population?
32. What is the starfish population after 4 years?
SUNFISH GROWTH An ocean sunfish, the mola mola, is about 0.006 foot
long when it hatches. By the time it reaches adulthood, the largest of the
mola mola will have tripled its length about 7 times.
33. What is the growth factor for the length of a mola mola?
34. What is the maximum length of an adult mola mola?
BICYCLE RACING In Exercises 35 and 36, use the following information.
The air intake b (in liters per minute) of a cyclist on a racing bike can be modeled
by b = 6.37(1.11/, where s is the speed of the bike (in miles per hour).
35. Use a calculator to find the cyclist’s air intake if the racing bike is
traveling 7 miles per hour, 19 miles per hour, or 25 miles per hour.
36. GRAPHING Graph the exponential growth model.
EXPONENTIAL GROWTH MODELS Match the description with its graph.
37. C = $300 r = 6% 38. C = $300 r = 12% 39. C = $300 r = 20%
Chapter 8 Exponents and Exponential Functions
40. CRITICAL THINKING Graph the exponential growth models you found in
Exercises 13-15. Which business would you rather own? Explain.
Standardized Test
Practice
Mixed Review
Maintaining Skills
41. CHALLENGE What is the value of an $1000 investment after 5 years if it
earns 6% annual interest compounded quarterly (four times a year).
HINT: Use the compound interest formula A = P[1 + — I , where A is the
value of the account, P is the initial investment, r is the interest rate, n is the
number of times per year the interest is compounded, and t is the time period
(in years).
42. IVIULTIPLE CHOICE The hourly rate of your new job is $5.00 per hour. You
expect a raise of 9% at the end of each year. What will your hourly rate be at
the end of your fifth year?
(A) $5.45 CD $7.25 CD $7.69
CD $9.50
43. MULTIPLE CHOICE What is the equation
of the graph?
CD y = (2 • 1.3)* CD y = 1.3(2)"
CD y = 2(1 - 0.3)" CD y = 2(1.3)"
VARIABLE EXPRESSIONS Evaluate the expression for the given value of
the variable. (Lesson 1.3)
44. 24 + m 2 when m = 5 45. 6x — 1 when x = 1
46. 3 • 15y wheny = 2
47. 1 — ^ when a — 9
SOLVING EQUATIONS Solve the equation. (Lesson 3.5)
48. -2(4 - 3jc) = 6(2x + 1) + 4 49. lx - (Ax + 3) = 4(3x + 15)
50. |(6 m - 3) + 10 = —8 (m + 2) 51. |(12 y - 4) - 2 y = -3 (y - 5)
52. BAGELS AND DONUTS You buy 6 bagels and 8 donuts for a total of $8.60.
Then you decide to buy 3 extra bagels and 3 extra donuts for a total of $3.75.
How much did each bagel and donut cost? (Lesson 7.4)
PRODUCT OF POWERS Write the expression as a single power of the
base. (Lesson 8.1)
53. 2 2 • 2 2 54. 7 6 * 7 2 55. 3 5 * 3 2
56. y 3 • y 57. r 2 • r 4 58. a 9 * a 4
SIMPLIFYING FRACTIONS Write the fraction in simplest form.
(Skills Review p. 763)
59.
25
100
60.
215
645
61
53
424
62
71
355
8.6
Exponential Growth Functions
Goal
Write and graph
exponential decay
functions.
Key Words
• exponential decay
• decay rate
• decay factor
Exponential Decay Functions
What will your car be worth after 8 years?
In Lesson 8.6 you used exponential
functions to model values that
were increasing. Exponential
functions can also be used to
model values that are decreasing.
In Examples 1-3 you will analyze
a car’s value that is decreasing
exponentially over time.
A quantity is decreasing exponentially if it decreases by the same percent r in
each unit of time t. This is called exponential decay. Exponential decay can be
modeled by the equation
y — C(1 — ry
where C is the initial amount (the amount before any decay occurs), r is the
decay rate (as a decimal), t represents time, and where 0 < r < 1. The expression
(1 — r) is called the decay factor.
1 Write an Exponential Decay Model
CARS You bought a car for $16,000. You expect the car to lose value, or
depreciate, at a rate of 12% per year. Write an exponential decay model to
represent this situation.
Solution
Let y be the value of the car and let t be the number of years of ownership. The
initial value of the car C is $16,000. The decay rate r is 12%, or 0.12.
y = C(1 — r)* Write exponential decay model.
= 16,000(1 — 0.12)* Substitute 16,000 for C and 0.12 for r.
= 16,000(0.88)* Subtract.
ANSWER ► The exponential decay model is y = 16,000(0.88)*.
Write an Exponential Decay Model
1. Your friend bought a car for $24,000. The car depreciates at the rate of 10%
per year. Write an exponential decay model to represent the car’s value.
Chapter 8 Exponents and Exponential Functions
2 Use an Exponential Decay Model
Use the model in Example 1 to find the value of your car after 8 years.
Solution To find the value after 8 years, substitute 8 for t.
y — 16,000(0.88/ Write exponential decay model.
= 16,000(0.88) 8 Substitute 8 for t.
~ 5754 Use a calculator.
ANSWER ► Your car will be worth about $5754 after 8 years.
3 Graph an Exponential Decay Model
a. Graph the exponential decay model in Example 1.
b. Use the graph to estimate the value of your car after 5 years.
Solution
a. Make a table of values, plot the points in a coordinate plane, and draw a
smooth curve through the points.
t
0
2
4
6
8
y
16,000
12,390
9595
7430
5754
o
o
VO
‘( 0 , 16 , 000 )
\
00
1 —
_cc
12,000
N
.( 2 , 12 , 390 )
( 4 , 9595 )
"o
■g
8000
y =
= 16 , 000 ( 0 . 88 )
M
( 6 , 7430 )
=3
I
a r\r\r\
|
(8 R7R41
L +K.
nju
n
°(
3
l i
X
6
8 *
Years of ownership
b. According to the graph, the value of your car after 5 years will be about
$8400. You can check this answer by using the model in Example 1.
Graph and Use an Exponential Decay Model
Use the model in Checkpoint 1.
2 . Find the value of your friend’s car after 6 years.
3. Graph the exponential decay model.
4. Use the graph to estimate the value of your friend’s car after 5 years.
8.7 Exponential Decay Functions
In Lesson 8.3 you learned that for b > 0 a function of the form y = ab x is an
exponential function. In the model for exponential growth, b is replaced by 1 + r
where r > 0. In the model for exponential decay, b is replaced by 1 — r where
0 < r < 1. Therefore you can conclude that an exponential model y = Cb f
represents exponential growth if b > 1 and exponential decay if 0 < b < 1.
J 4 Compare Growth and Decay Models
Classify the model as exponential growth or exponential decay. Then identify
the growth or decay factor and graph the model.
a. y — 30(1.2/, where t > 0 b. y — 3o( j j, where t > 0
Solution
a. Because 1.2 > 1, the
model y = 30(1.2/is an
exponential growth model.
The growth factor (1 + r)
is 1.2. The graph is shown
below.
b. Because 0 < -- < 1, the model
/ 3 y
y = 301 ^ I is an exponential
decay model. The decay factor
3
(1 — r) is The graph is shown
below.
Compare Growth and Decay Models
Classify the model as exponential growth or exponential decay. Then
identify the growth or decay factor and graph the model.
5- >’ = (2)'
6.y = (0.5)' 7. y = 5(0.2)'
8 .y = 0.7(1.1)'
* -
EXPONENTIAL j
yy
jf EXPONENTIAL j
—
yy
GROWTH MODEL
DECAY MODEL
JbC)
y= C( 1 + r) f ,
where 1 + r > 1
ToT C)
/= C(1 - r)\
_^ where 0 < 1 - r < 1
and t > 0 ,
and t > 0
t
t
k _
_>
Chapter 8 Exponents and Exponential Functions
z:! Exercises
Guided Practice
Vocabulary Check
Skill Check
CARS You buy a used car for $7000. The car depreciates at the rate of
6% per year. Find the value of the car after the given number of years.
3- 2 years 4. 5 years 5. 8 years 6- 10 years
7. CHOOSE A MODEL Which model best
represents the decay curve shown in the
graph at the right?
A. y = 60(0.08)' B. y = 60(1.20)'
C. y = 60(0.40)' D. y = 60(1.05)'
Classify the model as exponential growth or exponential decay.
8. y = 0.55(3)' 9. y = 3(0.55)' 10. y = 55(3)' 11. y = 55(0.3)'
1. In the exponential decay model, y — C(1 — r)', what is the decay factor?
2. BUSINESS A business earned $85,000 in 1990. Then its earnings decreased
by 2% each year for 10 years. Write an exponential decay model to represent
the decreasing annual earnings of the business.
Practice and Applications
EXPONENTIAL DECAY MODEL Identify the initial amount and the decay
factor in the exponential function.
12 . y = 10(0.2) f
13. y = 18(0.11)* 14. y
15. y = 0.5
i
WRITING EXPONENTIAL MODELS Write an exponential model to
represent the situation. Tell what each variable represents.
16. A $25,000 car depreciates at a rate of 9% each year.
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Example 1: Exs. 12-21
Example 2: Exs. 22-30
Example 3: Exs. 31-41
Example 4: Exs. 42-53
v _>
17. A population of 100,000 decreases by 2% each year.
18. A new sound system, valued at $800, decreases in value by 10% each year.
FINANCE Write an exponential decay model for the investment.
19. A stock is valued at $100. Then the value decreases by 9% per year.
20. $550 is placed in a mutual fund. Then the value decreases by 4% per year.
21. A bond is purchased for $70. Then the value decreases by 1% per year.
8.7 Exponential Decay Functions
PHARMACISTS must
understand the use,
composition, and effects
of pharmaceuticals.
More about
pharmacists at
www.mcdougallittell.com
TRUCKS You buy a used truck for $20,000. The truck depreciates 7% per
year. Find the value of the truck after the given number of years.
22. 3 years 23. 8 years 24. 10 years 25. 12 years
PHARMACEUTICALS In Exercises 26-28, use the following information.
The amount of aspirin y (in milligrams) in a person’s blood can be modeled by
y = A(0.8) ? where A represents the dose of aspirin taken (in milligrams) and t
represents the number of hours since the aspirin was taken. Find the amount of
aspirin remaining in a person’s blood for the given dosage and time.
26. Dosage: 250 mg 27. Dosage: 500 mg 28. Dosage: 750 mg
Time: after 2 hours Time: after 3.5 hours Time: after 5 hours
BASKETBALL In Exercises 29 and 30, use the following information.
At the start of a basketball tournament consisting of six rounds, there are
64 teams. After each round, one half of the remaining teams are eliminated.
29. Write an exponential decay model showing the number of teams left in the
tournament after each round.
30. How many teams remain after 3 rounds? after 4 rounds?
GRAPHING Graph the exponential decay model.
31. y = 15(0.9) f 32. y = 72(0.85) f 33. y = lo(^j 34. y = 55^|J
GRAPHING AND ESTIMATING Write an exponential decay model for the
situation. Then graph the model and use the graph to estimate the value
at the end of the given time period.
35. A $22,000 investment decreases in value by 9% per year for 8 years.
36. A population of 2,000,000 decreases by 2% per year for 15 years.
37. You buy a new motorcycle for $10,500. It’s value depreciates by 10% each
year for the 10 years you own it.
CABLE CARS In Exercises 38-41, use the following information.
From 1894 to 1903 the number of miles of cable car track in the United States
decreased by about 11% per year. There were 302 miles of track in 1894.
38. Write an exponential decay model showing the number of miles of cable car
track left each year.
39. Copy and complete the table. You may want to use a calculator.
Year
1894
1896
1898
1900
1902
Miles of track
?
?
?
?
?
40. Graph the results.
41. Use your graph to estimate the number of miles of cable car track in 1903.
Chapter 8 Exponents and Exponential Functions
MATCHING Match the equation with its graph.
42. y = 4 - 3t 43. y = 4(0.6/
COMPARING MODELS Classify the model as exponential growth or
exponential decay. Then identify the growth or decay factor and graph
the model.
44. y = 24(1.18/ 45. y = 14(0.98/ 46. y = 97(1.01/
47. y = 112(f)' 48. y = 9(f)' 49. y = 35(f)'
B EXPONENTIAL FUNCTIONS Use a calculator to investigate the effects
of a and b on the graph of y = ab x .
50. In the same viewing rectangle, graph y = 2(2) x , y = 4(2) x , and y = 8(2) x .
How does an increase in the value of a affect the graph of y = ab x l
51.
52.
53. LOGICAL REASONING Choose a positive value for b and graph y = b x and
y = . What do you notice about the graphs?
54. CHALLENGE A store is having a sale on sweaters. On the first day the price
of the sweaters is reduced by 20%. The price will be reduced another 20%
each day until the sweaters are sold. On the fifth day of the sale will the
sweaters be free? Explain.
StBndBfdiZ&d T®St 55. MULTIPLE CHOICE In 1995 you purchase a parcel of land for $8000. The
Practice value of the land depreciates by 4% every year. What will the approximate
value of the land be in 2002?
(A) $224 CD $5760 CD $6012
56. MULTIPLE CHOICE Which model best
represents the decay curve shown in the
graph at the right?
CE)y = 50(0.25/ (G) y = 50(0.75/
(FT) y = 50(1.5/ CD y = 50(2 /
CD) $7999
30
i n
1U
]
[
3
5
t
In the same viewing rectangle, graph y = 2 X , y = 4 X , and y = 8 X . How does
an increase in the value of b affect the graph of y = ab x when b > 1 ?
In the same viewing rectangle, graph y = ( ^ I , y
7 ) ,andy = (-
How does a decrease in the value of b affect the graph of y — ab x when
0 < 6 < 1 ?
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8.7 Exponential Decay Functions
Mixed Review
Maintaining Skills
Quiz 3
VARIABLE EXPRESSIONS Evaluate the expression for the given value of
the variable(s). (Lesson 1.3 )
57. x 2 — 12 when x = 6 58. 49 — 4w when w = 2
59. 100 — rs when r — 4, s — 1 60. b 2 — 4 ac when a — l, b — 5, c — 3
SOLVING EQUATIONS Solve the equation. Round the result to the
nearest hundredth. (Lesson 3.6)
61. 1.29x = 5.22x + 3.61 62. 1.33x - 7.42 = 5.48x
63. 10.52x + 1.15 = -1.12jc - 6.35 64. 8.75x + 2.16 = 18.28x - 6.59
WRITING EQUATIONS Write in point-slope form the equation of the line
that passes through the given point and has the given slope. (Lesson 5.2)
65. (2, 5),m = 3
66 . (0, —3), m — 5
67. (-1, -4 ),m = 4
68 . (6, 3), m = — 1
69. (—1, 7), m = —6
70. (-4, -5 ),m= -2
DIVIDING DECIMALS Divide. (Skills Review p. 760)
71. 0.5 - 0.2 72. 4.62 - 0.4 73. 0.074 - 0.37
74. 0.084 - 0.007 75. 0.451 - 0.082 76. 0.6064 - 0.758
COMPOUND INTEREST You deposit $250 in an account that pays
8% interest compounded yearly. Find the balance at the end of the given
time period. (Lesson 8.6)
1. 1 year 2. 3 years 3. 5 years 4. 8 years
5. POPULATION GROWTH An initial population of 50 raccoons doubles each
year for 5 years. What is the raccoon population after 5 years? (Lesson 8.6)
CAR DEPRECIATION You buy a used car for $15,000. The car depreciates
at a rate of 9% per year. Find the value of the car after the given number
of years. (Lesson 8.7)
6 . 2 years 7. 4 years 8 . 5 years 9. 10 years
10. CAMPERS You buy a camper for $20,000. The camper depreciates at a rate
of 8% per year. Write an exponential decay model to represent this situation.
Then graph the model and use the graph to estimate the value of the camper
after 5 years. (Lesson 8.7)
Classify the model as exponential growth or exponential decay. Then
identify the growth or decay factor and graph the model. (Lesson 8.7)
'I'l.y = 6(0.1) f 12 . y = 10(1.2) f 13.y = 3^|j 14. y = 2^|
Chapter 8 Exponents and Exponential Functions
<9 Chapter Summary
® and Review
• exponential function, p. 455 • growth rate, p. 476 • decay rate, p. 482
• scientific notation, p.469 • growth factor, p. 476 • decay factor, p. 482
• exponential growth, p. 476 • exponential decay, p. 482
\ _>
Multiplication Properties of Exponents
Examples on
pp. 443-445
Use multiplication properties of exponents to simplify expressions.
a. 4 2 • 4 7 = 4 2 + 7 = 4 9
b. (x2) 4 = X 2 - 4 = X 8
c. (6a) 3 = 6 3 • a 3 = 216a 3
d. w 3 (v 2 w ) 4 = w 3 • (v 2 ) 4 • w 4
3 8 4
= w • V • w
= vV
Use product of powers property.
Use power of a power property.
Use power of a product property.
Use power of a product property.
Use power of a power property.
Use product of powers property.
Simplify the expression.
1. 2 2 • 2 5 2. x 3 • x 3
5- (3x) 4 6, Cst 2 ) 2
3- (4 3 ) 2
7. /?(2/?) 3
4. (n 4 ) 3
8 . (3a) 3 (2a) 2
Zero and Negative Exponents
Examples on
pp. 449-451
, 9° = 1
10“ 2 = —
10 2
1
100
Use the definition of zero and negative exponents to simplify expressions.
o° is equal to 1.
10 -2 is the reciprocal of 10 2 .
Evaluate power.
lx 3 y = 7 • • y
Use definition of negative exponents.
Multiply.
kT*T|1|
Chapter Summary and Review
Chapter Summary and Review continued
Evaluate the expression.
9.2° 10.5“ 3 11.
Rewrite the expression with positive exponents.
13.x 6 y“ 6 14.-ZJ 15.
J q
(-7)“ 2
12 -^r
a ~ 2
b - 5
16. (2)?)
-4
Graphs of Exponential Functions
Examples on
pp. 455-457
Graph the function y = 3 X .
Solution Make a table of values that includes both positive and
negative x-values.
X
-2
-1
0
1
2
3
II
CO
*
1
9
1
3
1
3
9
27
Draw a coordinate plane and plot the points
given by the table. Then draw a smooth curve
through the points.
Notice that the graph has a y-intercept of 1, and
that it gets closer to the negative side of the x-axis
as the x-values decrease.
Graph the exponential function.
17. y = 5 X 18. y = 2(3f
19. y
20 . y
Division Properties of Exponents
Use division properties of exponents to simplify expressions.
Examples on
pp. 462-464
j
a -^ =
5 4 -2
= 5 2 = 25
Use quotient of powers property.
(x\ 3
X 3
X 3
b - (3)
~ 3 3 '
A
= 27
Use power of a quotient property.
2 x'y
y 3
_ 2 xY _ x 4
e. 2
• -
4xv 5
4x 3 y 5 2y
Use multiplication and division properties of exponents.
Chapter 8 Exponents and Exponential Functions
Chapter Summary and Review continued^
Simplify the quotient.
Simplify the expression. Use only positive exponents.
~ 9x 6 y 2 __ m 1 3m 2 n 2 __ ( 2 a 4 b 5
y x 6 3 n 4 mn \ 5 a 2 b
Ss 4 t 2 3s 2 f
28 2sV * 2s~ l
Scientific Notation
Examples on
pp. 469-471
You can write numbers in decimal form and in scientific
notation. Use the properties of exponents to perform operations with numbers
in scientific notation.
a. 1.24 X 10 2 = 124
Move decimal point 2 places to the right.
Move decimal point 3 places to the left.
Move decimal point 4 places to the left.
Move decimal point 2 places to the right.
b. 1.5 X 1(T 3 = 0.0015
c. 79 000 = 7.9 X 10 4
d. 0.0588 = 5.88 X 10“ 2
SA 4
e. (7.4 X 10 2 )(5 X 10 3 ) = (7.4 • 5) X (lO 2 • 10 3 )
= 37 X 10 5
= (3.7 x 10 1 ) X 10 5
= 3.7 X 10 6
f 4.25 X 10~ 2 = 425 10~ 2
8.5 X 10 5 8-5 1 0 5
= 0.5 X 10“ 7
= (5 X 10 _1 ) X 10“ 7
= 5 X 10“ 8
Use properties of multiplication.
Use product of powers property.
Write in scientific notation.
Use product of powers property.
Write as a product.
Use quotient of powers property.
Write in scientific notation.
Use product of powers property.
Write the number in decimal form.
29.7 X 10 1 30. 6.7 X 10 3 31. 2 X 10 -4 32. 7.68 X 10" 5
Write the number in scientific notation.
33.52,000,000 34. 63.5 35. 0.009 36. 0.00000023
Perform the indicated operation. Write the result in scientific notation.
37.(5 X 10 4 )(3 X 10 2 ) 38. (4.1 X 10 _1 )(6 X 10 5 ) 39. (1.2 X 10 7 )(l.2 X 10°)
9 X 10 6 4.9 X 10 1 3.4 X 10“ 4
4 °' 3 X 10 3 41 ' 7 X 10“ 8 42 ' 6.8 X 10“ 3
IK L® jg
Chapter Summary and Review
Chapter Summary and Review continued
Exponential Growth Functions
You deposit $1200 in an account that pays 9% interest
compounded yearly. What is the account balance after 8 years?
Solution The initial amount P is $1200, the growth rate r is 0.09, and the
time period ns 8 years. Let A be the account balance.
A = P( 1 + rf Write compound interest model.
= 1200(1 + 0.09) 8 Substitute 1200 for P, 0.09 for r, and 8 for t.
= 1200(1.09) 8 Add.
~ 2391 Use a calculator.
ANSWER ► The balance after 8 years will be about $2391.
FITNESS PROGRAM You start a walking program. You start by walking
2 miles. Then each week you increase your distance 5% per week.
43, Write an exponential growth function to model the situation.
44, How far will you walk in the tenth week?
Exponential Decay Functions
You bought a 32-inch television for $600. The television is
depreciating (losing value) at the rate of 8% per year. What is the value of the
television after 6 years?
Solution The initial value of the television C is $600, the decay rate r is
0.08, and the time ns 6 years. Let y be the value of the television.
y — C(1 — r)* Write exponential decay model.
= 600(1 — 0.08) 6 Substitute 600 for C, 0.08 for r, and 6 for t.
= 600(0.92) 6 Subtract.
~ 364 Use a calculator.
ANSWER ► The value of the television after 6 years will be about $364.
TENNIS CLUB A tennis club had a declining enrollment from 1993 to
2000. The enrollment in 1993 was 125 people. Each year for 7 years, the
enrollment decreased by 3%.
45. Write an exponential decay model to represent the enrollment in each year.
46. Estimate the enrollment in 2000.
Chapter 8 Exponents and Exponential Functions
Examples on
pp. 476-478
Examples on
pp. 482-484
u.
Iiapi^r
Chapter Test
Simplify the expression. Use only positive exponents.
1. x 3 • x 4
2. (a 3 ) 7
3. (2d) 3
4.
5. 9°
6 -PI
7. 8x 2 y“ 4
8.
( mn ) 2
9p~ 3
n
4
Graph the exponential function.
9- y = 2* 10. y = -5(3)* 11. y = (§)* 12. y = 10^)*
13. RADIOACTIVE DECAY The time it takes for a radioactive substance to
decay to half of its original amount is called its half-life. If you start with
16 grams of carbon-14, the number of grams g remaining after h half-life
periods is g = 16(0.5y*. Copy and complete the table and graph the function.
Half-life periods, h
0
1
2
3
4
Grams of carbon-14, g
?
?
?
?
?
Simplify the expression. Use only positive exponents.
14.
5 4
15.
16.
Ty
Zl
x 5
Write the number in decimal form.
18.4 X 10 5 19. 8.56 X 10 3 20. 5 X 10 -2
17.
a l b 2
ah
crb 3
a
-2
21. 6.28 X 10“ 4
Write the number in scientific notation.
22.9,000,000 23. 6550 24. 0.012 25. 0.0000317
26. AMAZON RIVER Each second 4.2 X 10 6 cubic feet of water flow from the
Amazon River into the Atlantic Ocean. How much water flows from the
Amazon River into the Atlantic Ocean each year? HINT: There are about
3.2 X 10 7 seconds in one year.
SAVINGS In Exercises 27 and 28, use the following information.
You deposit $500 in an account that pays 7% interest compounded yearly.
27. Write an exponential growth model to represent this situation.
28. What is the account balance after 7 years?
SALES In Exercises 29 and 30, use the following information.
In 1996 you started your own business. In the first year your sales totaled
$88,500. Each year for the next 5 years your sales decreased by 10%.
29. Write an exponential decay model to represent this situation.
30. Estimate your sales in 2001.
Chapter Test
Chapter Standardized Test
Tip
<^^>CE>ClD
Be aware of how much time you have left, but
keep focused on your work.
1. Simplify the expression 7 4 • 7 7 .
(A) 7 11 CD 7 28
CD 49 11 CD 49 28
2 . Simplify the expression (a 3 ) 4 .
(A) < 3 _1 CD
® a 12 CD a 81
3. Simplify the expression (2x 2 y 3 ) 2 .
(A) 2x 4 y 5 CD 2x 4 y 6
CD 4x 4 y 6 CD 4x 4 y 9
4.
Simplify the expression
2 b 2 c 2
CD
CD
2(2 1
Z? - 2 c 2 ’
b 2 c 2
2a
CD
CD
b 2
lac 2
5. What is the equation of the graph?
® y = 4*
CD y = 5(4)*
® > ={jf
® ?= 5 (?)"
CD none of these
6, Which expression simplifies to x 3 l
— x 2
CD —
— * 2
CD —
X J
i 5
X 5
CD T
X
Chapter 8 Exponents and Exponential Functions
7. Simplify the expression I—
CD y
4 x 2 y 2
8. Simplify the expression ^ •
8xy 3
4y '
CD 2xy 2
CD 2xy 3
CD 2x 2 y 3
CD 2x 2 y 4
9- Which of the following numbers is not
written in scientific notation?
CD 8.62 X 10 4 CD 2.12 X 10
CD 21.2 X 10" 5 CD 9.9132 X 10 _1
1 55 X 10 4
10. Evaluate the expression —-
2.5 X 1(T 3
Write the result in scientific notation.
(D 0.62 X 10 1 CD 0.62 X 10 7
CD 6.2 X 10° CD 6.2 X 10 6
11. You deposit $450 in an account that pays
6% interest compounded yearly. What is
the account balance after 6 years?
CD $471.00 CD $612.00
CD $638.33 CD $2862.00
12. A business had a profit of $42,000 in 1994.
Then its profit decreased by 8% each year
for 6 years. How much did the business
earn in 2000?
CD $11,010 CD $20,160
CD $21,840 CD $25,467
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
Maintaining Skills
i Write the Prime Factorization of a Number
Write the prime factorization of 1078.
Solution
Use a tree diagram to factor
the number until all factors are
prime numbers. To determine
the factors, test the prime
numbers in order.
107* x
/ \
,2 534
X \
1' J .77
il
ANSWER ► The prime factorization of 1078 is 2 • 7 • 7 • 11. This may also be
written as 2 • 7 2 • 11.
Try These
Write the prime factorization of the number.
1.8 2.60 3.105
4. 700
Student ttcCp
t Extra Examples
More examples
and practice
exercises are available at
www.mcdougallittell.com
| 2 Rewrite Improper Fractions as Mixed Numbers
Rewrite the improper fraction as a mixed number.
16
a. 3
Solution
16
a. —
30
30
16-3
Write fraction as a division problem.
5 remainder 1
Divide 16 by 3.
4
Write remainder over divisor to form fraction.
30-4
Write fraction as a division problem.
7 remainder 2
Divide 30 by 4.
7 !
Write remainder over divisor to form fraction.
7 ^
Reduce fraction.
Try These
Rewrite the improper fraction as a mixed number.
5.
21
8
6 .
42
7.
27
15
8.
75
jKTfTV
Maintaining Skills
Quadratic Equations
and Functions
j
What is the path of a home run ball?
APPLICATION: Baseball
A baseball player usually scores a home run by
hitting a ball over the outfield wall. If the ball stays in
the air long enough, and drops in the outfield without
being caught, a batter can score an inside-the-park
home run.
The path of a baseball can be modeled with a quadratic
equation. In Chapter 9 you will use mathematical
models to solve different types of vertical motion
problems.
Think & Discuss
1. Use the graph to approximate the maximum
height the ball reaches.
2 . Use the graph to approximate the maximum
horizontal distance the ball travels.
Learn More About It
You will use a vertical motion model to learn more
about the path of a baseball in Exercise 79 on p. 538.
application link More about baseball is available at
www.mcdougallittell.com
Study Guide
PREVIEW
What’s the chapter about ?
• Evaluating and approximating square roots
• Simplifying radicals
• Solving quadratic equations
• Sketching graphs of quadratic functions and quadratic inequalities
Key Words
- N
•
square root, p. 499
• vertex, p. 521
•
radicand, p. 499
• axis of symmetry, p. 521
•
perfect square, p. 500
• roots of a quadratic equation, p. 527
•
radical expression, p. 501
• quadratic formula, p. 533
•
quadratic equation, p. 505
• discriminant, p. 540
•
quadratic function, p. 520
• quadratic inequalities, p. 547
•
parabola, p. 520
S_
PREPARE
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to p. 222)
1. Complete: The ? of the line
shown at the right is 1.
(A) origin (IT) v-intercept
Co) ^-intercept Co) slope
Skill Check (refer to pp. 15, 95, 367)
2 . Evaluate the expression 3x 2 — 108 when x = —4.
(A) -184 CD -156 ® -120 CD “60
3. Which ordered pair is a solution of the inequality 3x + 4y < 5?
(3) (0,3) CD (“1,2) C©(“2,2) (D) (1,1)
STUDY TIP
Explain Your Ideas
Talking about math and
explaining your ideas
to another person can
help you understand a
topic better.
e s, 9" of a prodm
1 know the squlre^f Tposit^mb 3 '^ P ° SitiVl
In Chapter 2 we learnt C t ^ ,S positive -
![ lf has an even number of neoatC^ ' S positive
the square of a negative nlmhCl actors - Si ^e
factors, it is positive also. ” ^ tW0 ne 9 af ive
Chapter 9 Quadratic Equations and Functions
Square Roots
Goal
Evaluate and
approximate square
roots.
Key Words
• square root
• positive square root
• negative square root
• radicand
• perfect square
• radical expression
How many squares are on each side of a chessboard?
T * - 1 ' ' 1 -M..-- —
A chessboard is a large square
made up of 64 small squares. In
Exercises 84 and 85, you will use
square roots to investigate whether
game boards of other sizes can
be constructed.
You know how to find the square of a number. For instance, the square of 3 is
3 2 = 9. The square of —3 is also 9. In this lesson you will learn about the inverse
operation: finding a square root of a number.
SQUARE ROOT OF A NUMBER If b 2 = a , then b is a square root of a.
Examples: 3 2 = 9, so 3 is a square root of 9.
(— 3) 2 = 9, so — 3 is a square root of 9.
All positive real numbers have two square roots: a positive square root
(or principal square root) and a negative square root. Square roots are written
with a radical symbol V~. The number or expression inside a radical symbol is the
radicand. In the following example, 9 is the radicand. As shown in part (a), the
radical symbol indicates the positive square root of a positive number.
wmzmrn *
Read Square Root Symbols
Write the equation in words.
Student McCp
► Reading Algebra
The symbol ± is read
as "plus or minus" and
is used to write the
positive and negative
square roots of a
positive number. .
a. V9 = 3
Solution
Equation
a. V9 = 3
b. -V9 =
► c. ±V9 =
-3
±3
b. —V9 = -3 c. ±V9 = ±3
Words
The positive square root of 9 is 3.
The negative square root of 9 is —3.
The positive and negative square roots of 9 are 3 and
-3.
Read Square Root Symbols
3. —Vl6 = -4 4. ±V36 = ±6
Write the equation in words.
1.V4 = 2 2 . V25 = 5
9.1
Square Roots
NUMBER OF SQUARE ROOTS Positive real numbers have two square roots.
Zero has only one square root: zero. Negative numbers do not have real square
roots because the square of every real number is either positive or zero.
Student MeCp
► Reading Algebra
Since negative
numbers do not have
real squar e roo ts, we
say that V-64 is
undefined.
v _ J
2 Find Square Roots of N umbers
Evaluate the expression.
a. V64 b. —V64 c. ±V64
d. VO
Solution
a. V64 = = 8
Positive square root
b. -V64 = -V8 5 = -8
Negative square root
c. ±V64 = ±\l& = ±8
Two square roots
d. VO = 0
Square root of zero is zero.
Find Square Roots of Numbers
Evaluate the expression.
5. ±VT00 6. -V25
7. V36
8 . Vl6
The square of an integer is called a perfect square. Of course a square root of a
perfect square is an integer. On the other hand, if n is a positive integer that is not
a perfect square, then it can be shown that Vn is an irrational number. An
irrational number is a number that is not the quotient of integers. In Lesson 12.9
you will use an indirect proof to prove that \fl is an irrational number.
V4 = 2 4 is a perfect square. V4 is an integer.
V2 ~ 1.414 2 is not a perfect square. V2 is neither an integer nor a
rational number.
Student Hedp
>
► Study Tip
You can use a
calculator or the Table
of Square Roots on
p. 801 to approximate
an irrational square
root.
v _/
3 Evaluate Square Roots of Numbers
Evaluate the expression. Give the exact value if possible. Otherwise,
approximate to the nearest hundredth.
a. -V49 b. V3
Solution a. -V49 = -V7^ = -7 49 is a perfect square.
b. V3 ~ 1.73 Round to nearest hundredth.
Evaluate Square Roots of Numbers
Evaluate the expression. Give the exact value if possible. Otherwise,
approximate to the nearest hundredth.
9 . VT00 10. -V5 II.V 23 12. -V 81
Chapter 9 Quadratic Equations and Functions
RADICAL EXPRESSIONS An expression written with a radical symbol is called
a radical expression, or sometimes just a radical.
■iTiuidia al Evaluate a Radical Exoression
Evaluate Vb 2 — 4ac when a = 1, b =
—2, and c = —3.
Solution
The radical symbol is a grouping symbol. You must evaluate the expression
inside the radical symbol before you find the square root.
\lb 2 -4 ac = V(— 2) 2 - 4(1)(— 3)
Substitute values for a, b, and c.
= V4 + 12
Simplify.
= Vl6
Add.
= 4
Find the positive square root.
Evaluate a Radical Expression
Evaluate Vfc 2 - 4ac for the given values.
13- a = 2, b = 3, c = —5 14- a = — 1, b = 8, c = 20
P Student HaCp
^ —\
► Keystroke Help
To find the square root
of 3 on your calculator
you may need to press
D El ° r
. Test your
calculator to find out
which order it uses.
_>
BSES3S 5 Use a Calculator to Evaluate an Expression
1 ± 2V3
Use a calculator to evaluate---. Round the results to the
nearest hundredth.
Solution
When the symbol ± precedes the radical, the expression represents
two different numbers.
KEYSTROKES
m 1 m 2 03
4
4
DISPLAY
U.116055M0M1
I-D.6T6655MD31
ANSWER ► The expression represents 1.12 and —0.62.
Use a Calculator to Evaluate an Expression
B Use a calculator to evaluate the expression. Round the results to the
nearest hundredth.
15. 6±V5 16. 4±V8
17.
2 ± V3
3
18.
2 ± 3V6
4
9.1
Square Roots
Exercises
Guided Practice
Vocabulary Check 1. Complete: Since (—2) 2 = 4, —2 is a ? of 4.
2 . State the meaning of the symbols \T, and ± V~ when applied to a
positive number n.
3. Identify the radicand in the equation V4 = 2.
Skill Check Evaluate the expression.
4. V81 5. ±Vl2T 6. -V36 7. -V5
Determine whether each expression is rational or irrational.
8 . V25 9 . V6 10. V100 n.vTo
Use a calculator or a table of square roots to evaluate the expression.
Round the results to the nearest hundredth.
12. 1 ± V2 13. 6 ± 5V3 14. 3 ± V7 15. 2 ± 4V8
Practice and Applications
Student HeCp
► Homework Help
Example 1: Exs. 16-24
Example 2: Exs. 25-40
Example 3: Exs. 53-64
Example 4: Exs. 65-74
Example 5: Exs. 75-83
\ _ J
READING SQUARE ROOT SYMBOLS Write the equation in words.
16. V625 = 25
17. ±Vl6 =
= ±4
18. ±V4 = ±2
19. V225 = 15
20. -VT21
= -11
21. -V289 = -17
22. V49 = 7
23. Vi = 1
24 /f = —
V 9 3
FINDING SQUARE ROOTS Evaluate the expression. Check the results by
squaring each root.
25. Vl44
26. ±V25
27. Vl96
28. ±V900
29. ±V49
30. VO
31. -V256
32. -VTOO
33. V400
34. -V225
35. Vl2T
36. V289
37. -Vl
38. ±V81
39. Vl69
40. -V625
PERFECT SQUARES Determine whether the number is a perfect square.
41. 10
42. 81
43. -5
44. 120
45. 16
46. 1
47. 111
48. 225
49. -4
50. 10,000
51 -f
52 l
T
Chapter 9 Quadratic Equations and Functions
Student HeCp
► Homework Help
Extra help with
"'4t h/ problem solving in
Exs. 53-64 is available at
www.mcdougallittell.com
EVALUATING SQUARE ROOTS Evaluate the expression. Give the exact
value if possible. Otherwise, approximate to the nearest hundredth.
53. V5
57. -V49
61. ±Vl5
54. V25
58. ±V70
62. -V400
55. Vl3
59. ±Vl
63. -V20
56. -VT 25
60. VTo
64. ±Vl44
EVALUATING RADICAL EXPRESSIONS Evaluate Vib 2 - 4ac for the given
values.
65. a = A, b = 5, c = 1 66. a = 2, b = 4, c = —6
67. a = -2, b = 8, c = -8 68. a = -5, b = 5, c = 10
EVALUATING RADICAL EXPRESSIONS Evaluate the radical expression
when a = 2 and b = 4.
69.
72.
\fb 2 + 10a
Vfc 2 + 42a
a
70. \fb 2 - 8a
73.
10 + 2Vb
a
71. Va 2 + 45
74.
36 - V8 a
b
B EVALUATING RADICAL EXPRESSIONS Use a calculator to evaluate
the expression. Round the results to the nearest hundredth.
75. 8 ± V5 76. 2 ± 5V3 77. -6 ± 4 V 2
Link to
History
CHESS This illustration of
Spanish women playing
chess is from a thirteenth
century manuscript written
for the King of Spain.
Historians believe the game
of chess originated in India in
the seventh century.
78.
81.
1 ± 6V8
6
5 ± 6V3
3
79.
82.
7 ± 3V2
-1
3 ± 4V5
4
80.
83.
4 ± 7V3
2
7 ± 3Vl2
-6
CHESSBOARD A chessboard has 8 small squares on a side and therefore
has a total of 64 small squares.
84. Could a similar square game board be constructed that has a total of 81
small squares?
85. If a square game board has a total of m small squares of equal size, what
can you say about ml
LOGICAL REASONING In Exercises 86-88, determine whether the
statement is true or false. If it is true, give an example. If it is false,
give a counterexample.
86 . All positive numbers have two different square roots.
87. No number has only one square root.
88 . Some numbers have no real square root.
89. CHALLENGE Evaluate 3 ± V(-3) 2 - 4(0.5)(-8).
9.1 Square Roots
Standardized Test
Practice
90. MULTIPLE CHOICE Evaluate
(a) —70 and 80
Cc) 20 and 30
15 ± 5V225
CD —20 and 30
CD 70 and 80
91. MULTIPLE CHOICE Which is an example of a perfect square?
CD -100 CD 10 CD 121 CD 150
Student HeCp
►Test Tip
Square each integer to
find which perfect
squares 200 falls
between to help you
estimate V200 in
Exercise 92.
I _
92. MULTIPLE CHOICE Which two consecutive integers does V200 fall
between?
(A) 10 and 11 CD 13 and 14
CD 14 and 15 CD 19 and 20
93. MULTIPLE CHOICE If a 2 = 36 and b 2 = 49, choose the greatest possible
value for the expression b — a.
CD -13 CD-I CD 1 CD 13
GRAPH AND CHECK Graph the linear system and estimate a solution.
Then check your solution algebraically. (Lesson 7.1)
94.)/ = -3 95. 2x — 4y — 12 96. 2x - y = 10
x — 4 y — —2 x + y — 5
97. BASKETBALL TICKETS The admission price for a high school basketball
game is $2 for students and $3 for adults. At one game, 324 tickets were
sold and $764 was collected. How many students and adults attended the
game? (Lesson 7.2)
98. FLOWERS You are buying a combination of irises and lilies for a flower
arrangement. The irises are $4 each and the lilies are $3 each. You spend
$50 for an arrangement of 15 flowers. How many of each type of flower did
you buy? (Lesson 7.2)
LINEAR COMBINATIONS Use linear combinations to solve the system of
linear equations. (Lesson 7.3)
99. 10* - 3y = 17 100. 12* -4 y= -32 101. 8x - 5y = 70
-lx 4- y = 9 x + 3y = 4 2x + y = 4
Maintaining Skills FRACTIONS AND DECIMALS Write the fraction as a terminating or
repeating decimal. (Skills Review p. 767)
102. |
AO 8
103 i5
... 6
104. —
11
105 1
106. |
i07 4
108. |
6
109. |
110 . f
lll.f
112. |
9
113 ' To
Chapter 9 Quadratic Equations and Functions
Solving Quadratic Equations
by Finding Square Roots
Goal
Solve a quadratic
equation by finding
square roots.
Key Words
• quadratic equation
• leading coefficient
How long does it take for an egg to drop?
An egg is placed in a container
and dropped from a height of
32 feet. Can you tell how long it
will take the egg to reach the
ground? In Example 5 you will
use a quadratic equation to find
the answer.
A quadratic equation is an equation that can be written in the standard form
ax 2 + bx + c — 0, where a A 0; a is called the leading coefficient.
When b = 0, this equation becomes ax 2 + c = 0. One way to solve a quadratic
equation of the form ax 2 + c = 0 is to isolate x 2 on one side of the equation.
Then find the square root(s) of each side. In Example 3 you will see how to use
inverse operations to isolate x 2 .
Student HeCp
1^ V
► Study Tip
Remember that
squaring a number and
finding a square root
of a number are
inverse operations,
v _ j
i Solve Quadratic Equations
Solve the equation. Write the solutions as integers if possible. Otherwise, write
them as radical expressions.
a. x 2 = 4 b. n 2 = 5
Solution a. x 2 = 4 Write original equation.
x = ±V4 Find square roots.
x = ±2 2 2 = 4and(-2) 2 = 4
ANSWER ► The solutions are 2 and —2.
b- n 2 — 5 Write original equation.
n = ±V5~ Find square roots.
ANSWER ^ The solutions are V5~ and — V5~.
Solve Quadratic Equations
Solve the equation. Write the solutions as integers if possible. Otherwise,
write them as radical expressions. Check the results by squaring each root.
1. x 2 — 81 2. y 2 = 11 3- n 2 = 25 4. x 2 = 10
9.2 Solving Quadratic Equations by Finding Square Roots
T
IESImEU 2 Solve Quadratic Equations
Solve the equation.
a. x 2 = 0 b. y 2 = — 1
Solution a. x 2 — 0 Write original equation.
x = 0 Find square roots.
ANSWER ► The only solution is zero.
b. y 2 = — 1 has no real solution because the square of a real
number is never negative.
ANSWER ► There is no real solution.
Student HeCp
► More Examples
More examples
are available at
www.mcdougallittell.com
=# 3 Rewrite Before Finding Square Roots
Solve 3x 2 - 48 = 0.
Solution 3x 2 - 48 = 0
3x 2 = 48
x 2 = 16
x = ±Vl6
x = ±4
Write original equation.
Add 48 to each side.
Divide each side by 3.
Find square roots.
4 2 = 16 and (-4) 2 = 16
ANSWER ► The solutions are 4 and —4. Check both solutions in the original
equation.
CHECK / 3(4) 2 - 48 1 0 3(16) - 48 = 0 /
3(—4) 2 - 48 i 0 3(16) - 48 = 0 /
Both 4 and —4 make the equation true, so 3x 2 — 48 = 0 has two solutions.
Rewrite Before Finding Square Roots
Solve the equation.
5. x 2 — 1 = 0 6- 2x 2 — 72 = 0
7. 27 - 3y 2 = 0
As Examples 1, 2, and 3 suggest, a quadratic equation can have no real solution,
one solution, or two solutions.
Solving x 2 = d by Finding Square Roots
• If d > 0, then x 2 = d has two solutions: x = ±Vd. (Examples 1 and 3)
• If d = 0, then x 2 = d has one solution: x = 0. (Example 2a)
• If d < 0, then x 2 = d has no real solution. (Example 2b)
Chapter 9 Quadratic Equations and Functions
Student HcCp
► Study Tip
The negative square
root, -V2 , does not
make sense in this
situation, so you can
ignore that solution.
^ _ J
FALLING OBJECT MODEL When an object is dropped, the speed with
which it falls continues to increase. Ignoring air resistance, its height h
can be approximated by the falling object model.
Falling object model: h — —16 1 2 + s'
Here h is measured in feet, t is the number of seconds the object has fallen, and
s is the initial height from which the object was dropped.
4 Write a Falling Object Model
An engineering student is a contestant in an egg dropping contest. The goal is
to create a container for an egg so it can be dropped from a height of 32 feet
without breaking. Write a model for the egg’s height. Disregard air resistance.
Solution
The initial height is s = 32 feet.
h = — 16 1 2 + s Write falling object model.
h = —16 1 2 + 32 Substitute 32 for 5 .
ANSWER ► The falling object model for the egg is h = —16 t 2 + 32.
5 Use a Falling Object Model
How long will it take the egg container in Example 4 to reach the ground?
Round your solution to the nearest tenth.
Solution
Ground level is represented by h = 0 feet. To find the time it takes for the egg
to reach the ground, substitute 0 for h in the model and solve for t.
h = -16 1 2 + 32
0 = -16 1 2 + 32
-32 = -16 1 2
Write falling egg model from Example 4.
Substitute 0 for h.
Subtract 32 from each side.
2 = r 2
>±V2 = t
1.4
Divide each side by -16.
Find square roots.
Use a calculator or table of square roots to
approximate the positive square root of 2.
ANSWER ► The egg container will reach the ground in about 1.4 seconds.
Write and Use a Falling Object Model
Suppose the egg dropping contest in Example 4 requires the egg to be
dropped from a height of 64 feet.
8_ Write a falling object model for the egg container when s = 64.
9_ According to the model, how long will it take the egg container to reach
the ground?
9.2 Solving Quadratic Equations by Finding Square Roots
Exercises
Guided Practice
Vocabulary Check
1. Is 2x — 7 = 15 a quadratic equation? Explain why or why not.
2 . Write lx 2 = 12 + 3x in standard form. What is the leading coefficient?
Skill Check
Determine the number of real solutions for each equation.
3. x 2 = 6 4. x 2 = 0 5. x 2 = — 17
6.x 2 - 8 = -8
7.x 2 - 15 = 5
8. x 2 + 2 = -2
Solve the equation or write no real solution.
9. y 2 = 49 10 .x 2 =-16
12. 3x 2 - 20 = -2 13. 5X 2 = -25
11. n 2 = 7
14. 2x 2 — 8 = 0
FALLING OBJECTS Use the falling object model, h = -16f 2 + s. Given
the initial height s, find the time it would take for the object to reach the
ground, disregarding air resistance. Round the result to the nearest tenth.
15. s = 48 feet 16 . s = 160 feet 17. s = 192 feet
Practice and Applications
QUADRATIC EQUATIONS Solve the equation or write no real solution.
Write the solutions as integers if possible. Otherwise, write them as
radical expressions.
18. x 2 = 9 19. m 2 = 1 20. x 2 = 17 21. k 2 = -44
22 . y 2 = 15
26. t 2 = 39
30. y 2 = 400
23. x 2 = 225
27. x 2 = 256
31.x 2 = 64
24. r 2 = — 81 25.x 2 =121
28. y 2 = 0 29. n 2 = 49
32. m 2 = -9 33. x 2 = 16
QUADRATIC EQUATIONS Solve the equation or write no real solution.
Write the solutions as integers if possible. Otherwise, write them as
radical expressions.
f Student HeCp
► Homework Help
Example 1: Exs. 18-33
Example 2: Exs. 18-33
Example 3: Exs. 34-48,
50-55
Example 4: Ex. 59
Example 5: Ex. 60
1
34. 5x 2 = 500
37. a 2 + 3 = 12
40. 2s 2 — 5 = 27
43. 5X 2 + 5 = 20
46. m 2 — 12 = 52
35. 3x 2 = 6
38. x 2 - 7 = 57
41. 3x 2 - 75 = 0
44. 5 1 2 + 10 = 135
47. 2 y 2 + 13 = 41
36. 5y 2 = 25
39. x 2 + 36 = 0
42. lx 2 + 30 = 9
45. 3x 2 - 50 = 58
48. 20 - x 2 = 4
Chapter 9 Quadratic Equations and Functions
49. ERROR ANALYSIS Find
and correct the error at
the right.
B SOLVING EQUATIONS Use a calculator to solve the equation. Round
the result to the nearest hundredth.
50.4x2- 3 = 57 51. 6y 2 + 22 = 34
53. 3X 2 + 7 = 31 54. In 2 - 6 = 15
52. 2x 2 — 4 = 10
55. 5X 2 - 12 = 5
LOGICAL REASONING In Exercises 56-58, decide whether the
statement is true or false. If it is true, give a reason. If it is false, give
a counterexample.
56. x 2 = c has no real solution when c < 0.
57. x 2 = c has two solutions when c > 0.
58. x 2 — c has no solution when c — 0.
FALLING ROCK In Exercises 59 and 60, a boulder falls off the top of an
overhanging cliff during a storm. The cliff is 96 feet high. Find how long it
will take for the boulder to hit the road below.
59. Write a falling object model when s = 96.
60. Solve the falling object model for h = 0. Round to the nearest tenth.
Link to
Careers
MINERALOGISTS Study the
properties of minerals. The
Vickers scale applies to thin
slices of minerals that can be
examined with a microscope.
You can read more
' about mineralogists at
www.mcdougallittell.com
Science La In Exercises 61-66, use the following information.
Mineralogists use the Vickers scale to measure the hardness of minerals. The
hardness H of a mineral can be determined by hitting the mineral with a pyramid¬
shaped diamond and measuring the depth d of the indentation. The harder the
mineral, the smaller the depth of the indentation. A model that relates mineral
hardness with the indentation depth (in millimeters) is Hd 2 = 1.89.
Use a calculator to find the depth of the indentation for the mineral with
the given value of H. Round to the nearest hundredth of a millimeter.
61. Graphite: H = 12 62. Gold: H = 50 63. Galena: H = 80
9.2 Solving Quadratic Equations by Finding Square Roots
Standardized Test
Practice
Mixed Review
Maintaining Skills
History Link, In Exercises 67 and 68, use the following information.
Population estimates for the 1800s lead a student to model the population of the
United States by P = 5,500,400 + 683,300 1 2 , where t — 0, 1, 2, 3,.. . represents
the years 1800, 1810, 1820, 1830,....
67. Use this population model to estimate the United States population in 1800,
1850, and 1900.
68 . Use this model to estimate the year in which the United States population
reached 50 million.
69. (MULTIPLE CHOICE Which quadratic equation is written in standard form?
(A) 8x + 5x 2 — 9 = 0 Cb) 5x 2 + 8x = 9
<3D 5x 2 + 8x - 9 = 0 (D) 9 - 8x - 5x 2 = 0
70. MULTIPLE CHOICE Consider the equation 3x 2 - 44 = x 2 + 84.
Which statement is correct?
CD The equation has exactly one solution.
CD The equation has two solutions.
Ch) The equation has no real solution.
CD The number of solutions cannot be determined.
EVALUATING EXPRESSIONS Evaluate the expression when x = -2.
(Lessons 1.3 , 2.3 , 2.5)
71.2x 3 + 2x + 2 72. 4x 2 + 3x + 5 73. 3x 2 + 4x + 8 74. x 2 + lx + 9
SLOPE AND Y-INTERCEPT Find the slope and /'intercept of the graph of
the equation. (Lesson 4.7)
75. y = 5x + 6 76. y = —4x + 5 77. y — 8x = 2 78. 2x + 3y = 6
SOLVING AND GRAPHING Solve the inequality. Then graph the solution.
(Lesson 6.1)
79. -9<x-7 80. —15 >jc — 8 81.2 + x<4 82. 6>x+l
SCIENTIFIC NOTATION Write the number in scientific notation.
(Lesson 8.5)
83.0.0000008 84.564 85.8721 86.23,000
SIMPLIFYING FRACTIONS Write the fraction in simplest form.
(Skills Review p. 763)
87. |
88. |
83 V5
™ 30
®°-48
20
91 -24
M 12
93 ‘ 16
28
M -35
Chapter 9 Quadratic Equations and Functions
Simplifying Radicals
Goal
Simplify radical
expressions.
Key Words
• radical
• simplest form of a
radical expression
• product property of
radicals
• quotient property of
radicals
What is the maximum speed of a sailboat?
ill' fft ’ — 1 i
S'
1 4
; '
1 ■' JHB
* V S55
;J 1 ;
‘“J ^
t \
^■1
* \
The design of a sailboat
affects its maximum speed.
In Example 4 you will use a
boat’s water line length to
estimate its maximum speed.
The simplest form of a radical expression is an expression that has no perfect
square factors other than 1 in the radicand, no fractions in the radicand, and no
radicals in the denominator of a fraction. Properties of radicals can be used to
simplify expressions that contain radicals.
PRODUCT PROPERTY OF RADICALS
Vab = Va • Vb where a > 0 and b > 0 Example: V4 • 5 = V4 • V5 = 2Vb
_ /
Student MeCp
—\
► Study Tip
There can be more
than one way to factor
the radicand. An
efficient method is to
find the largest perfect
square factor. For
example, you can
simplify V48 using
V48 = Vi 6 • 3 =
Vl6 • V3 = 4V3. ••••
\ _ J
8222ESB 1 Simplify with the Product Property
Simplify the expression,
a. V50 b. V48
Solution Look for perfect square factors to remove from the radicand.
a. V50 = V25 • 2
Factor using perfect square factor.
= V25 • V2
Use product property.
= 5V2
Simplify: V25 = 5.
>b. V48 = V4 • 12
Factor using perfect square factor.
= V4 • 4 • 3
Factor using perfect square factor.
= V42 • V3
Use product property.
= 4V3
Simplify: = 4.
Simplify with the Product Property
3. V75 4. Vl80
Simplify the expression.
1.VT2 2. V32
9.3 Simplifying Radicals
QUOTIENT PROPERTY OF RADICALS
Student HeCp
► More Examples
More examples
are available at
www.mcdougallittell.com
2 Simplify with the Quotient Property
Simplify
(32
50'
Solution
2 * 16
2 • 25
Vl6
V25
4
5
Factor using perfect square factors.
Divide out common factors.
Use quotient property.
Simplify.
In Example 3 you will see how to eliminate a radical from the denominator by
multiplying the radical expression by an appropriate value of 1. This process is
called rationalizing the denominator.
Student Hedp
^
► Study Tip
1 , V 2
and -y are
equivalent radical
expressions. The
second expression is
in simplest form with a
rational denominator.
K _/
3 Rationalize the Denominator
Simplify
Solution
VI
Vis
1
V9 • V2
1
3V2
1 # V2
3V2 *
V2
6
Use quotient property.
Use product property.
Remove perfect square factor.
V2
Multiply by a value of 1: - 1.
Simplify: 3V2 • V2 = 3 • 2 = 6.
Simplify with the Quotient Property
Simplify the expression.
Chapter 9 Quadratic Equations and Functions
Link_
SaiC'mg
BOAT SPEED Mathematical
formulas help designers
choose dimensions for a
boats water line length, sail
area, and displacement that
will produce the greatest
speed.
More about sailing
is available at
www.mcdougallittell.com
Simplify a Radical Expression
BOAT SPEED The maximum speed s' (in knots, or nautical miles per hour)
that certain kinds of boats can travel can be modeled by the quadratic equation
? 16
s = -g-x, where v is the boat’s water line length (in feet).
The water line of a boat is the
line on the main body of the
boat that the surface of the
water reaches.
Use this model to express the maximum speed of a sailboat with a 32 foot
water line in terms of radicals. Then find the speed to the nearest tenth.
Solution s 2
s 2 =
16
— . 32
9
/f.32
s =
Vl6
V9
• V32
I-4V5
16V2
• 7.5
Write quadratic model.
Substitute 32 for x.
Find square root of each side.
Use quotient and product properties.
Remove perfect square factors from radicands.
Multiply.
Use a calculator or square root table.
ANSWER ► The sailboat’s maximum speed is knots, or approximately
7.5 knots.
Simplify a Radical Expression
9. Use the model in Example 4 to express the maximum speed of a sailboat
with a 50 foot water line in terms of radicals. Then find the speed to the
nearest tenth.
Simplest Form of a Radical Expression
• No fractions are in the radicand.
• No radicals are in the denominator of a fraction.
V8
V4 • 2
2 V 2
r y
V5
V5
V 16
Vl6
4
1
1 V7
V7
V7
V7 ’ V7
7
9.3 Simplifying Radicals
-L3 Exercises
Guided Practice
Vocabulary Check
Determine whether the radical expression is in simplest form. Explain.
1.fV2
2 ^
V 16
3. 5V40
4 ‘ V2
Skill Check
Match the radical expression with its simplest form.
5. V45
6. V98
7. V75
8. V54
A. 3V6
B. 5V3
c. 7 V 2
D. 3V5
Simplify the expression.
9. V36
10. V 24
11 . V60
[fA
i2 -vf
13 M
1J - V 16
14. |V20
15
1!> - V5
W. 9 JI
Practice and Applications
SIMPLEST FORM Determine whether the radical expression is in simplest
form. Explain.
17. ^ 18. 3V20 19. 5V3T
20 .
[2
V 8
PRODUCT PROPERTY Simplify the expression
21.V44 22. V54 23. Vl8
25. V27 26. V63 27. V200
29. VI25 30. Vl32 31.V144
24. V56
28. V90
32. Vl96
Student HeCp
► Homework Help
Example 1: Exs. 21-32,
59-74
Example 2: Exs. 33-44,
59-74
Example 3: Exs. 47-58,
59-74
Example 4: Exs. 75, 76
QUOTIENT PROPERTY Simplify the expression.
ERROR ANALYSIS In Exercises 45 and 46, find and correct the error.
Chapter 9 Quadratic Equations and Functions
RATIONALIZING THE DENOMINATOR Simplify the expression.
SIMPLIFYING Write the radical expression in simplest form.
59. 4V25
60. 9VT00
61. -2V27
62. |V63
63. -6V4
64. 3V44
65. -yV^9
66. |V32
67. |V24
68. |V56
69. -|V360
/48
70 - V si
71 /^-~
,l- V35
72 —4 —
V 10
CO
rs
74 - 2 iI
TSUNAMI In Exercises 75-77, use the following information. A tsunami is
a destructive, fast-moving ocean wave that is caused by an undersea earthquake,
landslide, or volcano. Scientists can predict arrival times of tsunamis by using
water depth to calculate the speed of a tsunami.
A model for the speed s (in meters per second) at which a tsunami moves is
s = Vgd where d is the depth (in meters) and g is 9.8 meters per second
per second.
Asia
8 h
• 4 h
6 h Pacific Ocean
2 h
*
^Hawaii
• 2 h
North
l America
\
* :
• • • # ••
#
10 h *
Water elevation stations
4 h
6 h
8 h
10 h
South
‘AAierica
*
12 h*
Tsunami travel times
(in hours) to Hawaii
75. Find the speed of a tsunami in a region of the ocean that is 1000 meters deep.
Write your solution in simplest form.
76. Find the speed of a tsunami in a region of the ocean that is 4000 meters deep.
Write your solution in simplest form.
77. CRITICAL THINKING Is the speed of a tsunami in water that is 4000 meters
deep four times the speed of a tsunami in water that is 1000 meters? Explain
why or why not.
9.3 Simplifying Radicals
Student HeCp
► Skills Review
For help with finding
the area of geometric
figures, see p. 772.
^ _ )
Standardized Test
Practice
Geometry Lk In Exercises 78 and 79, use the formula A = SLwXo find
the area of the figure. Write your solution in simplest form.
78.
n
79 .
VTo
_C
V20
7V2
80. Find the length of a side s of a square that has the same area
as a rectangle that is 12 centimeters wide and 33 centimeters long. Write
your solution in simplest form.
s
s
12 cm
33 cm
81. LOGICAL REASONING Copy
and complete the proof of the
following statement:
If = 9, then x — ±6.
Solution Step
x z =36
x = ±6
Explanation
Original Equation
? Property of Equality
Definition of ? root
CHALLENGE Write the radical expression in simplest form.
82. 3V63 • V4 83. -2V27 • V3 84. V9 • 4V25
85. h/32 • V2 86. -VS • 7S 87. -5V2 • ./Jr
2 V36 V 50
88. MULTIPLE CHOICE Which is the simplest form of V80?
(a) 2 V 5 CD 4V5 CD 2 V 20 CD 20
Vl25
89. MULTIPLE CHOICE Which is the simplest form of -^=-7
CD V 5 CD 2 V 5 CD 5 CD 5 V 5
90. MULTIPLE CHOICE Which of the following does not equal V48?
® V 2 • V 24 CD 2 V 12 CD 4V3 CD 12 V 16
91. MULTIPLE CHOICE Which step would you use to rationalize the
V3
denominator of ^ — ?
CD Multiply by
vTo
VTo'
CG) Multiply by
VTo
V3 •
CD Multiply by VlO.
CD Multiply by 10.
Chapter 9 Quadratic Equations and Functions
Mixed Review
Maintaining Skills
Quiz 1
GRAPHING EQUATIONS Use a table to graph the equation. (Lesson 4.2)
92. y = x + 5 93. x + y = —4 94. y = 3x — 1 95. 2x + y = 6
POWER OF A PRODUCT Simplify the expression. (Lesson 8.1)
96. (5 • 2) 5 97. (3x) 4 98. (-5x) 3 99. (-3 • 4) 2
100 . (a&) 6 101. (8xy) 2 102. (-3 mn) 4 103 .(-abcf
DOMAIN AND RANGE Use the graph to describe the domain and the
range of the function. (Lesson 8.3)
FRACTION OPERATIONS Divide. Write the answer as a fraction or as a
mixed number in simplest form. (Skills Review p. 765)
106 .| = 4 107.|-3 108 .| = |
110. 1 " 5 " 10 111.|-h63 112.J-h!
J 3 O D
109.
113 -To"
_ 8 _
15
7
Evaluate the expression. (Lesson 9.1)
1.V81 2 . — V25 3. Vl6 4. -V4
5 . ±Vl 6 . Vloo 7 . ±V49 8 . Vl 21
Solve the equation or write no real solution. Write the solutions as
integers if possible. Otherwise, write them as radical expressions.
(Lesson 9.2)
9. x 2 = 64 10 . x 2 = 63 11 . — 8x 2 = -48
12. 12x2= -120 13.4x2 = 64 14.5x2 - 44 = 81
Write the expression in simplest form. (Lesson 9.3)
15. Vl 8
16. V60
17. iV75
■a
CO
1
Si
19. 2^120
20 . |V42
V45
21
^ ' 9
00 /~5~
22m V 20
__ l~5
23 -Vi 6
[32
24 —
V 4
IS)
01
w|?ol
9.3 Simplifying Radicals
DEVELOPING CONGE
For use with
Lesson 9.4
Goal
Use reasoning to discover
how the value of a affects
the graph of y = ax 2 .
Materials
• graph paper
• pencil
Question What is the shape of the graph of y — ax 2 and y — -ax 2 ?
In this Developing Concepts, you will explore the shape of a quadratic function
and how the value of the leading coefficient a affects the shape of the graph.
Explore
Q Complete the table of values for y = x 2 . The value of a is 1.
R
-2
-1
0
1
2
3
LI
?
?
?
?
?
?
© Complete the table of values for y = —x 2 . The value of a is — 1.
H
-2
-1
0
1
2
3
L_
?
?
?
?
?
?
The graphs of y = x 2 and y = — x 2 are shown below on the same coordinate
plane. Use them to help you answer the following questions.
Think About It
1. How would you describe the shape of
each graph?
2. In what direction (up or down ) does the
graph of y = x 2 open?
3. Does the graph of y = x 2 have a highest
point or a lowest point?
4. In what direction (up or down) does
the graph of y = —x 2 open?
5- Does the graph of y = — x 2 have a highest point or a lowest point?
6 . Use the tables to compare the values of y for y = x 2 and y = —x 2 .
What is the value of y for each function when x = 2? when x = 0?
when x = — 1?
7- Generalize your results and complete the statement: For every point (k, k 2 )
on the graph of y = x 2 , there is a corresponding point (k, ? ) on the graph
of y — —x 2 .
8- The graph of y = x 2 is a reflection , or mirror image, of the graph of y = —x 2 .
The line of reflection is y = ? .
Chapter 9 Quadratic Equations and Functions
Question What happens to the shape of the graph of y — ax 2
" when \ a\ increases?
Explore
..* ■ 1 ...
O Sketch the graphs of y = ^x 2 , y = x 2 , and y = 2x 2 on the same coordinate
plane by plotting points and connecting them with a smooth curve.
Think About It
1. Do the graphs open up or down?
2 . Identify the lowest point on each graph.
3. Describe how changing the value of a from ^ to 1 to 2 changes the shape of
the graph of y = ax 2 .
You have just explored how the graph of y = ax 2 changes when the value of a is
positive and increases. On page 518 you explored how the graphs of y = ax 2 and
y = —ax 2 are related. Use this information to help you in the next section.
Explore
Q Predict how changing the value of a from — ^ to — 1 to —2 changes the shape
of the graph of y = ax 2 . Check your prediction by sketching the graphs of
y = — ^x 2 , y = —x 2 , and y = — 2x 2 in the same coordinate plane that you
used for the Explore at the top of the page.
Think About It
1. Do the graphs open up or down?
2 . Identify the highest point on each graph.
3- Describe how changing the value of a from — ^ to — 1 to —2 changes the
shape of the graph of y = ax 2 -
4. Generalize your results and complete the statement: As | a | increases, the
graph of y = ax 2 becomes ? .
Determine whether the graph of the function opens up or down and
whether the graph is wider or narrower than the graph of y = x 2 .
5. y = 5x 2 6 - y = —4x 2 7. y = ^-x 2
Developing Concepts
Graphing Quadratic
Functions
Goal
Sketch the graph of a
quadratic function.
Key Words
• quadratic function
• parabola
• vertex
• axis of symmetry
How high was the shot put?
In Exercise 48 you will find
the highest point of a parabola
to estimate the highest point on
the path of a record-breaking
shot-put throw.
A quadratic function is a function that can be written in the standard form
y — ax 2 + bx + c, where a ^ 0.
Every quadratic function has a U-shaped graph called a parabola. As you saw in
Developing Concepts 9.4, pages 518-519, the parabola opens up if the value of a
is positive. The parabola opens down if the value of a is negative.
i Describe the Graph of a Parabola
a. The graph of y = x 2 opens up. b. The graph of y = —x 2 + 4 opens
The lowest point is (0, 0). down. The highest point is (0, 4).
Describe the Graph of a Parabola
Decide whether the parabola opens up or down.
1. y = —x 2 2. y = 2x 2 — 4 3. y = — 3x 2 + 5x — 1
Chapter 9 Quadratic Equations and Functions
The vertex is the highest or lowest point on a parabola. The vertical line passing
through the vertex that divides the parabola into two symmetric parts is called the
axis of symmetry. The two symmetric parts are mirror images of each other.
GRAPHING A QUADRATIC FUNCTION
The graph of / = ax 2 + bx + c, where a ± 0, is a parabola.
step Q Find the x-coordinate of the vertex, which is x = —
step 0 Make a table of values, using x-values to the left and right
of the vertex.
step © Plot the points and connect them with a smooth curve to
form a parabola.
J 2 Graph Quadratic Function with Positive a -Value
Sketch the graph of y = x 2 — 2x — 3.
Solution In this quadratic function, a = 1, b = —2, and c = — 3.
Find the x-coordinate of the vertex. — — = = ^
0 Make a table of values, using x-values to the left and right of x = 1.
—
" 2
-1
0
1
2
3
4
H
Ll
0
-3
-4
-3
0
5
Student HeCp
► Study Tip
If you fold the graph
along the axis of
symmetry, the two
halves of the parabola
will match up exactly.
V j
© Plot the points. The vertex is
(1, —4). Connect the points to
form a parabola that opens up
since a is positive.
The axis of symmetry passes
through the vertex (1, —4). The
x-coordinate of the vertex is 1,
and the axis of symmetry is the
vertical linex = 1.
The axis of symmetry of y = ax 2 + bx + c is the vertical line x =
b_
2a’
m
Graph a Quadratic Function with a Positive a-Value
Sketch the graph of the function. Label the coordinates of the vertex.
4. y = x 2 + 2 5. y = 2x 2 — 4x — 1 6 . y = x 2 + 2x
9.4 Graphing Quadratic Functions
3 Graph Quadratic Function with Negative a-Value
Sketch the graph of y = —x 2 — 3x + 1.
Student HeCp
► Study Tip
If the x-coordinate of
the vertex is a fraction,
you can still choose
whole numbers when
you make a table.
\ _ J
Solution
In this quadratic function, a = — 1 ,b= — 3, and c = 1 .
O Find the x-coordinate of the vertex: — — = ~ 2(-i) = ~2' or ~^2'
This tells you that the axis of symmetry is the vertical line x = — 1^-.
0 Make a table of values, using x- values to the left and right of x = — ly.
D
F
-3
-2
-±
-1
0
1
■
LZ
1
3
3 i
3
1
-3
0 Plot the points. The vertex is
^ — 1^-, 3^j. Connect the points
to form a parabola that opens
down since a is negative.
To find the y-intercept of
y — —x 2 — 3x + 1, letx = 0.
The y-intercept is 1.
Since y — c when x = 0 in y = ax 2 + bx + c, the y-intercept of the graph is c.
Graph a Quadratic Function with a Negative a-Value
Sketch the graph of the function. Label the coordinates of the vertex.
7. y = —x 2 +1 8- y = — x 2 + 3x 9. y = — lx 2 + 4x + 1
v -
Graph of a Qu
The graph of y =
• If a is positiv
• If a is negati\
• The vertex hi
• The axis of s
• The y-interce
L _
>
adratic Function
ax 2 + bx + c is a parabola.
e, the parabola opens up.
/e, the parabola opens down.
as an x-coordinate of —Jr-
2a
ymmetry is the vertical line x =
pt is c.
Chapter 9 Quadratic Equations and Functions
f Exercises
Guided Practice
Vocabulary Check 1 . Identify the values of a , b , and c for the quadratic function in standard form
y — —5x 2 + lx — 4.
2 . What is the U-shaped graph of a quadratic function called?
Skill Check Decide whether the graph of the quadratic function opens up or down.
3. y = x 2 + 4x — 1 4. y = 3x 2 + 8x + 6 5. y = — x 2 + 7x — 3
6- y — —x 2 — 4x + 2 7. y = 5x 2 — 2x + 4 8- y = — 8x 2 — 4
Sketch the graph of the function. Label the coordinates of the vertex.
Write an equation for the axis of symmetry.
9. y = —3x 2 10. y = —5x 2 + 10 11.y = x 2 + 4
12. y = x 2 — 6x + 8 13. y = —3x 2 + 6x + 2 14. y = 2x 2 — 8x + 3
Practice and Applications
DESCRIBING GRAPHS Decide whether the parabola opens up or down.
15. y = lx 2 16. y = ~5x 2 17. y = ~lx 2 + 5
18. y — 5x + 6x 2 — 1 19. y — — 8x 2 — 9 20 . y — 3x 2 — 2x + 7
21. y = —3x 2 + 24x 22. y = —6x 2 — \5x 23. y = 8x — x 2
PREPARING TO GRAPH Find the coordinates of the vertex. Make a table
of values, using x-values to the left and to the right of the vertex.
24. y = 3x 2 25. y = 6x 2 26. y = - I2x 2
27. y = 2x 2 — lOx 28. y = —lx 2 + 2x 29. y = 6x 2 + 2x + 4
30. y = 5x 2 + lOx + 7 31 . v = —4x 2 — 4x + 8 32. y = —x 2 + 8x + 32
■ Student HeCp
^ -
► Homework Help
Example 1: Exs. 15-23
Example 2: Exs. 24-32,
36-44
Example 3: Exs. 24-32,
36-44
\ _ J
GRAPHS OF FUNCTIONS Match the quadratic function with its graph.
9.4 Graphing Quadratic Functions
SKETCHING GRAPHS Sketch the graph of the function. Label the
coordinates of the vertex.
36. y = —2x 2
39. y = 4x 2 + 8x — 3
42. y = 2x 2 + 5x - 3
37. y = 4x 2
40. y = x 2 + x + 4
43. y = —4x 2 + 4x + l
38. y = x 2 + 4x - 1
41. v = 3x 2 — 2x — 1
44. y = —3x 2 —3x + 4
TABLE T
Use a Quadratic Model
S TABLE TENNIS The path of a
table-tennis ball that bounces
over the net can be modeled by
h — —4.9x 2 + 2.07x, where h is the
height above the table (in meters)
and x is the time (in seconds).
Estimate the maximum height
reached by the table-tennis ball.
Round to the nearest tenth.
Solution The maximum height of the table-tennis ball occurs at the vertex
of the parabolic path. Use a = —4.9 and b = 2.07 to find the x-coordinate of
the vertex. Round your solution to the nearest tenth.
b_
2 a
2.07
2(—4.9)
~ 0.2
Substitute 0.2 for x in the model and use a calculator to find the maximum
height.
h = —4.9(0.2) 2 + 2.07(0.2) = 0.218 - 0.2
ANSWER ► The maximum height of the table-tennis ball is about 0.2 meters.
Nature
I
DOLPHINS follow the path
of a parabola when they jump
out of the water.
More about dolphins
is available at
www.mcdougallittell.com
45- BASKETBALL You throw a basketball. The height of the ball can be
modeled by h = — 16 1 2 + 15^ + 6, where h represents the height of
the basketball (in feet) and t represents time (in seconds). Find the vertex
of the graph of the function. Interpret the result to find the maximum
height that the basketball reaches.
k.__
DOLPHINS In Exercises 46 and 47,
use the following information.
A bottle-nosed dolphin jumps out
of the water. The path the dolphin
travels can be modeled by
h = —0.2 d 2 + 2d, where h
represents the height of the dolphin
and d represents horizontal distance.
46- What is the vertex of the
parabola? Interpret the result.
47. What horizontal distance did
the dolphin travel?
Chapter 9 Quadratic Equations and Functions
Student HeCp
► Homework Help
Extra help with
problem solving in
Exercise 48 is available at
www.mcdougallittell.com
Standardized Test
Practice
48. 1 TRACK AND FIELD Natalya Lisovskaya holds the world record for the
women’s shot put. The path of her record-breaking throw can be
modeled by h — — 0.0137x 2 + 0.9325x + 5.5, where h is the height (in feet)
and x is the horizontal distance (in feet). Use a calculator to find the
maximum height of the throw by Lisovskaya. Round to the nearest tenth.
CHALLENGE In Exercises 49-51, sketch the graphs of the three functions
in the same coordinate plane. Then describe how the three parabolas are
similar to each other and how they are different.
49. y = -|x 2 + x + 1
y = -x 2 + x + 1
y = -2x 2 + x + 1
50. y = x 2 + x + 1
y = x 2 + 2x + 1
y = x 2 + 3x + 1
51. y = x 2 — x + 1
y = x 2 — x + 3
y = x 2 — x — 2
52. MULTIPLE CHOICE Which equation is represented by the graph below?
(a) y = x 2 - 2x + 1
CD y = —x 2 — 2x + 1
(C) y = x 2 + 2x + 1
CD y = —x 2 + 2x — 1
53. MULTIPLE CHOICE What are the coordinates of the vertex of the graph of
y = — 2x 2 + 8x — 5?
CD (-2, -29) ® (2, 3) (ED (2, 7) CD (4, -5)
54. MULTIPLE CHOICE What is the axis of symmetry of the graph of
y = x 2 + 3x — 2?
17 3 3 19
( a ) x = — 4- CD x = - 2 CD x = 2 CD * = -4-
Mixed Review GRAPHING A SYSTEM Graph the system of linear inequalities.
(Lesson 7.6)
55. x — 3y > 3 56. x + y < 5 57. x + y < 10
x — 3y < 12 x > 2 2x + y > 10
y>0 x — y <2
PRODUCT OF POWERS Write the expression as a single power of the
base. (Lesson 8.1)
58. 4 2 • 4 5 59. (—5) • (— 5) 8 60. x 2 • x 4 • x 6 61. x 3 • x 5
62. t • (z 3 ) 63. m • m 4 • m 3 64. 5 • 5 2 • 5 3 65. 2(2) 4
Maintaining Skills ORDERING FRACTIONS Write the numbers in order from least to
greatest. (Skills Review p. 770)
66 --— 67 - — 2 6 q_9_73
66. 2 , 3, 12 07. 3 , 15 , 5 08. 5 , 1Q , 15 69. 1Q , - -
9.4 Graphing Quadratic Functions
4.5
Solving Quadratic Equations
by Graphing
Goal
Use a graph to find or
check a solution of a
quadratic equation.
Key Words
• x-intercept
• roots of a quadratic
equation
How far apart are the Golden Gate Bridge towers?
The Golden Gate Bridge in
California hangs from steel cables
that are supported by two towers.
In Example 3 you will use the
graph of a parabola to estimate the
distance between the towers.
The x-intercepts of the graph of y = ax 2 + bx + c are the solutions of the related
equation ax 2 + bx + c = 0. Recall that an x-intercept is the x-coordinate of a
point where a graph crosses the x-axis. At this point, y = 0.
i Use a Graph to Solve an Equation
The graph of y = ^x 2 — 8 is shown
at the right. Use the graph to estimate
the solutions of ^x 2 — 8 = 0.
Solution
The graph appears to intersect the
x-axis at (—4, 0) and (4, 0). By
substituting x = — 4 and x = 4 in
—x z — 8 = 0, you can check that —4
and 4 are solutions of the given equation.
Use a Graph to Solve an Equation
1 - The graph of y = 2x 2 — 4x is shown at the right.
Use the graph to estimate the solutions of
2x 2 — 4x = 0. Check your solutions algebraically
by substituting each one for x in the given equation.
Chapter 9 Quadratic Equations and Functions
ESTIMATING SOLUTIONS BY GRAPHING
The solutions of a quadratic equation in one variable xcan be
estimated by graphing. Use the following steps:
step Q Write the equation in the standard form ax 2 + bx + c = 0.
step © Sketch the graph of the related quadratic function
y = ax 2 + bx + c.
step © Estimate the values of the x-intercepts, if any.
The solutions, or roots, of ax 2 + bx + c = 0 are the x-intercepts of
the graph.
Student HeCp
► More Examples
More examples
-^pv gre ava j| a |3| e a t
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V
J
2 Solve an Equation by Graphing
Use a graph to estimate the solutions of x 2 — x = 2. Check your
solutions algebraically.
Solution
Q Write the equation in the standard form ax 2 + bx + c = 0.
x 2 — x = 2 Write original equation.
x 2 — x — 2 = 0 Subtract 2 from each side.
© Sketch the graph of the related
quadratic function y — x 2 — x — 2.
© Estimate the values of the
v-intercepts. From the graph, the
v-intercepts appear to be — 1 and 2.
CHECK /
You can check your solutions algebraically using substitution.
CHECK x= -1:
x 2 — x = 2
(—l) 2 — (—1) = 2
1+1=2/
ANSWER ► The solutions are — 1 and 2.
CHECK x= 2:
x 2 — x = 2
2 2 - 2 I 2
4-2=2/
Solve an Equation by Graphing
2. Use a graph to estimate the solutions of x 2 — x = 6.
3_ Check your solutions algebraically.
9.5 Solving Quadratic Equations by Graphing
Student HeCp
^
► Study Tip
The lowest point of the
main cables at the
midpoint is about 8 feet
above the roadway.
► Source: Golden Gate
Bridge, Highway and
Transportation District
l J
3 Points on a Parabola
0 The main suspension cables of the Golden Gate Bridge form a parabola
that can be modeled by the quadratic function
y = 0.000112x 2 + 8
where x is the horizontal distance from the middle of the bridge (in feet) and
y is the vertical distance from the road (in feet).
The cables are connected to the towers at points that are 500 feet above the
road. How far apart are the towers?
Solution
You can find the distance between the towers by finding the x-values for which
y = 500, or 0.000112x 2 + 8 = 500. Use a graphing calculator to find the
solutions of the equation.
Write the equation in the standard form ax 2 + bx + c = 0.
0.000112x 2 + 8 = 500 Write original equation.
0.000112x 2 — 492 = 0 Subtract 500 from each side.
Sketch the graph of the related quadratic function
y = 0.000112x 2 — 492 using a graphing
calculator.
Estimate the values of the x-intercepts. From the
graphing calculator screen, you can see that the
x-intercepts are approximately —2100 and 2100.
Each tower is approximately 2100 feet from the
midpoint. Because the towers are on opposite sides
of the midpoint, the distance between the towers is
2100 + 2100 = 4200.
ANSWER ^ The towers are approximately 4200 feet apart.
4. The main suspension cables of the Royal Gorge Bridge can be modeled
by the quadratic function y = 0.0007748x 2 . In the equation, x is the
horizontal distance from the middle of the bridge (in feet) and y is the
vertical distance from the road (in feet). The cables are connected to the
towers at points that are 150 feet above the road. Approximately how far
apart are the towers?
Chapter 9 Quadratic Equations and Functions
r
Exercises
Guided Practice
Vocabulary Check 1 . What are the roots of a quadratic equation?
2. Explain how you can use a graph to check the solutions of a quadratic
equation.
Skill Check
Match the quadratic function with its graph.
Solve the equation algebraically. Check your solutions by graphing.
6. 3x2- 12 = 0 7. 5x2 - 5 = 0 8. -2x 2 = -18
Estimate the solutions of the equation by graphing. Check your solutions
algebraically.
9.3x2 = 48 10.x 2 -4 = 5 11.-x 2 + 7x - 10 = 0
Practice and Applications
WRITING IN STANDARD FORM Write the quadratic equation in standard
form.
12. 4x2=12 13. x 2 — 6x = —6 14. -x 2 = 15
15. 5 + x = 3x 2 16. 2x — x 2 = 1 17. 6 x 2 = 12x
Student HeCp
^Homework Help
Example 1: Exs. 18-21
Example 2: Exs. 22-45
Example 3: Exs. 47-50,
52
v _ j
IDENTIFYING THE ROOTS Use the graph to identify the roots of the
quadratic equation.
18. -x 2 + 3x - 2 = 0
19. — x 2 — 2x + 3 = 0
20 . x 2 — 2x — 8 = 0
21. CHECKING SOLUTIONS Use substitution to check the solutions of the
quadratic equations in Exercises 18-20.
9.5 Solving Quadratic Equations by Graphing
SOLVING GRAPHICALLY Use a graph to estimate the solutions of the
equation. Check your solutions algebraically.
22. x 2 + 2x = 3
23. —4x 2 — 8x = —12
24. —x 2 + 3x = —4
25. 2x 2 + 4x = 6
26. 3X 2 + 3x = 6
27. x 2 - 4x - 5 = 0
28. x 2 — x = 12
29. —x 2 — 4x = —5
30. x 2 + x = 2
31. —x 2 — x + 6 = 0
32. 2x 2 — 8x = 10
33. —x 2 + x = —2
CHECKING GRAPHICALLY Solve the equation
solutions by graphing.
algebraically. Check your
34. lx 2 = 32
35. Ax 2 = 100
36. Ax 2 = 16
37.x 2 - 11 = 14
38. x 2 - 13 = 36
39. x 2 - 4 = 12
40.x 2 - 53 = 11
41.x 2 + 37 = 118
42. lx 2 - 89 = 9
43. 2x 2 + 8 = 16
44. 3x 2 + 5 = 32
45. 2x 2 - 7 = 11
Link to
Science
MICROGRAVITY
Researchers can investigate
the effects of microgravity
aboard an airplane. A plane
can attain low gravity
conditions for 15-second
periods by repeatedly flying in
a parabolic path.
46. SWISS CHEESE The consumption of
Swiss cheese in the United States
from 1970 to 1996 can be modeled by
P = -0.002 t 2 + 0.056 1 + 0.889.
P is the number of pounds consumed
per person and t is the number of
years since 1970.
According to the graph of the model,
in what year would the consumption
of Swiss cheese drop to 0? Is this a
realistic prediction?
B APPROXIMATING SOLUTIONS Use a graphing calculator to
approximate the solutions of the equation.
47. — x 2 — 3x + 4 = 0 48. x 2 + 6x — 7 = 0
49. ~x 2 + 2x + 16 = 0 50. |x 2 + 15x + 40 = 0
Scienc e Link > In Exercises 51 and 52, use the following information.
Scientists use a state of free fall to simulate a gravity-free environment called
micro,gravity . In microgravity conditions, the distance d (in meters) that an object
that is dropped falls in t seconds can be modeled by the equation d = 4.9 t 2 .
In Japan a 490-meter-deep mine shaft has been converted into a free-fall facility.
This creates the longest period of free fall currently available on Earth. How long
is a period of free fall in this facility?
51. Solve the problem algebraically.
52. Use a graphing calculator to check your answer by graphing the related
function y = 4.9x 2 — 490.
Swiss Cheese
o P\
1 9
,
5 1-2
jO-9
? 0.6
1 0.3
£ 0
C
rS
r
\
) 20 40 't
Years since 1970
► Source: U.S. Department of Agriculture
Chapter 9 Quadratic Equations and Functions
Standardized Test
Practice
Mixed Review
Maintaining Skills
53. MULTIPLE CHOICE What are the x-intercepts of y = x 2 - 2x -3?
(A) 1 and —3 CM) 2 and —3 CM) 6 and — 1 CM) 3 and — 1
54. MULTIPLE CHOICE Choose the equation whose roots are shown in
the graph.
CE) 5x 2 - 1 = 0
QD |x 2 - 5 = 0
CE) x 2 - 5 = 0
GD |x 2 - l = 0
55. LUNCH TIME At lunch, you order 1 pasta dish and 1 type of salad. Your
friend orders 1 pasta dish and 2 types of salads. The restaurant charges the
same price for each pasta dish and the same price for each salad. Your bill is
$7.90 and your friend’s bill is $9.85. How much did each pasta dish and each
salad cost? (Lesson 7.4)
SOLVING LINEAR SYSTEMS Use the substitution method or linear
combinations to solve the linear system and tell how many solutions the
system has. (Lesson 7.5)
56. —2x + 8 y = 11
x + 6y = 2
57. —2x + 8 y = 10
x + 6y = 15
59. 8 x + 4 y= -4
4x — y — —20
60. 6x + 4y = -4
2x — y — —6
58. —2x + 2y = 4
x — y = —2
61. 5x + 4y=-3
I5x + 12y = 9
EVALUATING RADICAL EXPRESSIONS Evaluate the radical expression
when a = -1 and b = 5. (Lesson 9.1)
62. \4 2 - 11 a 63. Yb 2 + 9a 64. Va 2 + 8
65. Va 2 - 1
66 .
Yb 2 + 24a
67.
Vfc 2 - 75a
b
68 .
V65 - a 2
69.
V86 + aft
SIMPLIFYING RADICAL EXPRESSIONS Simplify the radical expression.
(Lesson 9.3)
70. V40 71.V24 72. V60 73. V200
74. ^V80 75. |V27 76. -W32 77. |V300
z 3 o 3
COMPARING FRACTIONS AND MIXED NUMBERS Complete the
statement using <, >, or =. (Skills Review pp. 763, 770, 771)
70 _ ? r _
78. ? . 1 ?
79. f >2}
80. Jf # :4
Ol ? li
81 - 6 • l 6
HO
100
22
92.11*2^ 94.1^#^ 89.i#2l
9.5 Solving Quadratic Equations by Graphing
USING A GRAPHING CALCULATOR
For use with
Lesson 9.5
Student HeCp
^Keystroke Help
^ ee ke V strokes f° r
several models of
graphing calculators at
www.mcdougallittell.com
uJjJ'iJU-TJ:
You can use the root or zero feature of a graphing calculator to approximate the
solutions, or roots, of a quadratic equation.
£amp(t
Approximate the roots of 2x 2 + 3x
4 = 0.
Solution
Enter the related function
y = lx 2 + 3x — 4 into the
graphing calculator.
0 Adjust the viewing window so
you can see the graph cross the
x-axis twice. Graph the function.
Y i02X 2 + 3X-4
Y 2 =
Y 3 =
Y4 =
Y 5 —
Y 6 =
Y 7 —
WINDOW
X m i n = -1 0
Xma x = 10
Xsc 1 = 1
Ymin=-10
Yma x = 10
Y s c l = 1
0 Choose the Root or Zero
feature.
0 Follow your graphing calculator’s
procedure to find one root.
1:value
fflzero
3:minimum
4 : ma ximum
5: intersect
6:dy/dx
The positive root is approximately 0.85. Follow similar steps to find the negative
root, —2.35.
Tty Tilts*
APPROXIMATING ROOTS In Exercises 1-4, use a graphing calculator to
approximate the roots of the quadratic equation to the nearest hundredth.
1 .x 2 — x — 2 = 0
3. —4x 2 + 6x + 7 = 0
2 . 6 x 2 + 4x — 12 = 0
4. —2x 2 + 3x + 6 = 0
5. Each equation in Exercises 1-4 has two solutions, or roots. How many
x-intercepts does each related function have?
6 . If a quadratic equation has one solution, how many times do you think the
graph of its related function will cross the x-axis? No real solution?
Chapter 9 Quadratic Equations and Functions
Solving Quadratic Equations
by the Quadratic Formula
Goal
Use the quadratic
formula to solve a
quadratic equation.
Key Words
• quadratic formula
• vertical motion model
When will a baseball hit the ground?
In Exercise 79 you will use the
quadratic formula to find how
long it takes a baseball to reach
the ground after being hit by a
batter.
The quadratic formula gives the solutions of ax 2 + bx + c = 0 in terms of the
coefficients a , b , and c. In Lesson 12.5 you will see how the quadratic formula is
developed from the standard form of a quadratic equation.
P Student HeCp
^ — - -
► Reading Algebra
The quadratic formula
is read as “x equals
the opposite of b, plus
or minus the square
root of b squared
minus 4 ac, all divided
by la"
v__^
THE QUADRATIC FORMULA
The solutions of the quadratic equation ax 2 + bx + c = 0 are
b ± Vb 2 - 4 ac . . , . o
x =- 2 a- when a + 0 and /r 4 ac > 0.
Use the Quadratic Formula
Solve x 2 + 9x + 14 = 0.
Solution lx 2 + 9x + 14 = 0 Identify o = 1, b = 9, and c = 14.
_ -9 ± V9 2 - 4(1)(14) Substitute values in the quadratic
X 2(1) formula: o = 1, b = 9, and c = 14.
x =
—9 ± V25
2
Simplify.
x =
-9 ±5
2
Solutions.
ANSWER ► The two solutions are x =
-9 + 5
2
—2 andx =
-9-5
2
-7.
Use the Quadratic Formula
Use the quadratic formula to solve the equation.
1. x 2 - 4x + 3 = 0 2. 2x 2 + x — 10 = 0 3. -x 2 + 3x + 4 = 0
9.6 Solving Quadratic Equations by the Quadratic Formula
Student HeCp
► More Examples
More examples
are available at
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2 Write in Standard Form
Solve 2x 2 — 3x = 8. Round the results to the nearest hundredth.
Solution
lx 2 — 3x = 8
2x 2 — 3x — 8 = 0
_ -(-3) ± V(— 3) 2 - 4(2)(— 8)
2 ( 2 )
X =
3 ± V9 + 64
4
3 ± V73
Write original equation.
Rewrite equation in standard form.
Substitute values in the quadratic
formula: a = 2, b = -3, c = -8.
Simplify.
Solutions.
'x + V73 3 — V73
ANSWER ► The two solutions are x = —--~ 2.89 and x = —--~ — 1.39.
Write in Standard Form
Use the quadratic formula to solve the equation. If the solution involves
radicals, round to the nearest hundredth.
4. x 2 + x = 1 5- —x 2 = 2x — 3 6. lx 2 — 1 = —2x
Student MeCp
—\
► Study Tip
Recall from Lesson 9.5
that the x-intercepts of
the graph of a quadratic
equation in one variable
are also called the
roots of the equation.
^ _ J
B32EG223M 3 Find the x-intercepts of a Graph
Find the x-intercepts, or roots, of the graph of y = x 2 + 4x — 5.
Solution The x-intercepts occur
y = x 2 + 4x — 5
0 = lx 2 + 4x — 5
—4 ± V(4) 2 — 4(1)(— 5)
2 ( 1 )
-4 ± Vl6 +20
when y = 0.
Write original equation.
Substitute 0 for y.
Substitute values in the quadratic
formula: a = 1, b = 4, c = -5.
Simplify.
x =--- Solutions
ANSWER ► The two solutions are x = 1
andx = —5.
CHECK y Use a graph to check your solutions.
You can see from the graph that
the x-intercepts of y = x 2 + 4x — 5
are —5 and 1.
Chapter 9 Quadratic Equations and Functions
Student HeCp
► Study Tip
Velocity indicates
speed and direction
(up is positive and
down is negative).
Speed is the absolute
value of velocity.
VERTICAL MOTION MODELS In Lesson 9.2 you studied the model for the
height of a falling object that is dropped. For an object that is thrown up or down,
the model has an extra variable v. It is called the initial velocity.
VERTICAL MOTION MODELS
Object is dropped: h = — 16f 2 + s Object is thrown: h = -16 1 2 + vt + s
h = height (feet) t = time in motion (seconds)
s = initial height (feet) v = initial velocity (feet per second)
03Z&SI9 4 Model Vertical Motion
HOT-AIR BALLOONS You are competing
in the field target event at a hot-air balloon
festival. From a hot-air balloon directly
over a target, you throw a marker with an
initial downward velocity of —30 feet per
second from a height of 200 feet. How long
does it take the marker to reach the target?
Solution The marker is thrown down , so
the initial velocity v is —30 feet per second.
The initial height s is 200 feet. The marker
will hit the target when its height is 0 .
200 ft
Not drawn to scale
h — — 16 1 2 + vt + s
h = -16? 2 + (-30)? + 200
Choose the vertical motion model
for a thrown object.
Substitute values for v and s in the
vertical motion model.
0 = — 16? 2 - 30 ? + 200
Substitute 0 for h. Write in
standard form.
-(-30) ± V(— 30) 2 - 4(— 16X200)
2 (— 16)
Substitute values for a, b, and c in
the quadratic formula.
30 ± Vl3,700
t = -- Simplify.
t ~ 2.72 or —4.60 Evaluate the radical expressions.
ANSWER ► The weighted marker will reach the ground about 2.72 seconds after it
was thrown. The solution —4.60 doesn’t make sense in this problem.
Model Vertical Motion
7. In Example 4, suppose you throw a marker with an initial downward velocity
of —60 feet per second. Do you think it would hit the ground in half the
time? Check your prediction using the quadratic formula.
9.6 Solving Quadratic Equations by the Quadratic Formula
Exercises
Guided Practice
Vocabulary Check 1 , Write the formula that you can use to solve any quadratic equation when
a 0 and b 2 — 4 ac > 0 .
2 . Describe how you can check the solutions of a quadratic equation by looking
at the graph of the related function.
3. What new feature was introduced in the vertical motion model used in
Example 4?
Skill Check Write the equation in standard form. Identify the values of a, b, and c
that you would use to solve the equation using the quadratic formula.
4. x 2 = 1 5- I6x — 32 = 2x 2 6- x 2 — lx + 42 = 6x
Use the quadratic formula to solve the equation. Write your solutions in
simplest form.
7. x 2 + 6x — 7 = 0 8 . x 2 — 2a — 15 = 0 9. x 2 + 12x + 36 = 0
10- 4x 2 — 8 x + 3 = 0 11 - 3x 2 + x — 1 = 0 12 . x 2 + 6x — 3 = 0
Write the equation in standard form. Then use the quadratic formula to
solve the equation.
13. 2x 2 = — x + 6 14. —3x = 2x 2 + 1 15. 2 = x 2 — x
16. — 14x = —2x 2 + 36 17. — x 2 + 4x = 3 18. 4x 2 + 4x = —1
Practice and Applications
STANDARD FORM Write the equation in standard form. Identify the
values of a, b, and c.
19. 3x 2 = 3x + 6
22 . 3x 2 = 27x
25. k 2 = f
20 . — 2 1 2 = -8
23. —24x + 45 = — 3x :
26. 2 x 2 — \ = —%x
21 . —x 2 = —5x + 6
24. 32 - 4m 2 = 28 m
27. \ — 2x — \x 2
FINDING VALUES Find the value of b 2 — 4ac for the equation.
28. x 2 — 3x — 4 = 0
Student HeCp
► Homework Help
Example 1: Exs. 40-48
Example 2: Exs. 49-57
Example 3: Exs. 58-66
Example 4: Exs. 67-80
a___ y
31. r 2 - llr + 30 = 0
34. 2x 2 + 4x — 1 = 0
37. 5x 2 + 5x + \ = 0
29. 4x 2 + 5x + 1 = 0
32. s 2 - 135 + 42 = 0
35. 3t 2 - St - 7 = 0
38. h 2 + 5t - 8 = 0
30. —5w 2 — 3w + 2 = 0
33. 3x 2 - 5x - 12 = 0
36. —8 m 2 — 6m + 3 = 0
39. |v 2 - 6 v - 3 = 0
Chapter 9 Quadratic Equations and Functions
SOLVING EQUATIONS Use the quadratic formula to solve the equation.
If the solution involves radicals, round to the nearest hundredth.
40. 4x 2 — 13x + 3 = 0 41. y 2 + lly + 10 = 0
Student HeCp
► Study Tip
You can use a
calculator or the Table
of Square Roots on
p. 801 to approximate
irrational square roots.
43. -3 y 1 + 2y + 8 = 0
46. 8 m 2 + 6m —1 = 0
44. 6 n 2 - lOn + 3 = 0
42. 7x 2 + 8 x + 1 = 0
45. 9x 2 + 14x + 3 = 0
47. ~x 2 + 6 x + 13 = 0 48. 2x 2 - 3x + 1 = 0
STANDARD FORM Write the quadratic equation in standard form. Then
solve using the quadratic formula.
49. 2x 2 = 4x + 30 50. x 2 + 3x = —2 51. 5 = x 2 + 6x
52. 5x + 2 = 2x 2 53. 5x — 2x 2 +15 = 8 54. —2 + x 2 = — x
55. x 2 - 2x = 3
56. 2x 2 + 4 = 6x
57. 12 = 2x 2 - 2x
61-y = jc 2 + lOx + 16 62 . y ■■
64. y — x 2 — 2x — 2 65. y ■
■ 2x 2 + 4x — 6 66 . y — ~3x 2
Link to
GcoCogy
URBAN BIRDS Cities
provide a habitat for many
species of wildlife including
birds of prey such as
peregrine falcons and red¬
tailed hawks.
More about urban
birds is available at
www.mcdougallittell.com
FIELD TARGET EVENT In Exercises 67-72, six balloonists compete in a
field target event at a hot-air balloon festival. Calculate the amount of
time it takes for the marker to reach the target when thrown down from
the given initial height (in feet) with the given initial downward velocity
(in feet per second). Round to the nearest hundredth of a second.
67. s = 200; v = -50 68. s = 150; v = -25 69. s = 100; v = -10
70. s = 150; v = -33 71. s = 50; v = -40
72. s = 50; v = -20
73. PEREGRINE FALCON A falcon
dives toward a pigeon on the ground.
When the falcon is at a height of
100 feet, the pigeon sees the falcon,
which is diving at 220 feet per
second. Estimate the time the pigeon
has to escape. Round your solution to
the nearest tenth of a second.
74. RED-TAILED HAWK A hawk dives
toward a snake. When the hawk is at
a height of 200 feet, the snake sees
the hawk, which is diving at 105 feet
per second. Estimate the time the
snake has to escape. Round your
solution to the nearest tenth of
a second.
9.6 Solving Quadratic Equations by the Quadratic Formula
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 75-78 is available at
www.mcdougallittell.com
VERTICAL MOTION In Exercises 75-78, use a vertical motion model to
find how long it will take for the object to reach the ground. Round your
solution to the nearest tenth.
75. You drop your keys from a window 30 feet above ground to your friend
below. Your friend does not catch them.
76. An acorn falls 45 feet from the top of a tree.
77. A lacrosse player throws a ball upward from her playing stick from an initial
height of 7 feet, with an initial velocity of 90 feet per second.
78. You throw a ball downward with an initial velocity of —10 feet per second
out of a window to a friend 20 feet below. Your friend does not catch the ball.
79. BASEBALL A batter hits a pitched baseball when it is 3 feet off the
ground. After it is hit, the height h (in feet) of the ball is modeled by
h = —16 1 2 + 80/ + 3, where / is the time (in seconds). How long will
it take for the ball to hit the ground? Round to the nearest hundredth.
80. Sc ience Link '$> An astronaut standing on the moon’s surface throws a rock
upward with an initial velocity of 50 feet per second. The height of the rock
can be modeled by m = —2.1t 2 + 50/ + 6 , where m is the height of the rock
(in feet) and / is the time (in seconds). If the astronaut throws the same rock
upward with the same initial velocity on Earth, the height of the rock is
modeled by e = —16/ 2 + 50/ + 6 . Would the rock hit the ground in less time
on the moon or on Earth? Explain your answer.
Standardized Test
Practice
81. MULTIPLE CHOICE Which expression gives the solutions of 2x 2 — 10 = x?
(A)
1 ± Vl — (4)(2)(—10)
CD
el ± Vi - (4)(2)(io)
4
4
CD
10 ± V100 - (4)(2)(1)
CD
10 ± V100 - (4)(2)(-1)
4
4
82. MULTIPLE CHOICE What are the roots of the quadratic equation in
Exercise 81?
CD 2 ® 4 ®> CD None of these
83. MULTIPLE CHOICE Which quadratic equation has the solutions
-9 ± V81 - 56 „
x 4
(A) 2x 2 + 9x - 7 = 0 CD 2x 2 - 9x + 7 = 0
CD 4x 2 + 9x + 14 = 0 CD 2 x 2 + 9x + 7 = 0
84. MULTIPLE CHOICE Which equation would you use to model the height of
an object that is thrown down with an initial velocity of —10 feet per second
from a height of 100 feet?
CD h = -16 1 2 +100 CD h = — 16i 2 + 1 Or + 100
CH) h = —16 1 2 - 10r +100 C Dh = -16 1 2 - 10 1 - 100
Chapter 9 Quadratic Equations and Functions
Mixed Review
EVALUATING EXPRESSIONS Evaluate the expression for the given value
of the variable. (Lesson 2.5)
Maintaining Skills
Quiz 2
85- — 3(x) when* = 9 86- —5(—ri)(—ri) when n — 2
87- 4(—6)(m) when m = — 2 88- 2(— 1)(— x) 3 when x = — 3
GRAPHING LINES Write the equation in slope-intercept form. Then graph
the equation. (Lesson 4.7)
89. — 3x + y + 6 = 0 90- — x + y — 7 = 0 91. 4x + 2y — 12 = 0
SOLVING INEQUALITIES Solve the inequality. Then graph the solution.
(Lesson 6.2)
92. 6x < -2 93. -3x > 15 94. |x > 12
95. RECREATION There were about 1.4 X 10 7 people who visited Golden
Gate Recreation Area in California in 1996. Find the average number of
visitors per month. ►Source: National Park Service (Lesson 8.5)
COMPARING FRACTIONS Complete the statement using <, >, or =.
(Skills Review pp. 770 ; 771)
T5 «■! *! < ? 5
99.|#^ 100.4} 101.2§#3i
Decide whether the graph of the function opens up or down. (Lesson 9.4)
1 . < y = x 2 + 2 x— 11 2 . y = 2 x 2 — 8 x — 6 3. y = —3x 2 + 6 x — 10
4. y = —x 2 + 5x — 3 5. y = —lx 2 — lx + 7 6. y — —x 2 + 9x
Sketch the graph of the function. Label the coordinates of the vertex.
(Lesson 9.4)
7. y = —x 2 + 2x — 3 8. y = — 3x 2 + 12x — 10 9. y = 2x 2 — 6 x + 7
Use a graph to estimate the solutions of the equation. Check your
solutions algebraically. (Lesson 9.5)
10 . x 2 — 3x = 10 11 .x 2 — 12x = —36 12 . 3x 2 + 12x = —9
Use the quadratic formula to solve the equation. If your solution involves
radicals, round to the nearest hundredth. (Lesson 9.6)
13. x 2 + 6 x + 9 = 0 14. 2x 2 + 13x + 6 = 0 15. —x 2 + 6 x + 16 = 0
16. —2x 2 + lx — 6 = 0 17. —3x 2 — 5x + 10 = 0 18. 3x 2 — 4x — 1 = 0
9.6 Solving Quadratic Equations by the Quadratic Formula
Using the Discriminant
Goal
Use the discriminant to
determine the number of
solutions of a quadratic
equation.
Key Words
• discriminant
Can you throw a stick high enough?
One way that campers protect food
from bears is to hang it from a high
tree branch. In the example on
page 545, you will determine
whether a stick and a rope were
thrown fast enough to go over a
tree branch.
In the quadratic formula, the expression inside the radical is the discriminant.
—b ± Vb 2 — 4 ac - Discriminant
X ” 2a
The discriminant b 2 — 4ac of a quadratic equation can be used to find the number
of solutions of the quadratic equation.
Student HeCp
► Study Tip
Recall that positive
real numbers have two
square roots, zero has
only one square root,
negative numbers have
no real square roots.
v __ )
THE NUMBER OF SOLUTIONS OF A QUADRATIC EQUATION
Consider the quadratic equation ax 2 + bx + c = 0.
• If the value of b 2 - 4ac is positive, then the equation has two
solutions.
• If the value of b 2 - 4 ac is zero, then the equation has one
solution.
• If the value of b 2 - 4 ac is negative, then the equation has no real
solution.
J i) Find the Number of Solutions
Find the value of the discriminant. Then use the value to determine whether
x 2 — 3x — 4 = 0 has two solutions , one solution , or no real solution.
Solution Use the standard form of a quadratic equation, ax 2 + bx + c = 0,
to identify the coefficients.
x 2 — 3x — 4 = 0 Identify a = 1, b = -3, c = -4.
b 2 — 4 ac = (— 3) 2 — 4(1)(— 4) Substitute values for a, b, and c.
= 9+16 Simplify.
= 25 Discriminant is positive.
ANSWER ^ The discriminant is positive, so the equation has two solutions.
Chapter 9 Quadratic Equations and Functions
Find the Number of Solutions
Find the value of the discriminant. Then use the value to determine whether the
equation has two solutions , one solution , or no real solution.
a. —x 2 + 2x —1=0 b. 2x 2 — 2x + 3 = 0
Solution
a. — x 2 + 2x — 1 = 0 Identify a = -1, b = 2, c = -1.
b 2 — 4ac = (2) 2 — 4(—1)(— 1) Substitute values for a, b, and c.
= 4-4 Simplify.
= 0 Discriminant is zero.
ANSWER ► The discriminant is zero, so the equation has one solution.
b. 2x 2 — 2x + 3 = 0 Identify a = 2, b = -2, c = 3.
b 2 — 4 ac — (—2) 2 — 4(2)(3) Substitute values for o, b, and c.
= 4 — 24 Simplify.
= —20 Discriminant is negative.
ANSWER ► The discriminant is negative, so the equation has no
real solution.
I_
Find the Number of Solutions
Find the value of the discriminant. Then use the value to determine
whether the equation has two solutions, one solution, or no real solution.
1 . x 2 — 3x + 4 = 0 2_ x 2 — 4x + 4 = 0 3. x 2 — 5x + 4 = 0
Because each solution of ax 2 + bx + c = 0 represents an x-intercept of
y = ax 2 + bx + c, you can use the discriminant to determine the number of
times the graph of a quadratic function intersects the x-axis.
3 Find the Number of x-Intercepts
Determine whether the graph of y = x 2 + 2x — 2 will intersect the x-axis in
zero , one , or two points.
Solution
Let y = 0. Then find the value of the discriminant of x 2 + 2x — 2 = 0.
x 2 + 2x — 2 = 0 Identify a = 1, b = 2, c - -2.
b 2 — 4 ac = ( 2) 2 — 4(1)(— 2 ) Substitute values for a, b, and c.
= 4 + 8 Simplify.
= 12 Discriminant is positive.
ANSWER ^ The discriminant is positive, so the equation has two solutions and
the graph will intersect the x-axis in two points.
9.7 Using the Discriminant
Student HeCp
^More Examples
M°r e examples
IJfcL 2 are available at
www.mcdougallittell.com
J 4 Find the Number of x-lntercepts
Determine whether the graph of the function will intersect the x-axis in zero,
one, or two points.
a. y = x 2 + 2x + 1 b. y = x 2 + 2x + 3
Solution
a. Let y = 0. Then find the value of the discriminant of x 2 + 2x + 1
0 .
x 2 + 2x + 1 = 0
4 ac = (2) 2 - 4(1)(1)
= 4 — 4
= 0
Identify a = 1, b = 2, c = 1.
Substitute values for o, b, and c.
Simplify.
Discriminant is zero.
ANSWER ► The discriminant is zero, so the equation has one solution and
the graph will intersect the x-axis in one point.
b_ Let y = 0. Then find the value of the discriminant of x 2 + 2x + 3 = 0.
x 2 + 2x + 3 = 0 Identify a = 1, b = 2, c = 3.
Substitute values for a, b, and c.
Simplify.
Discriminant is negative.
ANSWER ► The discriminant is negative, so the equation has no real solution
and the graph will intersect the x-axis in zero points.
b 2 - 4ac = (2) 2 - 4(1)(3)
= 4-12
= -8
Student HeCp
► Look Back
For help with sketching
the graph of a quadratic
function, see p. 521.
I J
5 Change the Value of c
Sketch the graphs of the functions in Examples 3 and 4 to check the number of
x-intercepts of y = x 2 + 2x + c. What effect does changing the value of c have
on the graph?
Solution By changing the value of c, you
can move the graph of y — x 2 + 2x + c up
or down in the coordinate plane.
a. y — x 2 + 2x — 2
b_ y — x 2 + 2x + 1
c. y — x 2 + 2x + 3
If the graph is moved high enough, it will not have
an x-intercept and the equation x 2 + 2x + c = 0
will have no real solution.
I_
Find the Number of x-lntercepts
Find the number of x-intercepts of the graph of the function.
4. y = x 2 - 4x + 3 5. y = x 2 - 4x + 4 6. y = x 2 - 4x + 5
■ —
r
Exercises
Guided Practice
Vocabulary Check 1 . Write the quadratic formula and circle the part that is the discriminant.
2 . What can the discriminant tell you about a quadratic equation?
3. Describe how the graphs of y = 4x 2 , y = 4x 2 + 3, and y — 4x 2 — 6 are alike
and how they are different.
Skill Check Use the discriminant to determine whether the quadratic equation has
two solutions , one solution , or no real solution.
4. 3x 2 — 3x + 5 = 0 5- — 3x 2 + 6x — 3 = 0 6- x 2 — 5x — 10 = 0
Give the letter of the graph that matches the value of the discriminant.
7. b 2 — 4ac — 2 8- b 2 — 4ac = 0 9. b 2 — 4ac = —3
Determine whether the graph of the function will intersect the x-axis in
zero, one, or two points.
10. y = x 2 + 2x + 4 11. y = —x 2 — 3x + 5 12. y = 6x — 3 — 3x 2
Practice and Applications
WRITING THE DISCRIMINANT Find the discriminant of the quadratic
equation.
13. -2x 2 - 5x + 3 = 0 14. 3X 2 + 6x - 8 = 0 15. x 2 + 10 = 0
16. 5x 2 + 3x = 12 17. 2x 2 + 8x = —8 18. 7 — 5x 2 + 9x = x
19. -x = 7x 2 + 4 20. 2x = x 2 -x 21.-2 -x 2 = 4x 2
USING THE DISCRIMINANT Determine whether the equation has two
solutions , one solution , or no real solution.
22. x 2 - 3x + 2 = 0 23. lx 2 - 4x + 3 = 0
R Student HeCp
Homework Help
Example 1: Exs. 13-33
Example 2: Exs. 13-33
Example 3: Exs. 38-43
Example 4: Exs. 38-43
Example 5: Exs. 44-46
25. 2x 2 + 3x — 2 = 0
28. 3x 2 — 6x + 3 = 0
31. ——x 2 + x + 3 = 0
26. x 2 — 2x + 4 = 0
29. 4x 2 — 5x + 1 = 0
32. “X 2 — 2x + 4 = 0
24. — 3x 2 + 5x - 1 = 0
27. 6 x 2 — 2x + 4 = 0
30. —5x 2 + 6 x — 6 = 0
33. 5x 2 + 4x + 4 — 0
9.7 Using the Discriminant
34. ERROR ANALYSIS For the
equation 3x 2 + 4x — 2 = 0,
find and correct the error.
INTERPRETING THE DISCRIMINANT In Exercises 35-37, consider the
quadratic equation y = 2 . x 2 + 6x - 3.
35. Evaluate the discriminant.
36. How many solutions does the equation have?
37. What does the discriminant tell you about the graph of y = 2x 2 + 6x — 3?
NUMBER OF X-INTERCEPTS Determine whether the graph of the
function will intersect the x-axis in zero, one, or two points.
38. y = 2x 2 + 3x — 2 39. y = x 2 — 2x + 4 40. y = — 2x 2 + 4x — 2
41. y = 2x 2 + 2x + 6 42. y = 5x 2 + 2x — 3 43. y = 3x 2 — 6x + 3
CHANGING THE VALUE OF C Match the function with its graph.
44. y = -x 2 - 2x - 1 45. y = ~x 2 - 2x - 3 46. y = -x 2 - 2x + 3
Linkjtp^
Careers
FINANCIAL ANALYSTS
use mathematical models to
help analyze and predict a
company's future earnings.
More about financial
analysts is available
at www.mcdougallittell.com
B FINANCIAL ANALYSIS In Exercises 47-49, use a graphing calculator
and the following information.
A software company’s net profit
for each year from 1993 to 1998
lead a financial analyst to model
the company’s net profit by
P = 6Mt 2 - 3.76? + 9.29,
where P is the profit in millions
of dollars and t is the number of
years since 1993. In 1993 the
net profit was approximately
9.29 million dollars (t = 0).
47. Give the domain and range of the function for 1993 through 1998.
48. Use the graph to predict whether the net profit will reach 650 million dollars.
49. Use a graphing calculator to estimate how many years it will take for the
company’s net profit to reach 475 million dollars according to the model.
Software Company Profits
Pi
co 400
SB o g
2=0
200
2 4 6 8
Years since 1993
Chapter 9 Quadratic Equations and Functions
CAMPING Bears have an
excellent sense of smell that
often leads them to campsites
in search of food. Campers
can hang a food sack from a
high tree branch to keep it
away from bears.
Standardized Test
Practice
Mixed Review
Maintaining Skills
M Using the Discriminant
CAMPING You and a friend want to get a rope over a tree branch that is
20 feet high. Your friend attaches a stick to the rope and throws the stick
upward with an initial velocity of 29 feet per second. You then throw it with
an initial velocity of 32 feet per second. Both throws have an initial height of
6 feet. Will the stick reach the branch each time it is thrown?
Solution
Use a vertical motion model for an object that is thrown: h = — 16 1 2 + vt + s,
where h is the height you want to reach, / is the time in motion, v is the initial
velocity, and s is the initial height.
h = — 16 1 2 + vt + s h = — 16 1 2 + vt + s
20 = -16 1 2 + 29/ + 6 20 = -16/ 2 + 32/ + 6
0 = -16/ 2 + 29/ - 14 0 = —16/ 2 + 32/ - 14
b 2 - 4 ac = ( 29) 2 - 4(— 16)(— 14) b 2 - 4 ac = ( 32) 2 - 4(— 16)(— 14)
The discriminant is —55. The discriminant is 128.
ANSWER ► The discriminant is ANSWER ► The discriminant is
negative. The stick positive. The stick
thrown by your thrown by you will
friend will not reach reach the branch,
the branch.
50- BASKETBALL You can jump with an initial velocity of 12 feet per second.
You need to jump 2.2 feet to dunk a basketball. Use the vertical motion
model h = —16/ 2 + vt + s to find if you can dunk the ball. Justify your
answer.
51. MULTIPLE CHOICE For which value of c will — 3x 2 + 6x + c = 0 not have
a real solution?
(A) c < — 3 (D c = -3 (® c>-3 C5) c = 3
52. MULTIPLE CHOICE How many real solutions does x 2 — lOx + 25 = 0 have?
Cf) No solutions CG) One solution CFT) Two solutions Cj) Many solutions
SOLVING AND GRAPHING INEQUALITIES Solve the inequality. Then
graph the solution. (Lesson 6.4)
53. 2<x+ 1<5 54. 8>2x>-4 55. -12 < 2x - 6 < 4
GRAPHING LINEAR INEQUALITIES Graph the inequality. (Lesson 6.8)
56. 3x + y < 9 57. y - 4x < 0 58. -2jc -y> 4
MULTIPLYING DECIMALS Find the product. (Skills Review p. 759)
59. 3 X 0.02 60. 0.7 X 0.8 61. 0.1 X 0.1
62. 0.05 X 0.003 63. 0.09 X 0.02 64. 0.06 X 0.0004
9.7 Using the Discriminant
ff'Tift
DEVELOPING CONGE
xj flJXJiixJi'ii'ijx: JiJixjUiiJj'iJi:
For use with
Lesson 9.8
Goal
Use reasoning to discover
a strategy for sketching
the graph of a quadratic
inequality.
Materials
• graph paper
• pencil
Student HeCp
^ Look Back
To review strategies
for graphing a linear
inequality, see
pp. 367-369.
Question How do you determine which portion of the graph of a
quadratic inequality to shade?
In Lesson 6.8 you learned how to graph a linear inequality in two variables. You
can use similar strategies to graph a quadratic inequality in two variables.
Explore
0 Consider the graphs of the following two quadratic inequalities,
a. y < x 2 — 2x — 3 b. y > x 2 — 2x — 3
Graph y = x 2 — 2x — 3. Use a dashed line for < and a solid line for >.
0 Use substitution to test points inside and outside the parabola. An ordered pair
(x, y) is a solution of a quadratic inequality if the inequality is true when the
values of x and y are substituted into the inequality. Try testing (0, 0).
O’
f
(0,0)
-?
1
i
5 x
\
/
\
i
\ /
/
f
—4
r y<x 2 -
ro
1
*
CM
The point (0, 0) ? a solution. The point (0, 0) ? a solution.
The solutions appear to be the
set of all points that lie outside
the graph of y — x 2 — 2x — 3.
The solutions appear to be the set of
all points that lie inside or on the
graph of y — x 2 — 2x — 3.
© Can the inequality y < x 2 — 2x — 3 be interpreted as “all points (x, y) that lie
below the parabola y = x 2 — 2x — 3”? Explain.
Think About It
Match the quadratic inequality with its graph. Explain your reasoning.
1.y<x 2 - 4
2 . y > x 2
B.
4x
5
o k
\
3
\
J
\
\
\
/
/
/
1
V
/
\ 1
3
5 x
3. y <(x — 4) 2
C.
V
i
i-y
\T
i
—
1
-1
v 1
V
3
/
/
X
K
\
7
J
/
-3
vj
Chapter 9 Quadratic Equations and Functions
Graphing Quadratic
Inequalities
Goal
Sketch the graph of a
quadratic inequality in
two variables.
Key Words
• quadratic inequalities
• graph of a quadratic
inequality
How does a flashlight work?
A flashlight has a parabolic
reflector that helps to focus the
light into a beam. In Exercise 38
you will use a quadratic inequality
to learn more about how a
flashlight works.
In this lesson you will study the following types of quadratic inequalities.
y < ax 2 + bx + c y < ax 2 + bx + c
y > ax 2 + bx + c y> ax 2 + bx + c
The graph of a quadratic inequality consists of the graph of all ordered pairs
(.x, y) that are solutions of the inequality.
J i Check Points
Sketch the graph of y = x 2 — 3x — 3.
Plot and label the points AO, 2), 5(1, 4),
and C( 4, — 3). Determine whether each
point lies inside or outside the parabola.
Solution
Q Sketch the graph of y = x 2 — 3x — 3.
0 Plot and label the points A{ 3, 2),
5(1, 4), and C(4, -3).
ANSWER ► Points A and 5 lie inside the
parabola. Point C lies outside
the parabola.
(37ra^f7raftit? l^ Check Points
Sketch the graph of y = x 2 - 4x + 3. Plot the point and determine
whether it lies inside or outside the parabola.
1 .A(—1,2) 2. 5(0, 0) 3. C(2, 1)
9.8 Graphing Quadratic Inequalities
The shaded part of the graph of a quadratic inequality contains all of the ordered
pairs that are solutions of the inequality. Checking points tells you which region
to shade. You can use the following steps to graph any quadratic inequality.
METHOD I: GRAPHING A QUADRATIC INEQUALITY
step Q Sketch the graph of y = ax 2 + bx + c that corresponds to
the inequality.
Sketch a dashed parabola for inequalities with < or > to
show that the points on the parabola are not solutions.
Sketch a solid parabola for inequalities with < or > to
show that the points on the parabola are solutions.
step © The parabola separates the coordinate plane into two
regions. Test a point that is not on the parabola to
determine whether the point is a solution of the inequality.
step © If the test point is a solution, shade its region. If not,
shade the other region.
H Graph a Quadratic Inequality
Sketch the graph of y < lx 1 — 3x.
Student HeCp
► Study Tip
If the point (0, 0) is not
on the parabola, then
(0, 0) is usually good to
use as a test point. For
help with checking
ordered pairs as
solutions, see p. 367.
v _
Solution
0 Sketch the graph of the equation y = 2x 2 — 3x
that corresponds to the inequality y < lx 2 — 3x.
Use a dashed line since the inequality
contains the symbol <.
The parabola opens up since a is positive.
The vertex is f —1-|
H
r\
\
\
2
1
J
/
/
-
i
-i
:
t
l J 4 *]
lx 2 - ■
\x
0 Test a point, such as (1, 2), that is not on the
parabola. The point (1,2) lies inside the parabola.
y < 2x 2 — 3x Write original inequality.
2 < 2(1) 2 — 3(1) Substitute 1 for x and 2 for y.
2)^—1 2 is not less than -1.
Because 2 is not less than — 1,
the ordered pair (1, 2) is not a solution.
© Shade the region outside the parabola.
The point (1, 2) is inside the parabola
and it is not a solution, so the graph of
y < 2x 2 — 3x is all the points that are
outside, but not on, the parabola.
Chapter 9 Quadratic Equations and Functions
Until now you have used the fact that a parabola divides the plane into two
regions, one of which is inside the parabola and one of which is outside. For
parabolas given by y = ax 2 + bx + c, these regions can also be described as
lying above and below the parabola and can be graphed using the following steps.
METHOD II: GRAPHING A QUADRATIC INEQUALITY
step Q Sketch the graph of y = ax* + bx + c, using a dashed or a
solid curve as in Method I.
step © If the inequality is y > ax* + bx + c or y > ax* + bx + c,
shade the region above the parabola.
If the inequality is y < ax* + bx + c or y < ax* + bx + c,
shade the region below the parabola.
.
Student Hedp
► More Examples
More examples
gre ava j| a |3| e at
www.mcdougallittell.com
3 Graph a Quadratic Inequality
Sketch the graph of y < — x 2 — 5x + 4.
Solution
Q Sketch the graph of the equation y = — x 2 5.x + 4 that corresponds to the
o b
inequality y < —x — 5x + 4. The x-coordinate of the vertex is ——, or
—2— Make a table of values, using x- values to the left and right of
Plot the points and connect them
with a smooth curve to form a
parabola. Use a solid line since
the inequality contains the
symbol <.
© Shade the region below the
parabola because the inequality
states that y is less than or equal
to —x* — 5x + 4.
Graph a Quadratic Inequality
Sketch the graph of the inequality.
4. y < x 2 + 2x + 2 5- y > ~x 2 — 2x + 3 6. y > 2x 2 — 4x + 2
9.8 Graphing Quadratic Inequalities
Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Give an example of each of the four types of quadratic inequalities.
2. True or Falsel For inequalities with < or >, you sketch a solid parabola to
show that the points on the parabola are not solutions.
Sketch the graph of the equation y = x 2 + 2x — 4. Plot the point and
determine whether it lies inside or outside the parabola.
3. A(0, 0) 4. B(- 1, 3) 5. C(2, -2)
Decide whether each labeled ordered pair is a solution of the inequality.
6 . y < -x 2
7. y > x 2
8- y < 2x 2 + 5x
\
y>
(-
-1,1)
•
/ <
(i,d
\
f 1
3
-
1
i
X
/
L
Sketch the graph of the inequality.
9- y < x 2
12 . y >x 2 - 2x
10 . y > ~x 2 + 3
13. y < —2x 2 + 6x
11. y < ~x 2 + 2x
14. y < 2x 2 - 4x + 3
Practice and Applications
SOLUTIONS Determine whether the ordered pair is a solution of the
inequality.
15. y > 2x 2 — x, (—2, 10)
17. y < x 2 + 9x, (—3, 18)
19. y > 4x 2 — 7x, (2, 0)
16. y < 3x 2 + 7, (4,31)
18. y < 5x 2 + 8, (3, 45)
20 . y > x 2 — 13x, (—1, 14)
Student HeCp
^— HL -—^
► Homework Help
Example 1: Exs. 21-24
Example 2: Exs. 29-38
Example 3: Exs. 29-38
CHECKING POINTS Sketch the graph of the function. Plot the
given point and determine whether the point lies inside or outside
the parabola.
21. y = x 2 — 2x + 5 22. y = —x 2 + 4x — 2
A(0,4) *(3,-2)
23. y = “X 2 + x — 4
C(l, 0)
24. y = 4x 2 — x + 1
D(~ 2, 5)
Chapter 9 Quadratic Equations and Functions
Standardized Test
Practice
LOGICAL REASONING Complete the statement with always, sometimes,
or never.
25 , If a > b, then a 2 is ? greater than b 2 .
26 , If a > b and b > 0, then a 2 is ? greater than b 2 .
27 , If a 2 = 4, then a is ? equal to 2.
28 , If a is a real number, then Vo 2 is ? equal to \a\.
MATCHING INEQUALITIES Match the inequality with its graph.
29. y > ~2x 2 - 2x + 1
i
^ j
\
4
/
/
\
/
fr
/
V 2
3
1 ,
t 1
X
30. y > —2x 2 + 4x + 3
31. y < 2x 2 + x + 1
4
/
\—
Y]
\
i
i
.vJ
H
— i
i
V I
‘ H
f 1
if
4
I- \
SKETCHING GRAPHS Sketch the graph of the
32. y < —x 2 + x
35. y > —x 2 — 3x — 2
33. y <x 2 — 4
36. y < —x 2 + 3x + 4
inequality.
34. y > x 2 - 5x
37. y > —3x 2 — 5x — 1
38. FLASHLIGHT Light rays from a flashlight
bulb bounce off a parabolic reflector inside
a flashlight. The reflected rays are parallel
to the axis of the flashlight. In this way,
flashlights produce narrow beams
of light.
A cross section of a flashlight’s parabolic
reflector is shown in the graph at the
right. An equation for the parabola is
y = ~^x L + 1. Choose the region of the
graph where the bulb is located.
A. y < ^x 2 +1 B. y > -^x 2 + 1
bulb
C.y<±x 2 + 1
39. MULTIPLE CHOICE Which ordered pair is not a solution of the inequality
y > lx 1 -lx - 10?
®(0, -4) CD (-1,-1) ©(4,-13) ©(5,15)
40. MULTIPLE CHOICE Choose the statement that is true about the graph of the
quadratic inequality y < 5x 2 + 6x + 2.
(A) Points on the parabola Cb) The vertex is
are solutions.
Cep The parabola opens down. (Tp (0, 0) is not a solution.
9.8 Graphing Quadratic inequalities
Mixed Review
Maintaining Skills
Quiz 3
FINDING EQUATIONS The variables xand / vary directly. Use the given
values to write an equation that relates x and y. (Lesson 4.6)
41.x = 6,y = 42 42. x = -9,y = 54 43. x = 14, y = 7
44. x = —13, y = —52 45. x = 3, y = —6 46. x = —5, y = 60
GRAPHING FUNCTIONS
47. y = 3*
Graph the exponential function.
48. y = 5* 49. y =
51.y = 2^ B2.y =
(Lesson 8.3)
3(2)*
PERCENTS AND FRACTIONS Write the percent as a fraction or as a
mixed number in simplest form. (Skills Review p. 768)
53. 4%
54. 392%
55. 45%
56. 500%
57. 3%
58. 6%
59. 24%
60. 10%
61. 390%
62. 225%
63. 175%
64. 8%
65. 91%
66. 2%
67. 25%
68. 16%
Determine whether the equation has two solutions , one solution , or
no real solution. (Lesson 9.7)
1.x 2 — 15x + 56 = 0 2. x 2 + 8x + 16 = 0 3. x 2 — 3x + 4 = 0
4. THROWING A BASEBALL Your friend is standing on a balcony that is
45 feet above the ground. You throw a baseball to her with an initial upward
velocity of 50 feet per second. If you released the baseball 5 feet above the
ground, did it reach your friend? Explain. HINT: Use a vertical motion model
for an object that is thrown: h = —16 1 2 + vt + s. (Lesson 9.7)
Match the inequality with its graph. (Lesson 9.8)
5. y < —2x 2 + 4x — 2 6. y < —2x 2 + 3x + 2 7. y > —2x 2 — 3x + 2
2
2
-2
/
l
i
i
X
\
\
\
A
'
Sketch the graph of the inequality. (Lesson 9.8)
8. y > 2x 2 + 5 9. y < x 2 + 3x 10 . y > —x 2 — 2
11.y<x 2 + 3x — 2 12. y > x 2 + 2x + 1 13. y < —x 2 + 2x — 3
Chapter 9 Quadratic Equations and Functions
Q Chapter Summary
” and Review
• square root, p. 499
• positive square root, p. 499
• negative square root, p. 499
• radicand, p. 499
• perfect square, p. 500
• radical expression, p. 501
• quadratic equation, p. 505
• simplest form of a
radical expression, p.511
• quadratic function, p. 520
• parabola, p. 520
• vertex, p. 521
• axis of symmetry, p. 521
• roots of a quadratic
equation, p. 527
• quadratic formula, p. 533
• discriminant, p. 540
• quadratic inequalities, p. 547
• graph of a quadratic
inequality, p. 547
Square Roots
Examples on
pp. 499-501
Positive real numbers have a positive square root and a negative square root.
The radical symbol V~~ indicates the positive square root of a positive number.
a. V36 = 6
b. -V81 = -9
36 is a perfect square: 6 2 = 36.
81 is a perfect square: 9 2 = 81, so V*Tf = 9 and -V8T = -9.
Evaluate the expression.
1. -V4 2. Vl44 3. VlOO 4. -V25
Solving Quadratic Equations By Finding Square Roots
Examples on
pp. 505-507
To find the real solutions of a quadratic equation in the form ax 2 + c = 0,
isolate x 2 on one side of the equation. Then find the square root(s) of each side.
2x 2 - 98 = 0
2x 2 = 98
x 2 = 49
x = ±V49
x — ±1
Write original equation.
Add 98 to each side.
Divide each side by 2.
Find square roots.
7 2 = 49 and (-7) 2 = 49
Solve the equation.
5. x 2 = 144 6. 8j 2 = 968 7. 5j 2 - 80 = 0 8. 3x 2 - 4 = 8
Chapter Summary and Review
Chapter Summary and Review continued
q.3 Simplifying Radicals
Examples on
pp. 511-513
You can
a. V28 = V4 • 7
= V4 • V7
= 2V7
Vl6
V3
4
V3
4 _ V3
V3 ’
4V3
3
use properties of radicals to simplify radical expressions.
Factor using perfect square factor.
Use product property.
Remove perfect square factor from radicand.
Use quotient property.
Remove perfect square factor from radicand.
V3
Multiply by a value of 1: = 1.
Simplify.
Simplify the expression.
9. V45 10. V28
4.4 Graphing Quadratic Functions
Examples on
pp. 520-522
Sketch the graph of y = x 2 — 4x — 3.
function, a — 1, b — —4, and c — — 3.
Q Find the x-coordinate of the vertex.
© Make a table of values, using x- values
to the left and right of x = 2.
In this quadratic
o Plot the points. The vertex is (2, —7). Connect the points to form a parabola that
opens up since a is positive. The axis of symmetry is the vertical line x = 2. The
y-intercept is —3.
Sketch the graph of the function. Label the coordinates of the vertex.
13.y = x 2 — 5x + 4 14. y = —x 2 + 2x — 1 15. y = 2x 2 — 3x — 2
Chapter 9 Quadratic Equations and Functions
Chapter Summary and Review continued^
Solving Quadratic Equations by Graphing
Examples on
pp. 526-528
Use a graph to estimate the solutions of
Q Write the equation in the standard form ax 2 + bx + c = 0.
—x 2 + 3x = 2 Write original equation.
—x 2 + 3x — 2 = 0 Subtract 2 from each side.
© Sketch the graph of the related quadratic function
y = — x 2 + 3x — 2. The x-intercepts of the graph are the
solutions of the quadratic equation.
© Estimate the values of the x-intercepts.
ANSWER ► From the graph, the x-intercepts appear to be 1 and 2. Check your
solutions algebraically by substituting each one in the original equation.
—x 2 + 3x = 2.
Use a graph to estimate the solutions of the equation. Check your
solutions algebraically.
16.x 2 — 3x = -2 17. —x 2 + 6x = 5 18. x 2 — 2x = 3
Solving Quadratic Equations by the Quadratic Formula
Examples on
pp. 533-535
You can solve equations of the form ax 2 + bx + c = 0 by substituting the
values of a , b, and c into the quadratic formula. Solve x 2 + 6x — 16 = 0.
Quadratic Formula:
—b ± Vb 2 - 4 ac
2 a
when a ^ 0 and b 2 — 4ac > 0.
The equation lx 2 + 6x — 16 = 0 is in standard form. Identify a = 1, b = 6, and c = —16.
= -6 ± V6 2 - 4(1)(— 16)
X 2(1)
-6 ± V36 + 64
X= - 2 -
-6 ± VlOO
X =
-6 ± 10
2
ANSWER ► The two solutions are x =
-6 + 10
2
2 and x =
- 8 .
Use the quadratic formula to solve the equation.
19. 3x 2 — 4x + 1 = 0 20. —2x 2 + x + 6 = 0
21 . 10x 2 - llx + 3 = 0
Chapter Summary and Review
Chapter Summary and Review continued
qj Using the Discriminant
Examples on
pp. 540-542
You can use the discriminant, b 2 — 4ac, to find the number of
solutions of a quadratic equation in the standard form ax 2 + bx + c = 0. A
positive value indicates two solutions, zero indicates one solution, and a negative
value indicates no real solution. The value of the discriminant can also be used to
find the number of x-intercepts of the graph of y = ax 2 + bx + c.
EQUATION
3x 2 — 6x + 2 = 0
2x 2 + 8x + 8 = 0
x 2 + lx +15 — 0
DISCRIMINANT
(-6) 2 - 4(3)(2) = 12
8 2 - 4(2)(8) = 0
7 2 - 4(1)(15) = -11
NUMBER OF SOLUTIONS
2
1
0
Determine whether the equation has two solutions, one solution, or
no real solution.
22. 3x 2 — 12x + 12 = 0 23. 2x 2 + lOx + 6 = 0 24. — x 2 + 3x — 5 = 0
Find the number of x-intercepts of the graph of the function.
25. y = 2x 2 — 3x — 1 26. y = —x 2 — 3x + 3 27. y = x 2 + 2x + 1
<\.Z Graphing Quadratic Inequalities
Examples on
pp. 547-549
Sketch the graph of y < x 2 — 9.
0 Sketch the graph of y = x 2 — 9 that corresponds to y < x 2 — 9.
b
The x-coordinate of the vertex is — —, or 0. Make a table of values, using x-values
to the left and right of x = 0
H
- 3
-2
-1
0
1
2
3
B
-5
-8
-9
-8
-5
0
© Plot the points and connect them with a smooth
curve to form a parabola. Use a dashed line
since the inequality contains the symbol <.
0 Shade the region below the parabola because
the inequality states that y is less than x 2 — 9.
Sketch the graph of the inequality.
28. y <x 2 - 4 29. y > -x 2 - 2x + 3 30. y > lx 2 - 4x - 6
Chapter 9 Quadratic Equations and Functions
u.
M'/Mf
Cf Chapter Test
Evaluate the expression.
1. V64 2.
-V25
Solve the equation or write no real solution.
5- x 2 = 1 6 . n 2 = 36
9. t 2 - 64 = 0 10. 5x 2 + 125 = 0 1
Simplify the expression.
13.VI50 14.5,
4_
25
15 .
±Vl69
4 . -VToo
4 y 2 = 16
8 . 8x 2 = 800
2x 2 + 1 = 19
12 . x 2 + 6 = -10
/Z7
[9
V 45
16 ‘ \7
Give the letter of the graph that matches the function.
17 m y = —x 2 — 2x + 3
19. y = 2x 2 + x - 3
Use a graph to estimate the solutions of the equation. Check your
solutions algebraically.
20.x 2 — 4 = 5 21. —x 2 + lx — 10 = 0 22. —2x 2 + 4x + 6 = 0
Use the quadratic formula to solve the equation.
23.x 2 - 6x - 27 = 0 24. -x 2 + 3x + 10 = 0
25. 3x 2 + 4x — 7 = 0
Find the value of the discriminant. Then determine whether the equation
has two solutions , one solution , or no real solution.
26. —3x 2 + x - 2 = 0 27. x 2 - 4x + 4 = 0 28. 5x 2 - 2x - 6 = 0
Sketch the graph of the inequality.
29. y < x 2 + 2x — 3 30. y < — x 2 + 5x — 4
31. _y > x 2 + 7x + 6
VERTICAL MOTION In Exercises 32 and 33, suppose you are standing on
a bridge over a creek, holding a stone 20 feet above the water.
32. You release the stone. How long will it take the stone to reach the water?
Use a vertical motion model for an object that is dropped: h = — 16 1 2 + s.
33. You take another stone and toss it straight up with an initial velocity of 30
feet per second. How long will it take the stone to reach the water? Use a
vertical motion model for an object that is thrown: h = —16 1 2 + vt + s.
Chapter Test
Chapter Standardized Test
Tip
If you are unsure of an answer, try to eliminate some
of the choices so you can make an educated guess.
1. Which number is a perfect square?
(A) -25 CD VToo
CD 55 CD 100
2. Which one of the following is not a
quadratic equation?
(A) x 2 4 — 0 Cb) -9 + x 2 = 0
CD —lx + 12 = 0 CD — 2 + 9x +X 2 = 0
3. Which value of / is a solution of
2* 2 - 21 = 51?
(A) -6 CD “4
CD Vl5 CD 4
4. What are the values of x when
3X 2 - 78 = 114?
(A) ±2\/3 CD ±6
CD ±4V3 CD —8
5. Which radical expression is in simplest
form?
® Vs ® V6
CE>Vl2
CD None of these
6- Find the area of the rectangle.
(A) 4Vl5
CD 12V5
CD 60
CD 240
Vi2
Y20
7. What is the x-coordinate of the vertex of
the graph of y = — 2x 2 — x + 8?
CS> “I CD
8_ What are the x-intercepts of the graph of
y — —x 2 — 6x + 40?
(A) —10 and 4 CD —4 and 10
CD 0 and 4 CD 4 and 10
9, Which function has a y-intercept of 6?
(a) y = 6x 2 + 2 CD y ~ 2x 2 + 6x
CD y = 2x 2 + 6 CD y = 2x 2 - 6
10, What is the value of the discriminant of the
equation 5x 2 + 2x — 7 = 0?
(A) -136 CD 2
CD 12 CD 144
11. Which inequality is represented by the
(A) y < — 4x 2 + 8x - 5
CD y > ~4x 2 + 8x — 5
CD y ^ -4x 2 + 8x - 5
CD y - ~4x 2 + 8x — 5
Chapter 9 Quadratic Equations and Functions
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
Use the Distributive Property
Use the distributive property to rewrite the expression without parentheses,
a. 6(14x + 9) b. -3(5* - 2)
Solution
a. 6(14* + 9) = 6(14*) + 6(9)
= 84x + 54
b. —3(5* - 2) = -3(5*) - (—3)(2)
= —15* + 6
Distribute 6 to each term of (14x + 9).
Multiply.
Distribute -3 to each term of (5x - 2).
Multiply.
Try These
Use the distributive property to rewrite the expression without
parentheses.
1.8(2*-12) 2. 4(3*+ 2) 3. -5(13 - m)
4. 8(—5 + 6 c) 5. 10(8 +3a) 6.-12(5 + 60
2 Combine Like Terms
Simplify the expression,
a. 5x — 9y + 6x — 8 y
b. 6 + 3(* - 1)
Solution
a. 5x — 9y + 6x — 8y = 5x + 6x — 9y — 8y
= (5 + 6)* + (-9 - 8)y
= 11* - 17y
Group like terms.
Use distributive property.
Add coefficients.
b. 6 + 3(* — 1) = 6 + 3(*) - 3(1)
Use distributive property.
— 6 + 3x — 3
Multiply.
= 3x + 6 — 3
Group like terms.
Student HeCp
► Extra Examples
More examples
anc j p rac tice
exercises are available at
www.mcdougallittell.com
= 3x + 3
Try These
Simplify the expression.
7. 8n — 2n + 18m + 3 m
9. 4 + 2(* + 3)
Combine like terms.
8. 25c — Id — 10 d + 5c
10. 2x + 4(2x — 5)
Maintaining Skills
jup'flfi
|-<} Cumulative Practice
Does the table represent a function? Explain. (Lesson 1.7)
1. I ■ * I „ I * [ „ I 2.
Input x
5
3
5
2
Output y
8
7
4
3
3
6
9
12
HL
8
5
8
Simplify the expression. (Lesson 2.8)
3.
21 x - 54
4.
66 r + 39
-3
5.
-72 + 16h
-8
6 .
-28 - 10;
-2
7. PRETZELS You sell pretzels at a baseball game for $1.25 each. Write and solve
an equation to find how many pretzels you need to sell to earn $60. (Lesson 3.2)
Solve the percent problem. (Lesson 3.9)
8- What number is 75% of 48? 9- 54 is 15% of what number?
10. 64 is what percent of 80? 11. 20 is what percent of 5?
Find the x-intercept and the /-intercept of the line. (Lesson 4.4)
12.x + 2y = 8 13. x — 6y = —3 14. y = 12x — 2
15. y — —5x + 14 16. —2x — ly = 20 17. — 14x — y — 28
Determine whether the graphs of the two equations are parallel lines.
Explain your answer. (Lesson 4.7)
18. line a: y — 2x + 3 19. line a: y — 4x + 1 = 0 20. line a: 2x — 5y — —30
lineb: y — 3x = 2 lineb: 2y = 8x + 6 line b: —4x + lOy = —10
Write in slope-intercept form the equation of the line that passes through
the given points. (Lesson 5.3)
21.(7, 3) and (6, 4)
22. (2, 5) and (11, 8)
23. (-4, 6) and (3, -8)
24. (0, -12) and (3, 3)
25. (5, 2) and (-5, 7)
26. (5, -10) and (8, 2)
Write the equation in standard form with integer coefficients. (Lesson 5.4)
27. 3x — 5_y + 6 = 0
28. 6y = 2x + 4
29. -2x + ly - 15 = 0
30. y = \x — 1
31. y = -\x + 6
32. y = + 5
Solve the inequality. (Lessons 6.1-6.3)
33. m + 5 < —4
34. 8 > c - 3
35. —5f > 40
36. \x<9
37. -\y < -7
38.-f>2
39. 5y + 6 > —14
40. 4 (a - 1) < 8
41. 6 + 2k<3k — 1
Chapter 9 Quadratic Equations and Functions
Solve the linear system.
42.x + 4y = 0
x= 12
45. 3x + y = —19
— 32x + 4y = 144
(Lessons 7.1-7.3)
43. x + y = 8
2x + y = 10
46. —2x + 20 y = 10
x - 5y = -5
44. lOx - 3y = -1
—5x + 3y = 2
47. 4x + 2y = 3
3x — Ay — 5
48. VEGETABLES You buy 13 bell peppers to use in a vegetable platter. Green
peppers cost $1.20 each and red peppers cost $1.50 each. If you spend a total
of $18, how many of each kind are you buying? (Lesson 7.4)
Graph the system of linear inequalities. (Lesson 7.6)
49.x >0
y>0
x < 5
y<2
50. x > 2
x — y <2
x + 2y < 6
51. 3x + 5y > 15
x — 2y < 10
x > 1
52. -x + 4y < 8
—4x + y > —4
2x + y>~4
Simplify the expression. Use only positive exponents. (Lessons 8.1,8.2,8.4)
53. x 3 • x 6
54. (c 5 ) 4
55. (8/) 2
56. —3(—5) 2
57. 3 2 • 3 3
58. 3x 5 _y -3
59. 4“ 2 • 4°
"■dr
0)
1
i
00
x 8
62. -T
x J
3x 2 v 6xy 2
63. —t- ~z
3,3 2y
Perform the indicated operation without using a calculator. Write the
result in scientific notation. (Lesson 8.5)
65. (5 X 10" 2 )(3 X 10 4 ) 66. (6 X 10 _8 )(7 X 10 5 ) 67. (20 X 10 6 )(3 X 10 3 )
68. (7 X 10 3 ) -3 69. 8,8 X 10 | 70. (2.8 X 10“ 2 ) 3
v ' 1.1 X 10 _1 v
Simplify the radical expression. (Lesson 9.3)
71.V40
72. V52
73. V72
74. V96
75.^V84
76
76 ‘ V 36
77. 3jf
78
78 ‘ V75
79 -v^0
80. - 2.4
V 0
00
“-75
Sketch the graph of the quadratic function or the quadratic inequality.
(Lessons 9.4, 9.8)
83. y = — 3x 2 + 6x — 1 84. y > 5x 2 + 20x + 15 85. y < 2x 2 — 5x + 2
SENDING UP FLARES In Exercises 86 and 87, a flare is fired straight up
from ground level with an initial velocity of 100 feet per second.
(Lesson 9.7)
86 . How long will it take the flare to reach a height of 150 feet? Use the vertical
motion model h = —I6t 2 + vt + s.
87. Will the flare reach a height of 180 feet? Explain.
Cumulative Practice
>hapt*rs
J-Cj Project
illl
Designing
Materials
• graph paper
• pencil
• ruler
• calculator
OBJECTIVE Compare step measurements to see how they affect
stairway design.
Investigating the Data
The horizontal part of a step is
the tread , and the vertical part is
the riser. The table gives the
tread and riser measurements of
four different stairways.
Stairway
Tread (in.)
Riser (in.)
A
10
7
B
11
7
C
9
8
D
12
6
1- Use the measurements in the table to draw three steps for each
Stairway A-D on a piece of graph paper.
2 . Analyze your drawings. Which stairway is the steepest? Which stairway
gives the most foot space on a step?
3. For each Stairway A-D, find the ratio of riser size to tread size
Then write each ratio as a decimal rounded to the nearest tenth.
What characteristic of a stairway do these ratios describe?
Two generally accepted rules for designing stairways are listed below.
Rule 1: The sum of one tread and one riser is from 17 inches to I? inches.
Rule 2: The sum of one tread and two risers is from 24 inches to 25 inches.
I riser
\ tread
4. You can use the following linear inequalities to represent Rule 1.
t + r > 17 and t + r < 18
Write linear inequalities to represent Rule 2. Then use the inequalities to
show that each Stairway A-D follows one of the rules.
5. Graph the system of four inequalities on the same coordinate plane. Use
the horizontal axis for t and the vertical axis for r. Then use the values in
the table to label the point ( t , r) for each Stairway A-D. What does each
solution of the system represent?
6. Name any other point E that is a solution of the system. Give tread and riser
measurements for a Stairway E that the point represents.
Chapter 9 Quadratic Equations and Functions
Presenting Your Results
Write a report about tread and riser
measurements for stairways.
• Include a discussion of how
various tread and riser
measurements create stairways
that are different.
• Compare Stairways A-E in terms
of steepness and foot space. Use
diagrams or numbers to support
your comparison.
• Include your answers to Exercises 1-6.
• Explain how the two rules for designing stairways limit the possible
measurements for treads and risers. Use the graph of the linear system to
give the range of possible measurements for treads and the range of possible
measurement for risers. HINT: You can use inequalities to represent these
ranges.
• Give some examples of tread and riser measurements that do not follow one
of the given rules for designing stairways. Explain how these measurements
might create stairways that are hard to use or unsafe. Draw diagrams to
support your explanation.
v
*
*
*
9
rfj
A step is one
unit of a
stairway.
Step measurements
can affect the
comfort and safety
of stairways.
Extending the Project
Design a stairway by determining its tread and riser
measurements. Suppose the vertical distance from one
floor to the next is 105 inches.
1. Decide on a riser measurement that will give
you a whole number of steps on your stairway.
HINT: You can choose fractional measurements
for your treads and risers.
2 . Decide on a tread measurement which, along
with your riser measurement, follows one of the
generally accepted rules for designing stairways.
3. Find the slope of your stairway.
4. On graph paper, make a scale drawing of your
whole stairway. Number the steps.
Project
Polynomials and
Factoring
How wide and how deep are each of
the dishes of the VLA radio telescope?
APPLICATION: RadioTelescopes
The Very Large Array (VLA) radio telescope in
New Mexico is the most powerful radio telescope in the
world. It consists of 27 mobile parabolic dishes that are
combined electronically to provide images that would
result from a single dish that is 22 miles (116,160 feet)
across.
A cross section of one of the VLA dishes is shown
below, where x and y are measured in feet. This cross
section of a dish can be modeled by a polynomial
equation.
Think & Discuss
Use the graph above to answer the following questions.
1. Find the x-intercepts. How can you use this
information to find the diameter of the dish?
2 . Estimate the depth of the dish.
Learn More About It
You will use an algebraic model of the VLA radio
telescope dishes in Exercises 46 and 47 on p. 592.
APPLICATION LINK More information about the VLA radio
telescope is available atwww.mcdougallittell.com
Study Guide
PREVIEW
What’s the chapter about?
• Adding, subtracting, and multiplying polynomials
• Factoring polynomials
• Solving quadratic and cubic equations by factoring
' Key Words
- 1
• monomial, p. 568
• FOIL pattern, p. 576
• degree of a monomial, p. 568
• factored form, p. 588
• polynomial, p. 569
• zero-product property, p. 588
• binomial, p. 569
• factor a trinomial, p. 595
• trinomial, p. 569
• perfect square trinomial, p. 609
• standard form, p. 569
• prime polynomial, p. 617
• degree of a polynomial, p. 569
• factor completely, p. 617
PREPARE
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to p. 100)
1 _ Which equation uses the distributive property correctly?
Ca) 3x(x + 6) = 3x 2 + 6 Cb) 3x(x + 6) = 3x 2 + 18x
Cc?) 3x(x + 6) = x + 18x = 19x Cp) 3x(x + 6) = 3x + 18x = 2lx
Skill Check (refer to pp. 444 , 540)
2_ Simplify the expression (x 6 ) 2 .
(a) x 8 Cb) x 4 (c?> x 12 C5) * 3
3. How many real solutions does the equation 3x 2 — 4x + 6 = 0 have?
Ca) Three solutions Cb) Two solutions
Cc?) One solution CD?) No solution
STUDY TIP
Make Property Cards
Be sure to express the
property in words and
in symbols.
leaS,me ^‘^tcr smstte!m
Ifab — 0, then a = o or b =
0 .
Chapter 10 Polynomials and Factoring
p
*, DEVELOPING CONCE
it A<
For use with
Lesson 10.1
Goal
Use algebra tiles to
model the addition
of polynomials.
Materials
• algebra tiles
Question
How can you model the addition of polynomials with algebra tiles?
Explore
Algebra tiles can be used to model polynomials.
1
-1
Each of these 1-by-1
square tiles has an
area of 1 square unit.
Each of these 1-by-x
rectangular tiles has
an area of xsquare units.
Each of these x-by-x
square tiles has an
area of x 2 square units.
Q You can use algebra tiles to add the polynomials x 2 * * 5 * * 8 + 4x + 2 and
2x 2 — 3x — 1.
+
+
+
+
+
+ +
+
+
■
x 2 + 4x +2
2X 2 3x - 1
Student MeCp
\
p Look Back
For help with zero
pairs, see p. 77.
^ _ J
Q To add the polynomials, combine
like terms. Group the x 2 -tiles,
the x-tiles, and the 1-tiles.
+
+
+
mm
mm
warn
mm
+
+
© Rearrange the tiles to form zero
pairs. Remove the zero pairs. The
sum is 3x 2 + x + 1.
■■
Think About It
In Exercises 1-6, use algebra tiles to find the sum. Sketch your solution.
1- ( x 2 + x — l) + (4x 2 + 2x — 3) 2 . (3x 2 + 5x — 6) + (—2x 2 — 3x — 6)
3- (5x 2 — 3x + 4) + (—x 2 + 3x — 2) 4. (2x 2 — x — l) + (— 2x 2 + x + l)
5. (4x 2 — 3x — l) + (— 2x 2 + x + l) 6- (4x 2 + 5) + (4x 2 + 5x)
7. Describe how to use algebra tiles to model subtraction of polynomials.
Use algebra tiles to find the difference. Sketch your solution.
8. (x 2 + 3x + 4) — (x 2 + 3) 9, (x 2 — 2x + 5) — (3 — 2x)
Developing Concepts
Adding and Subtracting
Polynomials
Goal
Add and subtract
polynomials.
How large is the walkway around a pool?
Key Words
• monomial
• degree of a monomial
in one variable
• polynomial
• binomial
• trinomial
• standard form
• degree of a polynomial
in one variable
In Example 5 you will use
subtraction of polynomials
to find the area of a walkway
around a pool.
A monomial is a number, a variable, or the product of a number and one or more
variables with whole number exponents. The following expressions are monomials.
8 —2x 3 x 2 y ~x 2
The degree of a monomial is the sum of the exponents of the variables in the
monomial. The degree of 3x 2 is 2. The degree of —6 z 4 is 4. The degree of 3 x 2 y
is 2 + 1, or 3.
I Student HeCp
► Reading Algebra
The monomial -5x 4 is
read as "negative five
times xto the fourth
power." The coefficient
is -5. .*
V _ >
i Find the Degree of a Monomial
State the degree of the monomial.
•►a. — 5x 4 b. ^ b 3 c. 12
Solution
a. The exponent of x is 4.
ANSWER ► The degree of the monomial is 4.
b. The exponent of b is 3.
ANSWER ► The degree of the monomial is 3.
c. Recall 12 = 12x°, so the exponent is 0.
ANSWER ^ The degree of the monomial is 0.
State the degree of the monomial.
1. 6x 3 2. 4 p 3- —10
4- —3a 5
Chapter 10 Polynomials and Factoring
Student HeCp
p Morel Examples
More examples
are available at
www.mcdougallittell.com
POLYNOMIALS A polynomial is a monomial or a sum of monomials. A
polynomial such as x 2 + (—■ Ax) + (—5) is usually written as x 2 — 4x — 5.
Each of the following expressions is a polynomial.
4x 3 x 3 — 8 lx 2 — 4x + 6
A polynomial of two terms is a binomial. A polynomial of three terms is a
trinomial. Polynomials are usually written in standard form, which means that
the terms are arranged in decreasing order, from largest exponent to smallest
exponent. The degree of a polynomial in one variable is the largest exponent
of that variable.
2 Identify Polynomials
POLYNOMIAL
DEGREE
IDENTIFIED
BY DEGREE
IDENTIFIED BY
NUMBER OF TERMS
a. 6
0
constant
monomial
b. 3x + 1
1
linear
binomial
c. —x 2 + 2x — 5
2
quadratic
trinomial
d. 4x 3 - 8x
3
cubic
binomial
Identify Polynomials
Identify the polynomial by degree and by the number of terms.
5. 8x 6- lOx — 5 7. x 2 — 4x + 4 8. —24 — x 3
To add polynomials, you can use either a vertical format or a horizontal format,
as shown in Example 3.
Student Hedp
p Look Back
For help with
combining like
terms, see p. 108.
ly _ J
3 Add Polynomials
Find the sum. Write the answer in standard form,
a. (5x 3 — 2x + x 2 + 7) + (3x 2 + 7 — 4x) b. (2x 2 + x — 5) + (x + x 2 + 6)
Solution
a. Vertical format: Write each 5x 3 + x 2 — 2x + 7
expression in standard form. _ 3x 2 — 4x + 7
Line up like terms vertically. 5 X 3 + q x 2 _ + ^4
b_ Horizontal format: Group like terms.
(2x 2 + x — 5) + (x + x 2 + 6) = (2x 2 + x 2 ) + (x + x) + (—5 + 6)
= 3x 2 + 2x + 1
Add Polynomials
Find the sum. Write the answer in standard form.
9- (x 2 + 3x + 2) + (2x 2 — 4x + 2) 10. (2x 2 — 4x + 3) + (x 2 — 4x — 4)
10.1
Adding and Subtracting Polynomials
Student HeCp
^
► Study Tip
Remember to change
signs correctly.
I ^
J33IE3I 4 Subtract Polynomials
Find the difference.Write the answer in standard form.
a_ (— 2x 3 + 5x 2 — 4x + 8) — (— 2x 3 + 3x — 4)
b- (3x 2 — 5x + 3) — ( 2x 2 — x — 4)
Solution
a. Use a vertical format. To subtract one polynomial from another, you add
the opposite. One way to do this is to multiply each term in the subtracted
polynomial by — 1 and line up like terms vertically. Then add.
(— 2x 3 + 5x 2 — 4x + 8) —2x 3 + 5x 2 — 4x + 8
— ( —2x 3 + 3x — 4) Add the opposite. + 2x 3 — 3x + 4
.....t 5x 2 — lx + 12
b. Use a horizontal format. Group like terms and simplify.
(3x 2 — 5x + 3) — (2x 2 — x — 4) = 3x 2 — 5x + 3 — 2jc 2 + x + 4
= (3x 2 — 2x 2 ) + (— 5x + x) + (3 + 4)
= x 2 — 4x + 7
lEmSU 5 Subtracting Polynomials
You are installing a swimming
pool. Write a model for the area x x + 6
of the walkway. 1 x
Solution
Verbal
Model
Labels
Algebraic
Model
Area of
Total
Area of
walkway
area
pool
Area of walkway = A (square inches)
Total area = (6x)(x + 6) (square inches)
Area of pool = (3x)(x) (square inches)
A = (6x)(x + 6) — (3x)(x)
= 6x 2 + 36x — 3x 2
= 3x 2 + 36x
ANSWER ► A model for the area of the walkway is A = 3x 2 + 36x.
Subtract Polynomials
Find the difference. Write the answer in standard form.
11- (2x 2 + 3x — 5) — (2x + 8 + x 2 ) 12 - (4x 3 + 4x 2 — x — 2) — (3x 3 — 2x 2 + l)
Chapter 10 Polynomials and Factoring
MB Exercises
Guided Practice
Vocabulary Check 1. Is —4x 2 + 5x — 3x 3 + 6 written in standard form? Explain.
2 . Is 9x 2 + 8x — 4x 3 + 3 a polynomial with a degree of 2? Explain.
Skill Check Identify the polynomial by degree and by the number of terms.
3- —9y + 5 4. 6x 3 5- 12x 2 + lx
6- 4w 3 — 8w + 9 7. 7y + 2y 3 — y 2 8- —15
ERROR ANALYSIS In Exercises 9 and 10, find and correct the error.
Find the sum or the difference of the polynomials.
11 . (2x - 9) + (jc - 7) 12 . (lx - 3) - (9x - 2)
13. (. x 2 — 4x + 3) + (3x 2 — 3x — 5) 14. (3x 2 + 2x — 4) — (2x 2 + x — l)
Practice and Applications
LOGICAL REASONING Complete the statement with always , sometimes ,
or never.
15. The terms of a polynomial are ? monomials.
16. Like terms ? have the same coefficient and same variable part.
17. The sum of two trinomials is ? a trinomial.
18. A binomial is ? a polynomial of degree 2.
19. Subtraction is ? addition of the opposite.
Student HeCp
► Homework Help
Example 1: Exs. 20-23
Example 2: Exs. 24-32
Example 3: Exs. 33-50
Example 4: Exs. 33-50
Example 5: Exs. 51, 52
v j
FINDING THE DEGREE State the degree of the monomial.
20.8 n 21.12 b 4 22. -c 3 23.-100w 4
CLASSIFYING POLYNOMIALS Write the polynomial in standard form.
Then identify the polynomial by degree and by the number of terms.
24. 2x 25. 20 m 3 26. 7 - 3w
27.-16 28. 8 + 5y 2 — 3y 29.-14+llj 3
30. -2x + 5x 3 - 6 31. —4b 2 + lb 3 32. 14w 2 + 9w 3
10.1
Adding and Subtracting Polynomials
VERTICAL FORMAT Use a vertical format to add or subtract.
33. (l2x 3 + x 2 ) — (l8x 3 — 3x 2 + 6) 34. (a + 3a 2 + 2a 3 ) — ( a 2 — a 3 )
35. (2m — 8 m 2 — 3) + (m 2 + 5m) 36. (8 y 2 + 2) + (5 — 3y 2 )
37. (3x 2 + lx — 6) — (3x 2 + lx) 38. (4x 2 — lx + 2) + (— x 2 + x — 2)
HORIZONTAL FORMAT Use a horizontal format to add or subtract.
39. (x 2 — 7) + (2x 2 + 2) 40. (—3a 2 + 5) + (— a 2 + 4a — 6)
41. (z 3 + z 2 + l) - Z 2 42. 12 - (y 3 + 10 y + 16)
43. (3 n 2 + 2n — l) — (n 3 — n — 2) 44. (3a 3 — 4a 2 + 3) — (a 3 + 3 a 2 — a — 4)
CONSTRUCTION
MANAGERS are responsible
for coordinating and
managing people, materials,
and equipment; budgets,
schedules, and contracts; and
the safety of employees and
the general public.
POLYNOMIAL ADDITION AND SUBTRACTION Use a vertical format or a
horizontal format to add or subtract.
45. (9x 3 + 12x) + (l6x 3 — 4x + 2) 46. (— 2t 4 + 6 1 2 + 5) — (— 2t 4 + 5 1 2 + l)
47. (3x + 2x 2 — 4) — (x 2 + x — 6) 48. ( u 3 — u) — (u 2 + 5)
49. (~lx 2 + 12) - (6 - 4x 2 ) 50. (lOx 3 + 2x 2 - ll) + (9x 2 + 2x - l)
BUILDING A HOUSE In Exercises 51 and 52, use the following information.
You plan to build a house that is 1.5 times as long as it is wide. You want the land
around the house to be 20 feet wider than the width of the house, and twice as
long as the length of the house, as shown in the figure below.
51. Write an expression for the
area of the land surrounding
the house.
52. If x = 30 feet, what is the area
of each floor of the house?
What is the area of the entire
property?
ENERGY USE In Exercises 53 and 54, use the following information. From
1989 through 1993, the amounts (in billions of dollars) spent on natural gas N
and electricity E by United States residents can be modeled by the following
equations, where t is the number of years since 1989.
► Source: U.S. Energy Information Administration
Gas spending model: N — 1.488^ 2 — 3.403^ + 65.590
Electricity spending model: E — — 0.107^ 2 + 6.897 1 + 169.735
53. Find a model for the total amount A (in billions of dollars) spent on natural
gas and electricity by United States residents from 1989 through 1993.
54. CRITICAL THINKING According to the models, will more money be spent
on natural gas or on electricity in 2020. HINT: It may be helpful to graph the
equations on a graphing calculator to answer this question.
Chapter 10 Polynomials and Factoring
Standardized Test
Practice
Mixed Review
Maintaining Skills
55. MULTIPLE CHOICE Which of the following polynomials is not written in
standard form?
(A) 8 n 2 - 16 n + 144 CD 3y 3 - y 2 - 15 + 4y
CD 3w 4 + 4w 2 — w — 9 (5) 3 p 4 — 6p 3 + 2p + 16
56. MULTIPLE CHOICE What is the degree of -6x 4 ?
CD 4 <3D -6 (H) -4 CD 6
57. MULTIPLE CHOICE Which of the following is classified as a monomial?
(A) x + 1 CD 5 — y 2 CD a 3 — a — l CD 2 y
DISTRIBUTIVE PROPERTY Simplify the expression. (Lesson 2.6)
58. -3(jc + 1) - 2 59. (2x - 1)(2) + x
60. llx + 3(8 -x) 61. (5x - l)(-3) + 6
62. -4(l-x) + 7 63. -12*- 5(11 - x)
64. GAS MILEAGE The table below shows mileage and gasoline used for 6
months. For each of these months, find the mileage rate in miles per gallon.
Round to the nearest tenth. (Lesson 3.8)
Mileage (miles)
295
320
340
280
310
355
Gas Used (gallons)
12.3
13.3
14.2
11.6
12.9
14.8
B EXPONENTIAL EXPRESSIONS In Exercises 65-70, simplify. Then use a
calculator to evaluate the expression. (Lesson 8. V
65. 2 2 • 2 3 66 . (3 2 • l 3 ) 2 67. [(— l) 8 • 2 4 ] 2
68. (- 1 • 3 2 ) 3 69. (2 2 • 2 2 ) 2 70. (3 2 • 2 3 ) 3
71 . ALABAMA The population P of Alabama (in thousands) for 1995 projected
through 2025 can be modeled by P = 4227(1.0104/, where t is the number
of years since 1995. Find the ratio of the population in 2025 to the
population in 2000. (Lesson 8.6) ►Source: U S. Bureau of the Census
ADDING FRACTIONS Add. Write the answer as a mixed number in
simplest form. (Skills Review p. 764)
12 3
72 — + 1 —
/Z " 11 A 11
2 3
73 -5 +3f
74 - >1 + 1
75. i+l|
76. +
3 6
3 19
77 2 — + —
"■ Z 4 20
78. ji + 4
2 11
79 . 9 7 + 3 ^
1 4
80 . 2 ?+ 3
S’. 2i + f
82. 14 + 4
83 ->S + 4
10.1
Adding and Subtracting Polynomials
_tjxj Pvlynvnnuiz
For use with
Lesson 10.2
Goal
Multiply two polynomials
using the distributive
property.
Quostfoii
How can you multiply two polynomials using the distributive property?
Materials
• paper
• pencil
The arithmetic operations for polynomials are very much like the corresponding
operations for integers. For example, you can multiply (x + 3)(2x + 1) by using
the distributive property.
2x + 1
X x + 3
6x + 3 - Multiply 2x + 1 by 3.
2x 2 + jc _ -* - Multiply 2x + 1 by x.
2x 2 + lx + 3
So (x + 3)(2x + 1) = 2x 2 + lx + 3.
Explore
0 To multiply (3x + 2)(x + 4), write
the multiplication vertically.
© Multiply 4 X (3x + 2).
© Multiply v X (3x + 2).
Q Add the terms by using a vertical format.
Align like terms. Then add.
3X-F2
X X-F4
? -F ?
?+-?+- ?
Try Thoso
In Exercises 1-10, multiply the polynomials using the method
shown above.
1. (x + 3)(x + 7)
3. (x — 5)(x + 7)
5. (3jc - 1)(5jc - 2)
7- (x + 4)(x 2 + 2x + 3)
9- (3x + l)(v 2 + 3x + 5)
2 . (2x + 5)(3v + 4)
4. (4jc + 1)(5jc + 2)
6. (3x + 7)(2x + 9)
8. (x — 2)(x 2 — 4x + 6)
10 - ( 4x — l)(x 2 + 5x — 7)
11 - Explain how you can use the distributive property to multiply
(3x + 2)(x + 4) horizontally.
HINT: Use (3x + 2 )(jc + 4) = (3jc + 2)jc + (3x + 2)4 to do so.
Chapter 10 Polynomials and Factoring
Multiplying Polynomials
Goal
Multiply polynomials.
Key Words
• FOIL pattern
What is the total area of a window?
First degree polynomials are often
used to represent length and width.
In Example 5 you will multiply
two polynomials to find the area of
a window.
In Lesson 2.6 you learned how to multiply a polynomial by a monomial by using
the distributive property.
3(2x - 3) = (3)(2jc) - (3)(3) = 6x - 9
In this lesson you will learn how to multiply two binomials by using the
distributive property twice to multiply (x + 4){x + 5).
First distribute the binomial (x + 5) to each term of (x + 4).
(.x + 4)(x + 5) = x(x + 5) + 4(x + 5)
Then distribute the x and the 4 to each term of (x + 5).
= x(x) + x(5) + 4(x) + 4(5)
= x 2 + 5x + 4x + 20 Multiply.
= x 2 + 9x + 20 Combine like terms.
Student HeCp
p Look Back
For help with the
distributive property,
see p. 101.
\ _ )
i Use the Distributive Property
Find the product (.x + 2)(x — 3).
Solution
(x + 2)(x — 3)
x(x — 3) + 2(x — 3) Distribute (x - 3) to each term of (x + 2).
x(x) + x( — 3) + 2(x) + 2(— 3) Distribute x and 2 to each term of (x - 3).
x 2 — 3x + 2x — 6 Multiply.
x 2 — x — 6 Combine like terms.
I __
Use the Distributive Property
Use the distributive property to find the product.
1- (x + l)(x + 2) 2. (x — 2)(x + 4) 3- (2x + l)(x + 2)
10.2 Multiplying Polynomials
Student Hadp
► More Examples
More examples
are ava j| a bie a t
www.mcdougallittell.com
FOIL PATTERN In using the distributive property for multiplying two binomials,
you may have noticed the following pattern. Multiply the First, Outer, Inner, and
Last terms. Then combine like terms. This pattern is called the FOIL pattern.
Product of Product of Product of Product of
First terms Outer terms Inner terms Last terms
\ I
(3x + 4)(x + 5) — 3x 2 + 15x + 4x + 20
= 3x + 19x + 20 Combine like terms.
J 2 Multiply Binomials Using the FOIL Pattern
FOIL
I l J I
(2x + 3)(2x + 1) = 4x 2 + 2x + 6x + 3
= 4x 2 + 8x + 3 Combine like terms.
Multiply Binomials Using the FOIL Pattern
Use the FOIL pattern to find the product.
4. (jc + 1)(jc - 4) 5. (2x - 3)(x - 1)
6 . (jc - 2)(2x + 1)
To multiply two polynomials that have three or more terms, remember that each
term of one polynomial must be multiplied by each term of the other polynomial.
Use a vertical or a horizontal format. Write each polynomial in standard form.
HUES!# 3 Multiply Polynomials Vertically
Find the product (x — 2)(5 + 3x — x 2 ).
Solution
Line up like terms vertically. Then multiply as shown below.
— x 2 + 3x + 5 Standard form
X x — 2 Standard form
2x 2 — 6x — 10 ** - -2(-x 2 + 3x + 5 )
—x 3 + 3x 2 + 5x * - x(-x 2 + 3x + 5 )
—x 3 + 5x 2 — x — 10 Combine like terms.
Multiply Polynomials Vertically
Use a vertical format to find the product.
7. (x + l)(x 2 + 3x — 2) 8- (2x — l)(2x 2 + x — 3) 9. (2x — 3)(3x 2 + x — 4)
Chapter 10 Polynomials and Factoring
4 Multiply Polynomials Horizontally
Find the product ( 4x 2 — 3x — l)(2x — 5).
Solution Multiply 2x — 5 by each term of 4x 2 — 3x — 1.
(4x 2 - 3x - l)(2x - 5)
4x 2 (2x - 5) - 3x(2x - 5) - l(2x - 5)
8x 3 — 20x 2 — 6x 2 + 15x — 2x + 5
8x 3 + (—20x 2 — 6x 2 ) + (15x — 2x) + 5
8x 3 — 26x 2 + 13x + 5
Use distributive property.
Use distributive property.
Group like terms.
Combine like terms.
5 Multiply Binomials to Find an Area
The glass has a height-to-width ratio
of 3 : 2. The frame adds 6 inches to the
width and 10 inches to the height.
Write a polynomial expression that
represents the total area of
the window, including the frame.
Solution
The window has a total height of 3x + 10
and a total width of 2x + 6. The area of
the window is represented by the product
of the height and width.
3x
l3f
-lx-
-H
A = height • width
A = (3x + 10)(2x + 6)
Write area model for a rectangle.
Substitute (3x +10) for height and (2x + 6)
for width.
= 6x 2 + 18x + 20x + 60
= 6x 2 + 38x + 60
Use FOIL pattern.
Combine like terms.
ANSWER ► The area of the window can be represented by the
model A = 6x 2 + 38x + 60.
Multiply Polynomials
In Exercises 10-12, use a horizontal format to find the product.
10 - (x — 4)(x 2 + x + 1) 11- (x + 5)(x 2 — x — 3) 12 - ( 2x + l)(3x 2 + x — 1)
13. Suppose the height-to-width ratio of the glass portion of the window in
Example 5 above were 5:3. Write a model to represent the total area.
10.2 Multiplying Polynomials
H Exercises
Guided Practice
Vocabulary Check 1 - How do the letters in “FOIL” help you remember how to multiply two
binomials?
2 . Give an example of a monomial, a binomial, and a trinomial.
Skill Check Copy the equation and fill in the blanks.
3. (jc - 2)(jc + 3) = x(j_) + (-2)CD 4. (3x + 4)(2x - 1) = 3x(j_) + 4(_2_)
5- (x — 3)(x + 1) = x 2 — 2x — _?_ 6. (x + 2)(x + 6) = x 2 + ? +12
7. (x — 4)(x — 5) = x 2 — 9x + _?_ 8. (x + 2)(2x + 1) = ? + 5x + 2
Use the distributive property to find the product.
9. (4x + 7)(— 2x) 10- 2x(x 2 + x — 5) 11- — 4x 2 (3x 2 + 2x — 6)
12. (a + 4 )(a + 5) 13. (y — 2)(y + 8) 14. (2x + 3)(4x + 1)
Use the FOIL pattern to find the product.
15 . (w — 3 )(yv + 5) 16 . (x + 6)(x + 9) 17 . (x — 4)(8x + 3)
18 . (x — 3)(x + 4) 19 . (x + 8)(x — 7) 20. (3x — 4)(2x — 1)
Practice and Applications
MULTIPLYING EXPRESSIONS Find the product.
21. (2x - 5)(—4x) 22. 3t\lt - f 3 - 3) 23. 2x(x 2 - 8x + 1)
24. ( -y)(6y 2 + 5 y) 25. 4w 2 (3w 3 - 2w 2 - w) 26. ~b\6b 3 - 16 b + 11)
Student HeCp
► Homework Help
Example 1: Exs. 21-35
Example 2: Exs. 36-47
Example 3: Exs. 48-51
Example 4: Exs. 52-55
Example 5: Exs. 56-60
v _>
DISTRIBUTIVE PROPERTY Use the distributive property to find the
product.
27. (t + 8)0 + 5)
30. (a + 8 )(a — 3)
33. (3s ~ l)(s + 2)
28. (x + 6)(x - 2)
31. (j + 2)(2y + 1)
34. (2d + 3)(3 d + 1)
29. (d - 5)(d + 3)
32. (m - 2)(4 m + 3)
35. (4y ~ l)(2y - 1)
USING THE FOIL PATTERN Use the FOIL pattern to find the product.
36. (a + 6 )(a + 7) 37. (y + 5 )(y - 8) 38. (x + 6)(x - 6)
39. (2w - 5)(w + 5) 40. (4b - l)(b - 6) 41. (jc - 9)(2x + 15)
42. (3a - 1 )(a- 9) 43. (2 z + 7)(3z + 2) 44. (4 q - 1)(3 q + 8)
45. (5 1 - 3)(2f + 3) 46. (4x + 5)(4x - 3) 47. (9 w - 5)(7 w - 12)
Chapter 10 Polynomials and Factoring
MULTIPLYING EXPRESSIONS Use a vertical format to find the product.
48. (x + 2)(x 2 + 3x + 5) 49. (d — 5 ){d 2 — 2d — 6)
50. (a — 3)(a 2 — 4a — 6)
51. ( 2x + 3)(3x 2 — 4x + 2)
Unk to
Careers
PICTURE FRAMERS use
math when deciding on the
dimensions of the frame, the
matting, and the glass.
MULTIPLYING EXPRESSIONS Use a horizontal format to find the product.
52. (.x + 4)(x 2 — 2x + 3) 53. (a — 2 )(a 2 + 6a — 7)
54. (m 2 + 2m — 9)(m — 4)
55. (Ay 2 -3 y- 2 )(y + 12)
56. PICTURE FRAME The diagram at
the right shows the dimensions of
a picture frame. The glass has a
height-to-width ratio of 2 : 3. The
frame adds 4 inches to the width
and 4 inches to the height. Write
a polynomial expression that
represents the total area of the
picture, including the frame.
FOOTBALL In Exercises 57 and 58, a football field's dimensions are
represented by a width of (3x +10) feet and a length of (7x +10) feet.
57. Find an expression for
the area A of the football
field. Give your answer
as a quadratic trinomial. ( 3x + 1
58. An actual football field
is 160 feet wide and
360 feet long. For what + 10 )
value of x do the expressions
3x + 10 and lx + 10 give these dimensions?
(lx + 10) ft
Student HeCp
► Homework Help
Help with problem
solving in Exs. 59
and 60 is available at
www.mcdougallittell.com
VIDEOCASSETTES In Exercises 59 and 60, use the following information
about videocassette sales from 1987 to 1996, where t is the number of
years since 1987. The number of blank videocassettes B sold annually in the
United States can be modeled by B = 15^ + 281, where B is measured in
millions. The wholesale price P for a videocassette can be modeled by
P = —0.2If + 3.52, where P is measured in dollars.
► Source: EIA Market Research Department
59. Find a model for the revenue R from sales of blank videocassettes. Give the
model as a quadratic trinomial.
60. What conclusions can you make from your model about the revenue
over time?
61. LOGICAL REASONING Find the product (2x + l)(x + 3) using the
distributive property and explain how this leads to the FOIL pattern.
10.2 Multiplying Polynomials
Standardized Test
Practice
Mixed Review
Maintaining Skills
62. MULTIPLE CHOICE Find the product 2 a 2 (a 2 - 3 a + 1).
(A) 2 a 2 — 6a + 2 Cb) 2 a 4 — 6 a 3 + 2a
CC) 2 a 2 — 3 a 3 + 2a 2 CD) 2a 4 — 6a 3 + 2a 2
63. MULTIPLE CHOICE Find the product (x + 9)(x - 2).
CD x 2 + lx — 18 CG) x 2 — 1 lx — 18
Ch) x 2 — 18 CD x 2 — lx
64. MULTIPLE CHOICE Find the product (x - l)(2x 2 + x + 1).
(A) 2x 3 — 3x 2 — 1 CM) 2x 3 — x 2 — 2x — 1
Cc) 2x 3 — x 2 — 1 Cp) 2x 3 + 3x 2 + 2x + 1
SIMPLIFYING EXPRESSIONS Simplify the expression. Write your answer
as a power. (Lesson 8.1)
65. (lx) 2 66. [|mj 2 67. 68. (0.5w) 2
69. 9 3 • 9 5 70. (4 2 ) 4 71 . b 2 • b 5 72.(4 c 2 ) 4
73. (2t) 4 • 3 3 74. (-w 4 ) 3 75. (~3xy) 3 (2y) 2 76. (8x 2 /) 3
USING THE DISCRIMINANT Tell whether the equation has two solutions,
one solution, or no real solution. (Lesson 9.7)
77. x 2 - 5x + 6 = 0 78 x 2 + lx + 12 = 0 79. x 2 - 2x - 24 = 0
80. 2x 2 — 3x — 1 = 0 81. 4x 2 + 4x + 1 = 0 82. 3x 2 — lx + 5 = 0
83. lx 2 — 8x — 6 = 0 84. 10x 2 - 13x - 9 = 0 85. 6x 2 - 12x - 6 = 0
SKETCHING GRAPHS In Exercises 86-88, sketch the graph of the
inequality. (Lesson 9.8)
86. y > 4x 2 — lx 87. y < x 2 — 3x — 10 88. y > —2x 2 + 4x + 16
89. ASTRONOMY The distance from the sun to Earth is approximately
1.5 X 10 8 km. The distance from the sun to the planet Neptune is
approximately 4.5 X 10 9 km. What is the ratio of Earth’s distance from
the sun to Neptune’s distance from the sun? (Lesson 8.4)
DIVIDING FRACTIONS Divide. Write the answer in simplest form.
(Skills Review p. 765)
90.
93.
96.
n
16
n
12
91.
94.
97. 1
4 '
13
15
1
_9_
24
^ _7_
' 10
^ 3
' 4
„ 7 . 5
92 ‘ 8 * 2
95.
29
32
98. 2
1
23
24
1 _
27
Chapter 10 Polynomials and Factoring
Special Products
of Polynomials
Goal
Use special product
patterns to multiply
polynomials.
Key Words
• special product
• area model
What color will the offspring of two tigers be?
In Checkpoint Exercise 14 you will
use the square of a binomial pattern
to determine the possible coat colors
of the offspring of two tigers.
Some pairs of binomials have special products. If you learn to recognize such
pairs, finding the product of two binomials will sometimes be quicker and easier.
For example, to find the product of (y + 3)(y — 3), you could multiply the two
binomials using the FOIL pattern.
(y + 3)(y — 3) = y 2 + ( — 3y) + 3y — 9 Use FOIL pattern.
= y 2 — 9 Combine like terms.
Notice that the middle term is zero. This suggests a simple pattern for finding the
product of the sum and difference of two terms:
(i a + b)(a ~ b) = a 2 — b 2
Also, to find the product of (x + 4) 2 , you could multiply (.x + 4)(x + 4).
(x + 4){x + 4) = x 2 + 4x + 4x + 16 Use FOIL pattern.
= x 2 + 8v + 16 Combine like terms.
Notice that the middle term is twice the product of the terms of the binomial.
This suggests a simple pattern for finding the product of the square of a binomial:
(i a + b) 2 = a 2 + lab + b 2 or (a — b) 2 — a 2 — lab + b 2
SPECIAL PRODUCT PATTERNS
Sum and Difference Pattern
(a + b)(a — b) = a 2 — b 2 Example: (3x — 4)(3x + 4) = 9x 2 — 16
Square of a Binomial Pattern
(a + b) 2 = a 2 + lab + b 2 Example: (x + 5) 2 = x 2 + lOx + 25
(a — b) 2 — a 2 — lab + b 2 Example: (2x — 3) 2 = 4x 2 — 12x + 9
10.3 Special Products of Polynomials
Student HeCp
► Study Tip
When you use these
special product
patterns, remember
that a and b can be
numbers, variables, or
variable expressions.
J 1 Use the Sum and Difference Pattern
Find the product (5 1 — 2)(5 1 + 2).
Solution
(a — b)(a + b) = a 2 — b 2 Write pattern.
(5t — 2){5t + 2) = (5f) 2 — 2 2 Apply pattern.
= 251 2 - 4 Simplify.
CHECK y You can use the FOIL pattern to check your answer.
(5 1 - 2)(5 1 + 2) = (50(50 + (5t)(2) + (-2)(50 + (-2)(2)
= 25 t 2 + 10/+ (-100 + (-4)
= 25 1 2 - 4
V,
Use the Sum and Difference Pattern
Use the sum and difference pattern to find the product.
1- (x + 2)(x — 2) 2. (n — 3 ){n + 3) 3, (p + 8 )(p -
4. (2x - l)(2x + 1) 5. (3x + 2)(3x - 2) 6. (2x + 5)(2x
2 Use the Square of a Binomial Pattern
Find the product.
a. (3 n + 4) 2 b. (2x - lyf
Solution
(i a + b ) 2 = a 2 + 2 ab + b 2
Write pattern.
(3 n + 4) 2 = (3 n) 2 + 2(3/i)(4) + 4 2
Apply pattern.
= 9w 2 + 24 n + 16
Simplify.
(i a — b ) 2 — a 2 — 2 ab + Z? 2
Write pattern.
(2x - 7j) 2 = (2x) 2 - 2{2x){ly) + ily) 2
Apply pattern.
— 4x 2 — 28xy + 49 y 2
Simplify.
Use the Square of a Binomial Pattern
Use the square of a binomial pattern to find the product.
7. (x + l) 2 8. (t - 3) 2 9. (a - l) 2
10. (2x+l) 2 11. (4x — l) 2 12. (3a-4) 2
Use FOIL.
Simplify.
Combine
like terms.
8 )
-5)
Chapter 10 Polynomials and Factoring
Student HeCp
► More Examples
More examples
are available at
www.mcdougallittell.com
AREA MODELS Area models may be helpful when multiplying two binomials
or using any of the special patterns.
The square of a binomial pattern (a + b ) 2 = a 2 + lab + b 2 can be modeled as
shown below.
The area of the large square is (<a + b) 2 ,
which is equal to the sum of the areas of
the two small squares and two rectangles.
Note that the two rectangles with area ab
produce the middle term lab.
a 2
ab
ab
b 2
a b
3 Find the Area of a Figure
GEOMETRY LINK Write an expression
for the area of the blue region.
H
K
j
L
I * I 3 I
Solution
Student HeCp
► Look Back
For help subtracting
polynomials, see
p. 570.
^ _ )
Verbal
Area of
Area of
Area of
Model
blue region
entire square
red region
Labels Area of blue region — A (square units)
Area of entire region = (x + 3) 2 (square units)
Area of red region = (x + l)(x — 1) (square units)
Algebraic A = (x + 3) 2 — (x + l)(x — 1) Write algebraic model.
Model
= (x 2 + 6x + 9) — (x 2 — 1) Apply patterns.
= x 2 + 6x + 9 — x 2 + 1 Use distributive property.
= 6x + 10 Simplify.
ANSWER ^ The area of the blue region is 6x + 10 square units.
Find the Area of a Figure
13, Write an expression for the area of the figure
at the right. Name the special product pattern
that is represented.
x 11
x
i hr
i
10.3 Special Products of Polynomials
Link to
Science
PUNNETT SQUARES are
used in genetics to model the
possible combinations of
parents 7 genes in offspring.
4 Use a Punnett Square
PUNNETT SQUARES The Punnett
square at the right shows the possible
results of crossing two pink snapdragons,
each with one red gene R and one white
gene W. Each parent snapdragon passes
along only one gene for color to its
offspring. Show how the square of a
binomial can be used to model the
Punnett square.
Solution
RW
R W
RR
(red)
RW
(pink)
RW
(pink)
ww
(white)
Each parent snapdragon has half red genes and half white genes. You can
model the genetic makeup of each parent as follows:
0.5 R + 0.5 W
The genetic makeup of the offspring can be modeled by the product
(0.5 R + 0.5VP) 2
Expand the product to find the possible colors of the offspring.
0 a + b) 2 = a 2 + 2 ab + b 2 Write pattern.
(0.5 R + 0.5 W) 2 = (0.5 R) 2 + 2(0.5R)(0.5W) + (0.5W) 2 Apply pattern.
= 0.25 R 2 + 0.5 RW + 0.25 VP 2 Simplify.
t t t
Red Pink White
ANSWER ^ Given a sufficiently large number of offspring, 25% will be red, 50%
will be pink, and 25% will be white.
Use a Punnett Square
14. SCIENCE LINK In tigers, the normal color gene C is dominant and the gene for
white coat color c is recessive. This means that a tiger whose color genes are
CC or Cc will have normal coloring. A tiger whose color genes are cc will be
white. Note: The recessive gene c that results in a white tiger is extremely rare.
a. The Punnett square at the right
shows the possible results of
crossing two tigers, each with one
dominant gene C and one
recessive gene c. Find a model
that can be used to represent the
Punnett square. Write the
model as a polynomial.
b_ What percent of the offspring are
likely to have normal coloring?
What percent are likely to be white?
Exercises
Guided Practice
Vocabulary Check 1. What is the sum and difference pattern for the product of two binomials?
2. Complete: (x + 3) 2 = x 2 + 6x + 9 is an example of the ? pattern.
Skill Check Use a special product pattern to find the product.
3- (x — 6 ) 2 4. (w + 11 )(w — 11) 5- (6 + p ) 2
6. (3>’ - l) 2 7. (t - 6)(t + 6) 8. (a - 2)(a + 2)
LOGICAL REASONING Tell whether the statement is true or false. If the
statement is false, rewrite the right-hand side to make the statement true.
9. (3x + 4) 2 = 9x 2 + 12x + 16
10 . (3 + 2 y) 2 = 9+12 j + 4y 2
11 . (5x- l) 2 = 25x 2 — lOx + 1
12 . (2x - 6)(2x + 6) = 4x 2 - 12
Practice and Applications
DIFFERENCE PATTERN Tell whether the expression is a difference of
two squares.
13.x 2 -9 14. b 2 — 36 15. a 2 + 16 16. n 2 - 50
SQUARE OF A BINOMIAL Tell whether the expression is the square of
a binomial.
17, a 2 + 8a + 16 18. m 2 — 12 m — 36 19. y 2 — 10j + 25
20 . x 2 - 3x + 9 21 . n 2 — 18n + 81 22 . b 2 + 22b + 121
Student HeCp
► Homework Help
Example 1: Exs. 13-16,
23-28, 35-46
Example 2: Exs. 17-22,
29-46
Example 3: Exs. 51-53
Example 4: Exs. 56, 57
\ _ j
SUM AND DIFFERENCE PATTERN Write the product of the sum
and difference.
23. (x + 5)(x - 5) 24. ( y - 1 )(y + 1) 25. (2m + 2)(2m - 2)
26. (3 b - 1)(3 b + 1) 27. (3 + 2x)(3 - 2x) 28. (6 - 5n)(6 + 5 n)
SQUARE OF A BINOMIAL Write the square of the binomial as a trinomial.
29. (x + 5) 2 30. (a + 8) 2 31.(3x+l) 2
32. (2 y - 4) 2 33. (4b - 3) 2 34. (x - l) 2
10.3 Special Products of Polynomials
SPECIAL PRODUCT PATTERNS Find the product.
35. (x + 4)(x - 4)
36. (x — 3)(x + 3)
37. (3x - l) 2
38. (4 - n ) 2
39. (2y + 5)(2 'y ~ 5)
40. (An - 3) 2
41. (a + 2 b)(a — 2b)
42. (4x + 5) 2
43. (3x - 4v)(3x + Ay)
44. (3y + 8) 2
45. (9 - 40(9 + 40
46. (a - 2b) 2
CHECKING PRODUCTS Tell whether the statement is true or false. If the
statement is false, rewrite the right-hand side to make the statement true.
47. (a + 2b) 2 = a 2 + 2ab + 4b 2 48. (3s + 2t)(3s — 2t) = 9s 2 + 4 1 2
49. (9x + 8)(9x - 8) = 8 lx 2 - 64 50. (6 y - Iw ) 2 = 36y 2 - 49 w 2
Student fteCp
► Homework Help
Extra help with
problem solving in
Exs. 54-55 is available at
www.mcdougallittell.com
AREA MODELS Write two expressions for the area of the figure. Describe
the special product pattern that is represented.
54, Geome try Link s The area of a square is given by 4x 2 — 20x + 25.
Express its perimeter as a function of x.
55. Geometry Link / The side of a square is (3x — 4) inches. What is its area?
GENETICISTS study the
biological inheritance of traits
in living organisms.
Science Lit In Exercises 56 and 57, use the following information.
In chickens, neither the normal-feathered gene N nor the extremely rare
frizzle-feathered gene F is dominant. So chickens whose feather genes are
NN will have normal feathers. Chickens with NF will have mildly frizzled
feathers. Chickens with FF will have extremely frizzled feathers.
56. The Punnett square at the right
shows the possible results of
crossing two chickens with mildly
frizzled feathers. Find a model that
can be used to represent the results
shown in the Punnett square. Write
the model as a polynomial.
57. What percent of the offspring are
likely to have normal feathers?
What percent are likely to have
mildly frizzled feathers? What
percent are likely to have extremely
frizzled feathers?
Chapter 10 Polynomials and Factoring
Standardized Test
Practice
58. MULTIPLE CHOICE Find the product (2x + 3)(2x - 3).
(A) 2x 2 — 6x — 9 CM) 4x 2 — 9
Cc) 2x 2 — 9 Cd) 4x 2 + I2x + 9
59. MULTIPLE CHOICE Find the product of (3x + 5) 2 .
Cf) 3x 2 + 15x + 5 Cg) 9x 2 + 25
CH) 3x 2 + 25 Q) 9x 2 + 30x + 25
Mixed Review
SIMPLIFYING EXPRESSIONS
exponents. (Lesson 8.4)
Simplify the expression. Use only positive
62.x 7
1
SKETCHING GRAPHS Sketch the graph of the function. Label the vertex.
(Lesson 9.4)
64. y = lx 1 + 3x + 6 65. y = 3x 2 - 9x - 12 66. y = ~x 2 + 4x + 16
Maintaining Skills
MULTIPLYING FRACTIONS
67.
1
2
68 .
I
4
l
3
l
3
2
5
Multiply the fractions. (Skills Review p. 765)
2
3
4
9
2
• —
5
3 3 3
— • — •
4 4
5
8
5
8
Quiz 7 -
State the degree of the monomial. (Lesson 10.1)
1.6x 2 2. —8 3. —a 3 4. 25m 5
Use a vertical or a horizontal format to add or subtract. (Lesson 10.1)
5. (2x 2 + lx + 1) + (x 2 — 2x + 8)
6 . (—4x 3 — 5x 2 + 2x) - (2x 3 + 9x 2 + 2)
7. (It 2 - 3t + 5) - (At 2 + lOt - 9)
8 . (5x 3 — x 2 + 3x + 3) + (x 3 + 4x 2 + x)
Find the product. (Lesson 10.2)
9. (x + 8)(x - 1) 10. (y + 2)(y + 9) 11. -x 2 (12x 3 - 1 lx 2 + 3)
12. (3x — _y)( 2x + 5_y) 13. (An + 7)(4n — 7) 14. (2x 2 + x — 4)(x — 2)
Use a special product pattern to find the product. (Lesson 10.3)
15. (x - 6)(x + 6) 16. (4x + 3)(4x - 3) 17. (5 + 3b)(5 - 3b)
18. (2x - ly)(2x + ly) 19. (3x + 6) 2 20. (-6 - 8x) 2
■
10.3 Special Products of Polynomials
Solving Quadratic Equations
in Factored Form
Goal
Solve quadratic
equations in factored
form.
Key Words
• factored form
• zero-product property
How deep is a crater?
In Exercises 50 and 51
you will solve a quadratic
equation to find the depth of
the Barringer Meteor Crater.
A polynomial is in factored form if it is written as the product of two or more
factors. The polynomials in the following equations are written in factored form.
x(x - 7) = 0 (x + 2 )(jc + 5) = 0 (x + l)(x - 3)(x + 8) = 0
A value of x that makes any of the factors zero is a solution of the polynomial
equation. That these are the only solutions follows from the zero-product
property, stated below.
ZERO-PRODUCT PROPERTY
Let a and b be real numbers. If ab = 0, then a = 0 or b = 0.
If the product of two factors is zero , then at least one of the factors
must be zero.
Student HeCp
^
►Study Tip
The fact that the
solutions 2 and -3
in Example 1 are the
only solutions is a
consequence of the
zero-product property.
I _>
(3Z!mZZI 9 1 Using the Zero-Product Property
Solve the equation (x — 2)(x + 3) = 0.
Solution
(x — 2)(x + 3) = 0 Write original equation,
x — 2 = 0 or x + 3 = 0 Set each factor equal to 0.
x = 2 | x = —3 Solve for x.
ANSWER ► The solutions are 2 and —3. Check these in the original equation.
Solve the equation and check the solutions.
1- (x + l)(x — 3) = 0 2 . x (x — 2) = 0
3- (x — 5)(x + 7) = 0
Chapter 10 Polynomials and Factoring
Student HeCp
p More Examples
More examples
are available at
www.mcdougallittell.com
2 Solve a Repeated-Factor Equation
Solve (x + 5) 2 = 0.
Solution
This equation is a square of a binomial, so the factor (x + 5) is a repeated
factor. Repeated factors are used twice or more in an equation. To solve this
equation you set (x + 5) equal to zero.
(.x + 5) 2 = 0 Write original equation.
x + 5 = 0 Set factor equal to 0.
x = — 5 Solve for x.
ANSWER ► The solution is —5.
CHECK / Substitute the solution into the original equation to check.
(x + 5) 2 = 0
(-5 + 5) 2 = 0
0 = O/
Write original equation.
Substitute -5 for x.
Simplify. Solution is correct.
Solve a Repeated-Factor Equation
Solve the equation and check the solutions.
4 . (jc — 4) 2 = 0 5 - (jc + 6) 2 = 0
6 . (2x - 5) 2 = 0
3 Solve a Factored Cubic Equation
Solve (2x + 1 )(3x - 2)(x - 1) = 0.
Solution
(2x + 1)(3jc - 2)(x - 1) = 0
2x + 1 = 0
2x — —
or 3x — 2 = 0 or x — 1=0
\
'2
3x = 2
2
X= 3
1
Write original
equation.
Set factors equal to 0.
Solve for x.
1 2
ANSWER ► The solutions are — —, —, and 1. Check these in the original equation.
Solve a Factored Cubic Equation
Solve the equation and check the solutions.
7 . (x — 4)(x + 6)(4x + 3) = 0 8 - (x — 3)(x + 6)(3x + 2) = 0
9 . (2x + l)(x - 8) 2 = 0 10 . (y - 3) 2 (3_y - 2) = 0
10.4 Solving Quadratic Equations in Factored Form
4 Graph a Factored Equation
Sketch the graph of y = (x — 3)(x + 2).
O Find the x-intercepts. Solve (x — 3)(x + 2) = 0 to find the
x-intercepts: 3 and —2.
0 Use the x-intercepts to find the coordinates of the vertex.
• The x-coordinate of the vertex is the average of the x-intercepts.
3 + (-2) 1
* =- 2 - = 2
• Substitute the x-coordinate into
the original equation to find the
^-coordinate.
• The vertex is at
© Sketch the graph using the
vertex and the x-intercepts.
Graph a Factored Equation
Find the x-intercepts and the vertex of the graph of the function. Then
sketch a graph of the function.
11 - y = x(x + 2) 12 , y = (x + 4)(x — 5) 13 - y = (x — l)(x — 6)
Student HeCp
^
► Skills Review
For help with
multiplying decimals,
see p. 759.
v J
ms Use a Quadratic Model
An arch is modeled by y = —0.15(x — 8)(x + 8), with x and y measured in
feet. How wide is the arch at the base? How high is the arch?
Q Find the x-intercepts: 8 and —8.
• The width of the arch at the base
is 8 + 8 =16.
© Use the x-intercepts to find the
coordinates of the vertex.
• Substitute 0 into the original equation:
y = -0.15(0 - 8)(0 + 8) = 9.6
• The vertex is at (0, 9.6).
ANSWER ► The arch is 16 feet wide at the base and 9.6 feet high.
kh_
Chapter 10 Polynomials and Factoring
Exercises
Guided Practice
Vocabulary Check
Skill Check
1. What is the zero-product property?
2. Is (.x — 2)(x 2 — 9) = 0 in factored form? Explain.
3. Are —5, 2, and 3 the solutions of 3(x — 2)(x + 5) = 0? Explain.
4. ERROR ANALYSIS Find and
correct the error at the right.
Does the graph of the function have x-intercepts of 4 and -5?
5. y = 2(x + 4)(jc — 5) 6. y = 4(x — 4)(x — 5)
7. y = —(x — 4)(x + 5) 8. y = 3(x + 5)(x — 4)
Use the zero-product property to solve the equation.
9. (b + 1 )(b + 3) = 0 10. (t - 3 ){t - 5) = 0
11. (x — 7) 2 = 0 12. {y + 9)(y — 2)(y — 5) = 0
13. Sketch the graph of y = (x + 2)(x — 2). Label the vertex and the x-intercepts.
Practice and Applications
! Student HeGp
► Homework Help
Example 1: Exs. 14-36
Example 2: Exs. 14-36
Example 3: Exs. 29-36
Example 4: Exs. 37-45
Example 5: Exs. 46-51
ZERO-PRODUCT PROPERTY Use the zero-product property to solve
the equation.
14. (x + 4)(x + 1) = 0
17. (y + 3) 2 = 0
20 . (>- - 2)(y + 1 ) = 0
15. (t + 8 )(t - 6) = 0
18. (b - 9)(b + 8) = 0
21. (z + 2 )(z + 3) = 0
16. x(x + 8 ) = 0
19. (d + 7) 2 = 0
22 . (v - 7)(v - 5) = 0
23. (w - 17) 2 = 0 24. p(2p + 1) = 0 25. 4(c + 9) 2 = 0
26. (z + 9)(z - 11) = 0 27. (a- 20 )(a + 15) = 0 28. (d + 6)(3rf - 4) = 0
SOLVING FACTORED CUBIC EQUATIONS Solve the equation.
29. (x + l)(x + 2)(x — 4) = 0
31. (a + 5 ){a — 6) 2 = 0
33. 5(d + 8 ){d - 12 ){d + 9) = 0
35. (b - 8)(2 b + l)(b + 2) = 0
30. >-(>- - 4)(y - 8) = 0
32. r(r - 12) 2 = 0
34. 8 (n + 9)(n - 9)(n + 12) = 0
36. (y - 5)(y - 6)(3y - 2 ) = 0
10.4 Solving Quadratic Equations in Factored Form
MATCHING FUNCTIONS AND GRAPHS Match the function with its graph.
37. y = (x + 2)(x — 4) 38. y = (x — 2)(x + 4) 39. y = (x + 4)(x + 2)
SKETCHING GRAPHS Find the x-intercepts and the vertex of the graph of
the function. Then sketch the graph of the function.
40 . y — {x — 4)(x + 2) 41 . y = (x + 5)(x + 3) 42 . y = (x — 3)(x + 3)
43 . y = (x — l)(x + 7) 44 . y = (x — 2)(x — 6) 45 . y = (x + 4)(x + 3)
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 46-51 is available at
www.mcdougallittell.com
VLA TELESCOPE In Exercises 46 and 47, use the cross section of one of
the Very Large Array's telescope dishes shown below.
The cross section of the telescope’s
dish can be modeled by the
polynomial function
y = ~^( x + 41)(x - 41)
where x and y are measured in feet,
and the center of the dish is at x = 0.
46. Find the width of the dish. Explain your reasoning.
47. Use the model to find the coordinates of the center of the dish.
THE BARRINGER METEOR
CRATER was formed about
49,000 years ago when a
nickel and iron meteorite
struck the desert at about
25,000 miles per hour.
GATEWAY ARCH In Exercises 48 and 49, use the following information.
The Gateway Arch in St. Louis, Missouri, has the shape of a catenary (a
U-shaped curve similar to a parabola). It can be approximated by the following
model, where x and y are measured in feet. ►Source: National Park Service
Gateway Arch model: y — — ^ qqq (x + 300)(x — 300)
48. How far apart are the legs of the arch at the base?
49. How high is the arch?
BARRINGER METEOR CRATER In Exercises 50 and 51, use the following
equation which models a cross section of the Barringer Meteor Crater,
near Winslow, Arizona. Note that x and y are measured in meters and the
center of the crater is at x = 0. ►Source: Jet Propulsion Laboratory
Barringer Meteor model: y = ^qq (x — 600)(x + 600)
50. Assuming the lip of the crater is at y = 0, how wide is the crater?
51. What is the depth of the crater?
Chapter 10 Polynomials and Factoring
Standardized Test
Practice
Mixed Review
Maintaining Skills
52. MULTIPLE CHOICE Solve 6(x - 3)0 + 5)0 - 9) = 0.
(A) 6, 3, 5, and 9 CM) 3, —5, and 9
CM) 6, 3, —5, and 9 CM) 6, 3, 5, and —9
53. MULTIPLE CHOICE Which function
represents the graph at the right?
CD y = 0 + 2)0 + 4)
CD y = O + 2)0 — 4)
CH) j = 0 - 2)0 - 4)
GD y = 0 - 2)0 + 4)
DECIMAL FORM Write the number in decimal form. (Lesson 8.5)
54. 2.1 X 10 5 55. 4.443 X 10“ 2 56. 8.57 X 10 8 57. 1.25 X 10 6
58. 3.71 X 10“ 3 59. 9.96 X 10 6 60. 7.22 X 10“ 4 61. 8.17 X 10 7
MULTIPLYING EXPRESSIONS Find the product. (Lesson 10.2)
62. O - 2)0 - 7) 63. O + 8)0 “ 8) 64. O “ 4)0 + 5)
65. (2x + 7)(3x - 1) 66. (5x - l)(5x + 2) 67. (3x + l)(8x - 3)
68. (2x - 4)(4x - 2) 69. 0 + 10)0 + 10) 70. (3x + 5)(2x - 3)
EXPONENTIAL MODELS Tell whether the situation can be represented by
a model of exponential growth or exponential decay. Then write a model
that represents the situation. (Lessons 8.6 , 8.7)
71. COMPUTER PRICES From 1996 to 2000, the average price of a computer
company’s least expensive home computer system decreased by 16% per
year.
72. MUSIC SALES From 1995 to 1999, the number of CDs a band sold
increased by 23% per year.
73. COOKING CLUB From 1996 to 2000, the number of members in the
cooking club decreased by 3% per year.
74. INTERNET SERVICE From 1993 to 1998, the total revenues for a company
that provides Internet service increased by about 137% per year.
FINDING FACTORS List all the factors of the number.
(Skills Review p. 761)
75. 12
76. 20
77. 18
78. 35
79. 51
80. 24
81. 36
82. 48
83. 64
84. 90
85. 84
86. 112
10.4 Solving Quadratic Equations in Factored Form
DEVELOPING CONCEPTS
-j
-r SJX -r
Goal
Use algebra tiles to model
the factorization of a
trinomial of the form
x 2 + bx + c.
Materials
• algebra tiles
Question
How can you use algebra tiles to factor x 2 + 5x + 6?
Explore
Factor the trinomial x 2 + 5x + 6.
© Use algebra tiles to model x 2 + 5x + 6.
v -V-' - y
x 2 4- 5x +6
For use with
Lesson 10.5
Q With the x 2 -tile at the upper left, arrange the x -tiles and 1-tiles around the
x 2 -tile to form a rectangle.
x 111
+
+
+ +
+
+ + +
+
+ + +
© The width of the rectangle is ( ? + ? ), and the length of the rectangle is
(? + ?). Complete the statement: x 2 + 5x + 6 = (? + ?)•(? + ?).
Think About It
Write the factors of the trinomial represented by the algebra tiles.
2 .
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ ++++++ + +++++++
■ ++++++ + +++++++
+ +++++++
In Exercises 3-8, use algebra tiles to factor the trinomial. Sketch
your model.
3- x 2 + lx + 6 4 . x 2 + 6x + 8 5- x 2 + 8x + 15
6- x 2 + 6x + 9 7. x 2 + 4x + 4 8. x 2 + lx + 10
9_ Use algebra tiles to show why the trinomial x 2 + 3x + 4 cannot be factored.
Chapter 10 Polynomials and Factoring
Factoring x 2 + bx + c
Goal
Factor trinomials of the
form x 2 + bx + c.
Key Words
• factor a trinomial
• factored form
How wide should the border of a garden be?
In Example 7 you will factor a
quadratic equation to find the
width of a border around a garden.
A trinomial of the form x 2 + bx +
c , where b and c are integers is shown below.
x 2 + 9x + 14, b = 9, c = 14
To factor a trinomial of this form
means to write the trinomial as the
product of two binomials (factored
form).
Trinomial
x 2 + 9x + 14
x 2 — x — 12
x 2 - 2x - 15
Factored Form
(x + 2)(x + 7)
(jc + 3)(x - 4)
(x + 3)(x — 5)
In order to write x 2 + bx + c in the form (x + p)(x + q), note that
(x + p)(x + q) = x 2 + (p + q)x + pq
This leads you to seek numbers p and q such that p + q — b and pq — c.
i Factor when b and c Are Positive
Factor x 2 + 6x + 8.
Solution
The first term of each binomial factor is x. For this trinomial, b = 6 and c = 8.
You need to find numbers p and q whose product is 8 and whose sum is 6.
pand q p + q
1,8 9
2, 4 6 The numbers you need are 2 and 4.
ANSWER ► x 2 + 6x + 8 = (x + 2)(x + 4). Check your answer by multiplying.
Factor when b and c Are Positive
Factor the trinomial.
1. x 2 + 4x + 3 2. x 2 + 5x + 6 3. x 2 + 8x + 7 4- x 2 + lx + 6
10.5 Factoring x 2 + bx + c
Student HeCp
->
► Study Tip
When the constant
term c of a trinomial is
positive, you will need
two numbers with the
same sign.
\ _ /
2 Factor when b Is Negative and c Is Positive
Factor x 2 — 5x + 6.
Solution
The first term of each binomial factor is x.
(x _)(* _)
For this trinomial, b = — 5 and c = 6. Because c is positive, you need to find
numbers p and q with the same sign. Find numbers p and q whose sum is —5
and whose product is 6.
pand q p + q
-1,-6 -7
-2,-3 —5 The numbers you need are-2 and-3.
ANSWER ► x 2 — 5x + 6 = (x — 2)(x — 3). Check your answer by multiplying.
L_
Factor when b Is Negative and c Is Positive
Factor the trinomial.
5. x 2 — 5x + 4 6- x 2 — 4x + 4 7. x 2 — 8x + 7 8- x 2 — lx + 12
Student HeCp na> Factor when b and c Are Negative
Factor x 2 — llx — 12.
Solution
The first term of each binomial factor is x.
(x _)(x _)
For this trinomial, b = — 11 and c = —12. Because c is negative, you need to
find numbers p and q with different signs. Find numbers p and q whose sum is
— 11 and whose product is —12.
pand q p + q
- 1 , 12 11
1,-12 —11 The numbers you need are 1 and - 12 .
ANSWER ^ x 2 — llx — 12 = (x + l)(x — 12). Check your answer by
multiplying.
r ->
► Study Tip
When the constant
term c of a trinomial is
negative, you will need
two numbers with
different signs.
\_ J
Factor when b and c Are Negative
Factor the trinomial.
9- x 2 — 5x — 6 10- x 2 — 3x — 10 11- x 2 — 13x — 14 12- x 2 — 6x — 7
Chapter 10 Polynomials and Factoring
Student HeCp
-N
► Study Tip
As soon as you find the
correct pair of numbers
for a trinomial, you can
stop listing all possible
pairs. For example, in
Example 4, you do not
need the pairs
-2 and 9, 2 and -9,
-3 and 6, or 3 and -6.
\ _ J
mmzm 4 Factor when b Is Positive and c Is Negative
Factor x 2 + 17x — 18.
Solution
The first term of each binomial factor is x.
(x _)(x __)
For this trinomial, b = 17 and c = —18. Because c is negative, you need to find
numbers p and q with different signs. Find numbers p and q whose sum is 17
and whose product is —18.
pand q p + q
1,-18 -17
— 1, 18 17 The numbers you need are-1 and 18.
ANSWER ^ x 2 + 17x — 18 = (x — l)(x + 18).
Factor when b Is Positive and c Is Negative
Factor the trinomial.
13. x 2 + x — 6 14. x 2 + 2x — 8 15. x 2 + 8x — 20 16. x 2 + 3x — 10
5 Check Using a Graphing Calculator
Factor x 2 — 2x — 8.
Solution
The first term of each binomial factor is x.
(x _)(x _)
For this trinomial, b = — 2 and c = — 8. Because c is negative, you need to find
numbers p and q with different signs. Find numbers p and q whose sum is —2
and whose product is —8.
p and q p + q
-1, 8 7
1, -8 -7
-2, 4 2
2, -4 —2 The numbers you need are 2 and-4.
ANSWER ^ x 2 - 2x - 8 = (jc + 2)(x - 4).
CHECK^Usea graphing calculator.
Graph y = x 2 — 2x — 8 and
y = (x + 2)(x — 4) on the same
screen. The graphs are the same,
so your answer is correct.
10.5 Factoring x 2 + bx + c
Student HeCp
► More Examples
M°r e examples
are available at
www.mcdougallittell.com
6 Solve a Quadratic Equation
Solve x 2 — 3x = 10 by factoring.
Solution
X 2 -
3x =
10
x 2
— 3x —
10 =
0
(x -
- 5)(x +
2) =
0
X —
5 = 0
or
x + 2 = 0
x = 5
x = —2
ANSWER ► The solutions are 5 and —
Write equation.
Write in standard form.
Factor left side.
Use zero-product property.
Solve for x .
Check these in the original equation.
Link to
Careers
LANDSCAPE DESIGNERS
plan and map out the
appearance of outdoor
spaces like parks, gardens,
golf courses, and other
recreational areas.
More about landscape
4^"* designers available at
www.mcdougallittell.com
7 Write a Quadratic Model
LANDSCAPE DESIGN You are
putting a stone border along two sides
of a rectangular Japanese garden that
measures 6 yards by 15 yards. Your
budget limits you to only enough stone
to cover 46 square yards. How wide
should the border be?
bH- 15
Solution
Total Garden
area area
46 = (x + 15)(x + 6) - (15)(6)
46 = x 2 + 6x + 15x + 90 — 90
46 = x 2 + 21x
0 = x 2 + 21x — 46
0 = (jc + 23)(x - 2)
v + 23 = 0 or x — 2 = 0
x = —23 I x — 2
Area of
border
Write quadratic model.
Multiply.
Combine like terms.
Write in standard form.
Factor.
Use zero-product property.
Solve for x .
The solutions are —23 and 2. Only x = 2 is a reasonable solution, because
negative values for dimension do not make sense.
ANSWER ^ The border should be 2 yards wide.
Solve a Quadratic Equation
Solve the equation by factoring.
17. 0 = x 2 + 4x + 3 18. 0 = x 2 — 5x + 4 19. 0 = x 2 — 5x — 6
20 . Suppose the garden in Example 7 above measured 7 yards by 12 yards and
the budget lets you cover 66 square yards. How wide should the border be?
Chapter 10 Polynomials and Factoring
Exercises
Guided Practice
Vocabulary Check
1. What does it mean to factor a trinomial of the form x 2 + bx + cl
Skill Check
Match the trinomial with a correct factorization.
2 . x 2 — x — 20
3. x 2 + x — 20
4. x 2 + 9x + 20
5. x 2 — 9x + 20
A. (x + 5)(x — 4)
B. (x + 4)(x + 5)
C. (x - 4)(x - 5)
D. (x + 4)(x - 5)
Solve the equation by factoring.
6- 0 = x 2 — 4x + 4 7. 0 = x 2 — 4x — 5 8- 0 = x 2 + x — 6
LOGICAL REASONING Complete the statement with always, sometimes,
or never.
9. Factoring ? reverses the effects of multiplication.
10, In the factoring of a trinomial, if the constant term is positive, then the signs
in both binomial factors will ? be the same.
11. In the factoring of a trinomial, if the constant term is negative, then the signs
in both binomial factors will ? be negative.
Practice and Applications
Student HeCp
—'N
► Homework Help
Example 1: Exs. 12-23
Example 2: Exs. 12-23
Example 3: Exs. 12-23
Example 4: Exs. 12-23
Example 5: Exs. 39-41
Example 6: Exs. 24-35
Example 7: Exs. 42-45
Choose the correct factorization.
13- x 2 — lOx + 16 14- x 2 + 1 lx — 26
FACTORED FORM
12.x 2 + lx + 12
A- (x + 6)(x + 2)
B. (x + 4)(x + 3)
21- m 2 — 1m — 30
A. (x - 4)(x - 4)
B. (x — 8)(x — 2)
19. r 2 + 8r + 16
22 . w 2 + 13 w + 36
A. (x - 13)(x + 2)
B. (x + 13)(x - 2)
17. b 2 + 5b - 24
20 . y 2 - 3y - 18
23. b 2 + 3b- 40
FACTORING TRINOMIALS Factor the trinomial.
15. z 2 + 6z + 5 16. x 2 + 8x — 9
18. a 2 — a — 20
SOLVING QUADRATIC EQUATIONS Solve the equation by factoring.
24- x 2 + 7x + 10 — 0
27. x 2 + 16x + 15 = 0
30. x 2 + 100 = 20x
33. x 2 + 8x = 65
25. x 2 + 5x — 14 = 0
28. x 2 - 9x = -14
31.x 2 - 15x + 44 = 0
34. x 2 + 42 = 13x
26. x 2 + 6x + 9 = 0
29. x 2 + 3x = 54
32. x 2 — 20x = —51
35. —x + x 2 = 56
10.5 Factoring x 2 + bx + c
tTV
Solve x 2 — 9x + 18 = 2x.
Solution
x 2 — 9x + 18 — 2x
x 2 — 9x + 18 — 2x = 0
x 2 — llx + 18 = 0
(jc - 2)(jc - 9) = 0
x — 2 = 0 or x — 9 = 0
x = 2
x = 9
Write original equation.
Add -2x to each side.
Combine like terms.
Factor.
Use zero-product property.
Solve for x.
ANSWER ► The solutions are 2 and 9. Check your answers.
Solve the equation by factoring.
36. x 2 — x — 8 = 82 37. n 2 + 8n + 32 = —An 38. c 2 + 10c — 48 = 12c
a CHECKING GRAPHICALLY Solve the equation by factoring. Then use
a graphing calculator to check your answer.
39. x 2 — llx + 30 = 0 40. x 2 — 20x + 19 = 0 41. x 2 + 3x — 18 = 0
link]
Architecture
TAJ MAHAL
It took more than 20,000 daily
workers 22 years to complete
the Taj Mahal around 1643 in
India. Built mainly of white
marble and red sandstone,
the Taj Mahal is renowned
for its beauty.
More about the Taj
Mahal is available at
www.mcdougallittell.com
MAKING A SIGN In Exercises 42 and 43, a triangular sign has a base that
is 2 feet less than twice its height. A local zoning ordinance restricts the
surface area of street signs to be no more than 20 square feet.
42. Write an inequality involving the height that represents the largest triangular
sign allowed.
43. Find the base and height of the largest triangular sign that meets the zoning
ordinance.
THE TAJ MAHAL In Exercises 44 and 45, refer to the illustration of the
Taj Mahal below.
44. The platform is about 38 meters wider
than the main building. The total area
of the platform is about 9025 square
meters. Using the fact that the
platform and the base of the building
are squares, find their dimensions.
Building
Platform
45. The entire complex of the Taj Mahal
is about 245 meters longer than it is
wide. The area of the entire complex
is about 167,750 square meters. What
are the dimensions of the entire
complex? Explain your steps in
finding the solution.
Chapter 10 Polynomials and Factoring
Standardized Test
Practice
Mixed Review
Maintaining Skills
46. MULTIPLE CHOICE Factor x 2 - lOx - 24.
Ca) (x — 4)(x — 6) CM) (x + 4)(x + 6)
CD (x + 2)(x - 12) CD (x - 2)(x + 12)
47. MULTIPLE CHOICE Solve x 2 - 9x = 36 by factoring.
CD 12 and —3 CD —12 and 3
(H) 4 and —9 (3) 9 and —4
48. MULTIPLE CHOICE The length of a rectangular plot of land is 24 meters
more than its width. A paved area measuring 8 meters by 12 meters is placed
on the plot. The area of the unpaved part of the land is then 880 square meters.
If w represents the width of the plot of land in meters, which of the following
equations can be factored to find the possible values of wl HINT: Begin by
drawing and labeling a diagram.
(A) w 2 + 24w = 880 CD w 2 + 24w + 96 = 880
CD> w 2 + 24w — 96 = 880 CD w 2 + 24w = 96
49. MULTIPLE CHOICE A triangle’s base is 16 feet less than 2 times its height.
If h represents the height in feet, and the total area of the triangle is 48 square
feet, which of the following equations can be used to determine the height?
©2/i + 2 (h + 4) = 48 CD h 2 - 8h = 48
CD h 2 + Sh = 48 GD 2 h 2 - 16 h = 48
FINDING THE GCF Find the greatest common factor. (Skills Review p. 761)
50.12,36 51.30,45 52.24,72
53. 49, 64 54. 20, 32, 40 55. 36, 54, 90
MULTIPLYING EXPRESSIONS Find the product. (Lessons 10.2 and 10.3)
57. (y + 9)(y - 4) 58. (7x - 11) 2
60. (3a - 2)(4a + 6) 61. (5 1 - 3)(4 1 - 10)
Solve the equation. (Lesson 10.4)
56. 3 q(q 2 — 5q 2 + 6)
59. (5 — w)(12 + 3w)
SOLVING FACTORED EQUATIONS
62. (x + 12)(x + 7) = 0
64. (t - 19) 2 = 0
66 . (y + 47)(y - 27) = 0
68 . (a — 3)(a + 5) 2 = 0
63. (z + 2)(z + 3) = 0
65. 5(x — 9)(x — 6) = 0
67. (z ~ 1)(4 z + 2) = 0
69. (b + 4 )(b - 3)(2 b - 1) = 0
ADDING DECIMALS Add
70. 3.7 + 1.04 + 5.2
72. 7.421 + 5 + 8.09
74. 6.012 + 2.9 + 5.6314
76. 3.2 + 5.013 + 0.0021
Review p. 759)
71. 6.7 + 0.356 + 4
73. 8.1 + 0.2 + 3.56
75. 7.9 + 3.0204 + 10
77. 100 + 9.81 + 5.0006
. (Skills
10.5 Factoring x 2 + bx + c
DEVELOPING CONCEPTS
SJ SJ.r -r V.
Goal
Use algebra tiles to model
the factorization of a
trinomial of the form
ax 2 + bx + c.
Materials
• algebra tiles
Question
How can you use algebra tiles to factor 2X 2 + 5x+ 3?
Explore
Factor the trinomial 2x 2 + 5x + 3.
Q Use algebra tiles to model 2x 2 + 5x + 3.
—V~ v ■V'
2x2+ 5x +3
For use with
Lesson 10.6
Q With the x 2 -tiles at the upper left, arrange the x-tiles and the 1-tiles around the
x 2 -tiles to form a rectangle.
X
1
X X 111
+
+
+
+ +
+
+
+ + +
€) The width of the rectangle is ( ? + ? ), and the length of the rectangle is
(? + ■).
Complete the statement: 2x 2 + 5x + 3 = (? + ?)•(? + ?).
Think About It
i ——— —
Use algebra tiles to factor the trinomial. Sketch your model.
1. 2x 2 + 9x + 9 2. 2x 2 + lx + 3 3. 3x 2 + 4x + 1
4. 3x 2 + lOx + 3 5- 3x 2 + lOx + 8 6- 4x 2 + 5x +1
ERROR ANALYSIS The algebra tile model shown below is incorrect.
Sketch the correct model, and use the model to factor the trinomial.
7. 2x 2 + 3x + 1 8- 2x 2 + 4x + 2 9. 4x 2 + 4x + 1
Chapter 10 Polynomials and Factoring
Factoring ax 2 + bx + c
Goal
Factor trinomials of the
form ax 2 + bx + c.
Key Words
• factor a trinomial
• FOIL pattern
• quadratic
How long will it take a cliff diver to enter the water?
In Example 5 you will use a
vertical motion model to find the
time it takes a cliff diver to enter
the water.
To factor a trinomial of the form ax 2 + bx + c, write the trinomial as the product
of two binomials (factored form).
factors of 6
Example: 6x 2 + 22x + 20
^ \
(3jc + 5)(2x + 4)
factors of 20
12 + 10 = 22
One way to factor ax 2 + bx + c is to find numbers m and n whose product is a
and numbers p and q whose product is c so that the middle term is the sum of the
Outer and Inner products of FOIL.
in x n = a
ax
2 + bx +
(mx + p)(nx + q)
p x q = c
b = mq + np
I 1
Factor when a and c Are Prime Numbers
Factor 2x 2 + lbc + 5.
0 Write the numbers m and n whose
product is 2 and the numbers
p and q whose product is 5.
© Use these numbers to write trial
factors. Then use the Outer and
Inner products of FOIL to
check the middle term.
mand n
1,2
Trial Factors
(x + 1)(2 jc + 5)
(2x + 1)(jc + 5)
ANSWER t 2x 2 + llx + 5 — (2x + l)(x + 5).
p and q
1,5
Middle Term
5x + 2x = lx
lOx + x — llx
Factor when a and c Are Prime Numbers
2. 2x 2 + 5x + 3 3. 3x 2 + lOx + 3
Factor the trinomial.
1. 2x 2 + lx + 3
10.6 Factoring ax 2 + bx + c
2 Factor when a and c Are not Prime Numbers
Factor 6x 2 — 19x + 15.
Student HeCp
► Study Tip
Once you find the
correct binomial
factors, it is not
necessary to continue
checking the remaining
trial factors.
I _/
Solution
For this trinomial, a = 6, b = —19, and c = 15. Because c is positive, you need
to find numbers p and q with the same sign. Because b is negative, only
negative numbers p and q need to be tried.
© Write the numbers m and n whose m and n
product is 6 and the numbers 1, 6
p and q whose product is 15. 2, 3
p and q
-1, -15
-3, -5
© Use these numbers to write trial
factors. Then use the Outer
and Inner products of FOIL
to check the middle term.
Trial Factors
(jc - l)(6x - 15)
(x - 15)(6x - 1)
(2x - 3)(3x ~ 5)
Middle Term
— 15x — 6x — —2 lx
—x — 90x = —91x
— 3 Ox — 3x = —19x
ANSWER ^ 6x 2 — 19x + 15 = (2x — 3)(3x — 5).
3 Factor with a Common Factor for a , b, and c
Factor 6x 2 + 2x — 4.
Solution
The coefficients of this trinomial have a common factor 2.
2(3x 2 + x — 2) Factor out the common factor.
It remains to factor a trinomial with a = 3, b = 1, and c = — 2. Because c is
negative, you need to find numbers p and q with different signs.
© Write the numbers m and n whose m and n
product is 3 and the numbers 1, 3
p and q whose product is —2.
p and q
- 1,2
1,-2
0 Use these numbers to write trial
factors. Then use the Outer and
Inner products of FOIL to
check the middle term.
Trial Factors
(x - l)(3x + 2)
(x + 2)(3x — 1)
(x + l)(3x - 2)
Middle Term
2x — 3x = —x
—x + 6x = 5x
— 2x + 3x = x
Remember to include the common factor 2 in the complete factorization.
ANSWER ^ 6x 2 + 2x — 4 = 2(x + l)(3x — 2).
Factor Trinomials
Factor the trinomial.
4. 2x 2 + 5x + 2 5- 5x 2 — lx + 2
7. 8^ - 6r — 9 8. 6x 2 - 14x + 4
6. 4x 2 + 8x + 3
9. 20x 2 + 5x — 15
Chapter 10 Polynomials and Factoring
Student HeQp
p More Examples
More examples
IJtL 2 are available at
www.mcdougallittell.com
4 Solve a Quadratic Equation
21n 2 + 14ft + 7 = 6ft + 11
21ft 2 + 8ft — 4 = 0
(3ft + 2)(7ft - 2) = 0
3/7 + 2 = 0 or In — 2 = 0
2
n= ~3
2
7
Write original equation.
Write in standard form.
Factor left side.
Use zero-product property.
Solve for n.
2 2
ANSWER The solutions are — — and Check these in the original equation.
Solve a Quadratic Equation
Solve the equation.
10 . 2x 2 + 7x + 3 = 0 11 . lx 2 — x — 3 = 0 12 . 4x 2 — I6x + 15 = 0
Student HeCp
^
► Look Back
For help with using a
vertical motion model
see p. 535.
L J
BSJJHSS 5 Write a Quadratic Model
When a diver jumps from a ledge, the vertical
component of his motion can be modeled by the
vertical motion model. Suppose the ledge is
48 feet above the ocean and the initial upward
velocity is 8 feet per second. How long will it
take until the diver enters the water?
Use a vertical motion model.
Let v = 8 and s = 48.
h = —16 1 2 + vt + s Vertical motion model
= —16^ 2 + 8/ + 48 Substitute values.
Solve the resulting equation for t to find
the time when the diver enters the water.
Let h = 0.
t = 0, v = 8 ft/sec
48 -r
height (ft)
0 ^
Not drawn to scale
f=?
J/
0= — 16l 2 + 8f + 48
0 = (—8)(2? 2 - / - 6)
0 = (-8)(f - 2)(2f + 3)
t — 2 = 0 or 2^ + 3 = 0
t = 2
Write quadratic model.
Factor out common factor -8.
Factor.
Use zero-product property.
Solve for 1
3
The solutions are 2 and ——. Negative values of time do not make sense for this
problem, so the only reasonable solution is t = 2.
ANSWER 4 It will take 2 seconds until the diver enters the water.
10.6 Factoring ax 2 + bx + c
fcgfl Exercises
Guided Practice
Vocabulary Check 1 , What is the difference between factoring quadratic polynomials of the form
x 2 + bx + c and ax 2 + bx + cl
Skill Check Copy and complete the statement.
2 . (2x + l)(x + 1 ) = 2x 2 _J_ + 1 3 . (3x + 2)(x - 3) = 3x 2 - lx _J_
4. (3x - 4)(x - 5) = 3x 2 _?_ + 20 5. (5x + 2)(2x + 1) = _?_ + 9x + 2
Match the trinomial with a correct factorization.
A. (3x + 2)(x + 3)
B. (3x + l)(x - 6)
C. (3x - l)(x + 6)
D. (3x - 2)(x + 3)
6- 3x 2 — \lx — 6
7. 3x 2 + lx — 6
8- 3x 2 + 1 lx + 6
9 - 3x 2 + 17x — 6
Factor the trinomial.
10. 2x 2 + 17x + 21
13. 12x 2 - 19x + 4
11. 2x 2 — 3x — 2
14. 6x 2 + lx - 20
12 . 6t 2 - t - 5
15. 3x 2 + 2x - 8
18. -7n 2 - 40 n = -12
Solve the equation.
16. 3& 2 + 26b + 35 = 0 17. 2z 2 + 15z = 8
Practice and Applications
I
Student HeCp
► Homework Help
Example 1: Exs. 19-39
Example 2: Exs. 19-39
Example 3: Exs. 19-39
Example 4: Exs. 42-54
Example 5: Exs. 55-57
FACTORIZATIONS Choose the correct factorization. If neither choice is
correct, find the correct factorization.
19. 3x 2 + 2x — 8 20. 6_y 2 - 29y - 5
A. (3x - 4)(x + 2) A. (2 y + l)(3y - 5)
B. (3x - 4)(x - 2) B. (6 y - l)(y + 5)
FACTORING TRINOMIALS Factor the trinomial.
23. 2? + 16f + 5
22 . 2x 2 - x - 3
25. 6 a 2 + 5a + 1
28. 8 b 2 + 2b - 3
31.2z 2 + 19z- 10
34. 4« 2 - 22 n - 42
37. 6t 2 + t- 70
26. 5w 2 — 9w — 2
29. 6x 2 — 9x — 15
32. 6 y 2 — 11 y — 10
35. 3c 2 - 37c + 44
38. 14y 2 - 15y + 4
21 . 4w 2 - 14w - 30
A. (2 w + 3)(2 w ~ 10)
B. (Aw + 15)(w - 2)
24. 5x 2 + 2x — 3
27. 6b 2 - lib - 2
30. 12/ - 20v + 8
33. Ax 2 + 21 x + 35
36. lAr 2 - 6r - 45
39. 8 y 2 — 26 y +15
Chapter 10 Polynomials and Factoring
ERROR ANALYSIS Find and correct the error.
SOLVING EQUATIONS Solve the equation by factoring.
42.
2x 2 —
9x -
-35 = 0
43. lx 2 -
45.
4x 2 -
21x
+
5 = 0
46. 2x 2 -
48.
2x 2 +
19x
=
-24
49. 4X 2 -
51.
8x 2 -
34x
+
24 = ■
-11
53.
28x 2 -
- 9x
—
1 = -
4x + 2
lOx + 3 = 0 44, 3x 2 + 34x + 11 =0
17* - 19 = 0 47. 5x 2 - 3x - 26 = 0
8x = —3 50. 6x 2 — 23x = 18
52. 6x 2 + I9x - 10 = -20
54. 10x 2 + x — 10 = —2x + 8
Student HeCp
► Homework Help
Extra help with
-^ 0 + pj-Qbiem solving in
Exs. 55-57 is available at
www.mcdougallittell.com
VERTICAL COMPONENT OF MOTION In Exercises 55-57, use the vertical
motion model h = -16f 2 + vt + s where h is the height (in feet), t is the
time in motion (in seconds), v is the initial velocity (in feet per second),
and s is the initial height (in feet). Solve by factoring.
55. GYMNASTICS A gymnast
dismounts the uneven parallel
bars at a height of 8 feet with
an initial upward velocity of
8 feet per second.
a. Write a quadratic equation
that models her height
above the ground.
b. Use the model to find the time
t (in seconds) it takes for the
gymnast to reach the ground. Is your answer reasonable?
56. CIRCUS ACROBATS An acrobat is shot out of a cannon and lands in a
safety net that is 10 feet above the ground. Before being shot out of the
cannon, she was 4 feet above the ground. She left the cannon with an initial
upward velocity of 50 feet per second.
a. Write a quadratic model to represent this situation.
b. Use the model to find the time t (in seconds) it takes for her to reach the
net. Explain why only one of the two solutions is reasonable.
57. T-SHIRT CANNON At a basketball game, T-shirts are rolled-up into a ball
and shot from a “T-shirt cannon” into the crowd. The T-shirts are released
from a height of 6 feet with an initial upward velocity of 44 feet per second.
If you catch a T-shirt at your seat 30 feet above the court, how long was it in
the air before you caught it? Is your answer reasonable?
10.6 Factoring ax 2 + bx + c
Standardized Test
Practice
Mixed Review
Maintaining Skills
Quiz 2
58. MULTIPLE CHOICE Factor 9x 2 - 6x - 35.
(A) (9x - 5)(x + 7) Cl) (3x + 5)(3x - 7)
CD (9x + 5)(x - 7) CD (3x - 5)(3x + 7)
59. MULTIPLE CHOICE Solve 2x 2 + 5x + 3 = 0.
CD “land—| CD - '|and-| CD ^ and —| CD 1 and-|
SOLVING SYSTEMS Use linear combinations to solve the linear system.
Then check your solution. (Lesson 7.3 )
60- 4x + 5y = 7 61- 6x — 5_y = 3 62. 2x + y = 120
6x — 2 v = — 18 — \2x + 8_y = 5 x + 2y = 120
SPECIAL PRODUCT PATTERNS Find the product. (Lesson 10.3)
63. (4 1 - l) 2 64. (b + 9 )(b - 9) 65. (3x + 5)(3x + 5)
66. (2a - 7)(2 a + 7) 67. (11 - 6x) 2 68. (100 + 27x) 2
OPERATIONS WITH FRACTIONS Simplify. (Skills Review p. 765)
69.
2 6
3 * 9
n_
3
70.
1
1 2
9 * 3
71.
1 4 5
2*9*6
72.
8
9 8
8 * 9
73.
2 4 6
- • - • -
3 5 7
74.
12 3
15 * 4
75.
5 9 1
6*4*3
1 , A I I
2 2 * 3
1 1
4 * 5
Solve the equation. (Lesson 10.4)
1. (x + 5)(2x + 10) = 0 2. (2x + 8) 2 = 0 3. (2x + 7)(3x - 12) = 0
4. x(5x — 2) = 0 5. 3(x — 5)(2x + 1) = 0 6. x(x + 4)(x — 7) 2 = 0
Find the x-intercepts and the vertex of the graph of the function. Then
sketch the graph of the function. (Lesson 10.4)
7. y = (x — 2)(x + 2) 8. y = (x + 3)(x + 5)
Factor the trinomial. (Lesson 10.5)
10 . y 2 + 3y - 4
13.x 2 + 7x + 24
19. y 2 + 5y — 6 = 0
22 . t 2 + Ut= -18
25. 3 b 2 - 10b - 8 = 0
11. w 2 + 13 w + 22
14. b 2 -6b- 16
17 x 2 + 17x + 66
23. 2a 2 + 1 la + 5 = 0
26. 4c 2 + 12c + 9 = 0
9. y = (x - l)(x + 3)
12 . n 2 + 16 n — 57
15. r 2 - 3r - 28
18. r 2 — 41r — 86
21 . z 2 - 14z + 45 = 0
24. 3p 2 - 4p + 1 = 0
27. 15 b 2 + 41 b = -14
16. nr — 4m — 45
Solve the equation by factoring. (Lesson 10.6)
20 - n 2 + 26 n + 25 = 0
Chapter 10 Polynomials and Factoring
Factoring Special Products
Goal
Factor special products.
Key Words
• perfect square
trinomial
What height can a pole-vaulter reach?
-
In Exercise 65 you will factor a
quadratic polynomial to find the
height a pole-vaulter can vault.
In Lesson 10.5 you learned to factor trinomials of the form x 2 + bx + c, where b
and c are integers. For example, to factor x 2 + 3x + 2, you looked for two
numbers whose product was 2 and whose sum was 3. The two numbers are 1
and 2, so you wrote x 2 + 3x + 2 = (x + l)(x + 2).
You can factor x 2 — 9 using the same reasoning. Since there is no middle term,
its coefficient must be zero. So you will need two numbers whose product is —9
and whose sum is 0. The two numbers are 3 and —3. Thus, you can write
x 2 - 9 = (x + 3)(x - 3).
This suggests a simple pattern for factoring the difference of two squares.
a 2 — b 2 = (a + b)(a — b )
If we rewrite the square of a binomial pattern (from page 581) as shown below,
two useful factoring patterns are created.
a 2 + lab + b 2 — (a + b) 2 or a 2 — lab + b 2 = (a — b ) 2
Consider factoring x 2 — lOx + 25, for example. You can try the second pattern
because the middle term is negative. Let a = x and b = 5. The pattern requires
that — lab be the constant term, which is true here because — 2(x)(5) = — lOx.
Therefore, x 2 — lOx + 25 = (x — 5) 2 .
Trinomials of the form a 2 + lab + b 2 and a 2 — lab + b 2 are called perfect
square trinomials because they can be factored as the squares of binomials.
FACTORING SPECIAL PRODUCTS
Difference of Two Squares Patterns
a 2 - b 2 = (a + b)(a — b) Example: 9x 2 S 25 = (3x + 5)(3x - 5)
Perfect Square Trinomial Pattern
a 2 + lab + b 2 = (a + b) 2 Example: x 2 + 14x + 49 = (x + 7) 2
a 2 — lab + b 2 = (a — b) 2 Example: x 2 — 12x + 36 = (x — 6) 2
10.7 Factoring Special Products
Student HeCp
► Study Tip
You can check your
work by multiplying
the factors.
^ _ )
1 Factor the Difference of Two Squares
Factor the expression.
a. m 2 — 4 b. 4p 2 — 25 c. 9q 2 — 64 d. a 2 — 8
Solution
a. m 2 — 4 = m 2 — 2 2
= (m + 2 ){m — 2)
b. 4p 2 - 25 = (2p) 2 - 5 2
= (2 p + 5)(2 p - 5)
c. 9q 2 - 64 = (3 qf - 8 2
= (3 q + 8)(3 q - 8)
Write as a 2 - b 2 .
Factor using pattern.
Write as a 2 - b 2 .
Factor using pattern.
Write as a 2 - b 2 .
Factor using pattern.
d. a 2 — 8 cannot be factored using integers because it does not fit the
difference of two squares pattern; 8 is not the square of an integer.
Factor the Difference of Two Squares
Factor the expression.
1. x 2 — 16 2 . n 2 — 36 3 - r 2 — 20
5. 8>’ 2 - 1 6. 4y 2 - 49 7. 9x 2 - 25
4. m 2 - 100
8 . 16 q 2 - 45
2 Factor Perfect Square Trinomials
Factor the expression.
a. x 2 — 4x + 4 b. a 2 — 18a + 81 c. 16j 2 + 24_y + 9
Solution
a. x 2 - 4x + 4 = x 2 - 2(x)(2) + 2 2
= (x - 2) 2
b. a 2 - 18a + 81 = a 2 - 2(a)(9) + 9 2
= (a - 9) 2
Write as o 2 - 2 ab + b 2 .
Factor using pattern.
Write as a 2 - lab + b 2 .
Factor using pattern.
c. 16y 2 + 24y+ 9 = (4 y) 2 + 2(4y)(3) + 3 2
= (4y + 3) 2
Write as a 2 + lab + b 2 .
Factor using pattern.
Factor Perfect Square Trinomials
Factor the expression.
9. x 2 + 6x + 9
12 . 4 b 2 - 4b + 1
10 . n 2 - + 16
13. 25 m 2 + 10m + 1
11. a 2 + 18 a + 81
14. 9 a 2 — 30 a + 25
T
Chapter 10 Polynomials and Factoring
3 Factor Out a Constant First
a. 50 - 98x 2 = 2(25 - 49x 2 )
= 2[5 2 - (7x) 2 ]
= 2(5 + 7x)(5 - lx)
Factor out common factor.
Write as o 2 - b 2 .
Factor using pattern.
b. 3x 2 — 30x + 75 = 3(x 2 — lOx + 25)
= 3[x 2 - 2(x)(5) + 5 2 ]
= 3(x - 5) 2
Factor out common factor.
Write as o 2 - 2 cib + b 2 .
Factor using pattern.
c. 4x 2 + 24x + 44 = 4(x 2 + 6x + 11) Factor out common factor.
Since 11 is not the square of any integer, you cannot factor 4(x 2 + 6x + 11)
with integers using the perfect square trinomial pattern.
Factor Out a Constant First
Factor the expression.
15. 2x 2 - 32
18. 8 n 2 - 24 n + 18
16. 3 p 2 + 36 p + 108
19. 1000 - 10m 2
17. 3 b 2 - 48
20 . 2 a 2 + 28 a + 98
4 Graphical and Analytical Reasoning
Solve the equation — 2x 2 + 12x
—2x 2 + 12x - 18 = 0
—2(x 2 — 6x + 9) = 0
-2[x 2 - 2(x)(3) + 3 2 ] = 0
—2(x - 3) 2 = 0
x — 3 = 0
x = 3
ANSWER ► The solution is 3.
18 = 0.
Write original equation.
Factor out common factor.
Write as a 2 - lab + b 2 .
Factor using pattern.
Set repeated factor equal to 0.
Solve for x.
CHECK y You can check your answer by substitution or by graphing.
Also, a graphing calculator will provide a graphical representation of the
solution x = 3.
Graph y = —2x 2 + 12x — 18.
Graph the x-axis, y = 0.
B Use your graphing calculator’s
Intersect feature to find the x-intercept,
where — 2x 2 + 12x — 18 = 0.
Whenx = 3, —2x 2 + 12x — 18 = 0,
so your answer is correct.
10.7 Factoring Special Products
Student HeCp
► More Examples
More exam Pl es
are available at
www.mcdougallittell.com
5 Solve a Quadratic Equation
Solve 4x 2 + 4x + 1 = 0.
Solution
4x 2 + 4x + 1 = 0
(2x) 2 + 2(2x) + l 2 = 0
(2x + l) 2 = 0
2x + 1 = 0
1
X= “2
ANSWER ► The solution is —
Write original equation.
Write as o 2 + 2 ab + b 2 .
Factor using pattern.
Set repeated factor equal to 0.
Solve for x.
Check this in the original equation.
Solve a Quadratic Equation
Solve the equation by factoring. Then use a graphing calculator to check
your solutions.
21. x 2 — 81 = 0 22. m 2 — Am + 4 = 0 23. In 2 — 288 = 0
Link to
Science
BLOCK AND TACKLE A
block and tackle makes it
easier to lift a heavy object.
For instance, using a block
and tackle with 4 pulleys, you
can lift 1000 pounds with only
250 pounds of applied force.
6 Write and Use a Quadratic Model
BLOCK AND TACKLE An object lifted with a rope or wire should not weigh
more than the safe working load for the rope or wire. The safe working load
S (in pounds) for a natural fiber rope is a function of C, the circumference of
the rope in inches.
Safe working load model: 150 • C 2 = S
You are setting up a block and tackle to lift a 1350-pound safe. What size
natural fiber rope do you need to have a safe working load?
Solution
150 C 2 = S
Write model.
150 C 2 = 1350
Substitute 1350 for 5.
150C 2 - 1350 = 0
Subtract 1350 from each side.
150(C 2 - 9) = 0
Factor out common factor.
150(C + 3)(C - 3) = 0
Factor.
C + 3 = 0 or C — 3 = 0
Use zero-product property.
C = — 3 ! C — 3 Solve for C.
ANSWER ► Negative values for circumference do not make sense, so you will
need a rope with a circumference of at least 3 inches.
Chapter 10 Polynomials and Factoring
EH Exercises
Guided Practice
Vocabulary Check
1. Write the three special product factoring patterns. Give an example of
each pattern.
Skill Check
Factor the expression.
2 . x 2 - 9
5. w 2 — 16 w + 64
8 . 18 - 2 b 2
3. b 2 + 10 b + 25
6 . 16 - c 2
9. 4X 2 — 4x + 1
4 .p 2 + 25
7. 6/ - 24
10. 4a 2 - b 2
Solve the equation by factoring.
11.x 2 + 6x +9 = 0 12. 144 — y 2 = 0 13. s 2 - 14s + 49 = 0
14. -25+x 2 = 0 1 5. 4y 2 — 24y + 36 = 0 16. 7x 2 + 28x + 28 = 0
17. VERTICAL COMPONENT OF MOTION You throw a ball upward from the
ground with an initial velocity of 96 feet per second. How long will it take
the ball to reach a height of 144 feet? HINT: Use the vertical motion model
on page 607.
Practice and Applications
DIFFERENCE OF TWO SQUARES Factor the expression.
18. n 2 - 16
19. q 2 - 64
20. b 2 - 48
21. 9c 2 - 1
22. 49 - a 2
23. 81 - x 2
24. 36x 2 + 25
25. w 2 - 9y 2
26. 25s 2 - 16 1 2
PERFECT SQUARES
Factor the expression.
27. x 2 + 8x + 16
28. x 2 - 20x + 100
29. b 2 - 14 b + 49
30. y 2 + 30y + 225
31. 9x 2 + 6x + 1
32. 4r 2 + 12r + 9
33. 25 n 2 — 20u + 4
34. 18x 2 + 12x + 2
35. 16w 2 — 80w +100
36- 36 m 2 — 84 m + 49 37. a 2 — 4 ab + 4 b 2
38. x 2 + 12xy + 36 y 2
Student HeCp
► Homework Help
COMMON FACTOR
Factor the expression.
Example 1 : Exs. 18-26
Example 2: Exs. 27-38
39. 4 n 2 - 36
40. -32 + 18x 2
41. 5c 2 + 20c + 20
Example 3: Exs. 39-50
Example 4: Exs. 51-58
42. 6b 2 - 54
43. 27 1 2 + 18f + 9
44. 28y 2 - 7
Example 5: Exs. 51-58
Example 6: Exs. 60-65
45. 3k 2 - 39 k + 90
46. 24a 2 - 54
47. 4 b 2 - 40 b + 100
V J
48. 32x 2 - 48x + 18
49. 16w 2 + 80w +100
50. 2x 2 + 28xy + 98y 2
10.7 Factoring Special Products
Link to
Sports
POLE-VAULTERS The pole-
vault is a track and field
event. From a running start,
the athlete uses a springy
pole to leap over a high
crossbar.
SOLVING EQUATIONS Solve the equation by factoring. Use a graphing
calculator to check your solution if you wish.
51. 4x 2 + 4x + 1 = 0 52. 25x 2 - 4 = 0
53. 3x 2 - 24x + 48 = 0 54. -27 + 3x 2 = 0
55. 6b 2 - 12b + 216 = 0
57. 16X 2 - 56x + 49 = 0
56. 90x 2 - 120x + 40 = 0
58. 50x 2 + 60x + 18 = 0
59. VERTICAL COMPONENT OF MOTION A model rocket is fired upward with
an initial velocity of 160 feet per second. How long will it take the rocket to
reach a height of 400 feet? Hint: Use the vertical motion model on p. 607.
SAFE WORKING LOAD In Exercises 60 and 61, the safe working load
S (in tons) for a wire rope is a function of D, the diameter of the rope
(in inches).
Safe working load model for wire rope: 4 • D 2 — S
60, What diameter of wire rope do you need to lift a 9-ton load and have a safe
working load?
61 _ When determining the safe working load S of a rope that is old or worn,
decrease S by 50%. Write a model for S when using an old wire rope.
What diameter of old wire rope do you need to safely lift a 9-ton load?
Hang time model: h = 41 1
62, If you jump 1 foot into the air,
what is your hang time?
63, If a professional player
jumps 4 feet into the air, what
is the hang time?
HANG TIME In Exercises 62 and 63, use the following information
about a basketball player's hang time, the length of time spent in the air
after jumping.
The maximum height h jumped
(in feet) is a function of t , where
t is the hang time (in seconds).
POLE-VAULTING In Exercises 64 and 65, use the following information.
In the sport of pole-vaulting, the height h (in feet) reached by a pole-vaulter can
be approximated by a function of v, the velocity of the pole-vaulter, as shown in
the model below. The constant g is approximately 32 feet per second per second.
v 2
Pole-vaulter height model: h = —■
64, To reach a height of 9 feet, what is the pole-vaulter’s velocity?
65- What height will a pole-vaulter reach if the pole-vaulter’s velocity is
32 feet per second?
Chapter 10 Polynomials and Factoring
Standardized Test
Practice
Mixed Review
Maintaining Skills
66. MULTIPLE CHOICE Which of the following is a correct factorization
of-12x 2 + 147?
(A) -3(2x + l ) 2 CD 3(2x - 7)(2x + 7)
CD —2(2x - 7)(2x + 7) CD -3(2* - 7)(2* + 7)
67. MULTIPLE CHOICE Which of the following is a correct factorization
of 72x 2 - 24x + 2?
CD -9(3* - l) 2 CD 2(6x - l) 2
CD 8(3x - l) 2 CD 9(3x - l) 2
68. MULTIPLE CHOICE Solve 9x 2 - 12* + 4 = 0.
CD -3 CD -f CD f CD 3
CHECKING FOR SOLUTIONS Determine whether the ordered pair is a
solution of the system of linear equations. (Lesson 7. 1)
69. x + 9y = — 11
-4x + y= -30 (7, -2)
70. 2x + 6 y = 22
-x - 4y = - 13 (-5, -2)
71. -2* + 7y = -41
3x + 5y = 15 (-10,3)
72. —5x — = 28
9x — 2 y = 48 (4, -6)
SOLVING LINEAR SYSTEMS Use the substitution method to solve the
linear system. (Lesson 7.2)
73. x — y = 2
2x + y = 1
74. x 2y = 10
3x — y = 0
75. —x + y = 0
2x + y = 0
76. x — 2y = 4
2x + y = 3
77. x - y = 0
3x + 4 y= 14
78. 2x + 3y = -5
x — 2 y = —6
SIMPLIFYING RADICAL EXPRESSIONS Simplify the expression.
(Lesson 9.3)
79. V216
83.
80. V5 • Vl5
10V8
84.
V25
8i. V 10 • V 20
12 V 4
85.
82. V4 • 3V9
-6V42
V9
86 .
V4
SOLVING EQUATIONS Use the quadratic formula to solve the equation.
(Lesson 9.6)
87. 9x 2 - 14x - 7 = 0 88. 9 d 2 - 5M + 24 = 0 89. ly 2 - 9y - 17 = 0
PRIME FACTORIZATION Write the prime factorization of the number if it
is not a prime number. If a number is prime, write prime.
(Skills Review p. 761)
90.8 91.20
94. 96 95. 80
98. 244 99. 345
92. 45 93. 57
96. 101 97. 120
100.250 1 01.600
10.7 Factoring Special Products
Factoring Cubic Polynomials
Goal
Factor cubic polynomials.
Key Words
• prime polynomial
• factor a polynomial
completely
What are the dimensions of a terrarium?
In Example 6 you will factor a
cubic polynomial to determine the
dimensions of a terrarium, which
is an enclosed space for keeping
small animals indoors.
You have already been using the distributive property to factor out constants that
are common to the terms of a polynomial.
9x 2 —15 = 3(3x 2 — 5) Factor out common factor.
You can also use the distributive property to factor out variable factors that are
common to the terms of a polynomial. When factoring a cubic polynomial, you
should factor out the greatest common factor (GCF) first and then look for
other patterns.
Student HeCp
► Skills Review
For help with finding
the GCF, see p. 761.
L )
i Find the Greatest Common Factor
Factor the greatest common factor out of 14x 3 — 21x 2 .
Solution
First find the greatest common factor of 14x 3 and 2 lx 2 .
14x 3 = 2 • 7 • x • x • x
21x 2 = 3 • 7 • x • x
GCF = 7 • x • x — lx 2
Then use the distributive property to factor out the greatest common factor
from each term.
ANSWER ^ 14x 3 — 21x 2 = 7x 2 (2x — 3).
Find the Greatest Common Factor
Factor out the greatest common factor.
1 . 1 lx — 22 2 . 6x 2 + 12x + 18
4. 3 n 3 - 36 n 2 + 12 n 5. 4y 3 - 10 y 2
3. 8X 3 — 16x
6 . 9X 3 + 6x 2 + 18x
Chapter 10 Polynomials and Factoring
PRIME FACTORS A polynomial is prime if it cannot be factored using integer
coefficients. To factor a polynomial completely, write it as the product of
monomial and prime factors.
J 2 Factor Completely
Factor 4x 3 + 20x 2 + 24x completely.
Solution
4x 3 + 20x 2 + 24x = 4x(x 2 + 5x + 6)
4x(x + 2)(x + 3)
^-- *
Factor out GCF.
Factor trinomial.
Monomial factor
Prime factors
Factor Completely
Factor the expression
7. 2 n 3 + 4n 2 + 2 n
10. x 3 + 4x 2 + 4x
completely.
8- 3x 3 — I2x
11 . 2X 3 - 10x 2 + 8x
9. 5m 3 — 45m
12 . 6 p 3 + 21 p 2 + 9p
FACTORING BY GROUPING Another use of the distributive property is in
factoring polynomials that have four terms. Sometimes you can factor the
polynomial by grouping the terms into two groups and factoring the greatest
common factor out of each term.
H3ZSE2H 3 Fact or by Grouping
Factor x 3 — 2x 2 — 9x + 18 completely.
Solution
x 3 - 2x 2 - 9x + 18 = (x 3 - 2x 2 ) + (—9x + 18)
= x 2 (x - 2) + (—9)(x - 2)
= (x - 2)(x 2 - 9)
= (x - 2)(x - 3)(x + 3)
Group terms.
Factor each group.
Use distributive
property.
Factor difference of
two squares.
Factor by Grouping
Use grouping to factor the expression completely.
13. 2X 3 - 8x 2 + 3x - 12 14. x 3 + 5x 2 - 4x - 20 15. x 3 - 4x 2 - 9x + 36
10.8 Factoring Cubic Polynomials
Student HeCp
p More Examples
More examples
are available at
www.mcdougallittell.com
SUM OR DIFFERENCE OF TWO CUBES In Lessons 10.3 and 10.7, you used the
difference property to study the special product pattern of the difference of two
squares. You can also use the distributive property to confirm the following
special product patterns for the sum or difference of two cubes.
FACTORING MORE SPECIAL PRODUCTS
Sum of Two Cubes Pattern
a 3 + b 3 = (a + b)(a 2 - ab + bi 2 ) Example: (x 3 + 1) = (x + 1)(x 2 - x + 1)
Difference of Two Cubes Pattern
a 3 - b 3 = (a - b)(a 2 + ab + b 2 ) Example: (x 3 - 8) = (x - 2)(x 2 + 2x + 4)
4 Factor the Sum of Two Cubes
Factor x 3 + 27.
Solution
x 3 + 27 = x 3 + 3 3 Write as sum of cubes.
= (x + 3)(x 2 — 3x + 9) Use special product pattern. Notice that
x 2 - 3x + 9 is prime and does not factor.
Factor the Sum of Two Cubes
Factor the expression.
16. x 3 + 125 17. n 3 + 8 18. 2m 3 + 2 19. 4X 3 + 32
(B3332EQM 5 Factor the Difference of Two Cubes
Factor n 3 — 64.
Solution
n 3 — 64 = n 3 — 4 3 Write as difference of cubes.
— (n —4)(n 2 + 4 n + 16) Use special product pattern. Notice that
n 2 + 4n + 16 is prime and does not factor.
Factor the Difference of Two Cubes
Factor the expression.
20 . x 3 - 27 21 . p 3 - 216 22 . In 3 - 250
23. 4z 3 - 32
Chapter 10 Polynomials and Factoring
SPACE REQUIREMENTS
Generally, an adult bearded
dragon lizard will need a
terrarium or cage that is at
least 4 to 6 feet in length,
2 to 3.5 feet in height and
2 to 3.5 feet in depth.
6 Write and Use a Polynomial Model
SPACE REQUIREMENTS A terrarium
has a volume of 12 cubic feet. Find the
dimensions of the terrarium. Do the
dimensions meet the space requirements
of an adult bearded dragon lizard?
(x + 4)ft
(X-I)ft
Solution
V = height • width • length
12 = jc(x — l)(x + 4)
12 = x 3 + 3x 2 — 4x
0 = (x 3 + 3x 2 ) + (—4x —12)
0 = x 2 (x + 3) +(—4)(x +3)
0 = (jc + 3)(x 2 - 4)
0 = (jc + 3)(x - 2)(x + 2)
Write volume model for a prism.
Substitute for height, width and length.
Multiply.
Write in standard form and group terms.
Factor each group of terms.
Use distributive property.
Factor difference of two squares.
By setting each factor equal to zero, you can see that the solutions are —3, 2,
and —2. The only positive solution is x = 2.
ANSWER ^ The dimensions of the terrarium are 2 feet by 1 foot by 6 feet.
Because the height must be between 2 and 3.5 feet, the dimensions
do not meet the space requirements of an adult bearded dragon lizard.
Patterns Used to Solve Polynomial Equations
graphing: Can be used to solve any equation, but gives only
approximate solutions. Examples 2 and 3, pp. 527-528
the quadratic formula: Can be used to solve any quadratic
equation. Examples 1-3, pp. 533-534
factoring: Can be used with the zero-product property to solve
an equation that is in standard form and whose polynomial is
factorable.
• Factoring x 2 + bx + c: Examples 1-7, pp. 595-598
• Factoring ax 2 + bx + c\ Examples 1-5, pp. 603-605
•Special Products: Examples 1-6, pp. 610-612 and Examples 4
and 5, p. 618
a 2 - b 2 = (a + b)(a - b)
a 2 + lab + b 2 = {a + b) 2
a 2 - lab + b 2 = (a - b) 2
a 3 + b 3 = (a + b)(a 2 - ab + b 2 )
a 3 - b 3 = (a - b)(a 2 + ab + b 2 )
• Factoring Completely: Examples 1-3, pp. 616-617
10.8 Factoring Cubic Polynomials
Exercises
Guided Practice
Vocabulary Check 1 . What does it mean to say that a polynomial is prime?
Skill Check ERROR ANALYSIS Find and correct the error.
> 3 +■ 12b 2 - 14b
= -2b(b 2 •+■ 6b - 7)
==-2b(b+-7)(b>tt
Find the greatest common factor of the terms and factor it out of
the expression.
4. 5 n 3 - 20 n 5. 6x 2 + 3x 4 6. 6y 4 + 14y 3 - 10y 2
Factor the expression.
7. x 3 - 1 8. x 3 + 64 9. 27X 3 + 1 10. 125x 3 - 1
Factor the expression completely.
11 . 2b 3 — ISb 12 . la 3 — 14a 2 — 21a
14. y 3 — 6y 2 + 5y 15. x 3 — 16x
13. 3t 3 + m 2 + 27 1
16. 5b 3 - 25b 2 - 10b
Practice and Applications
FACTORING THE GCF Find the greatest common factor of the terms and
factor it out of the expression.
17. 6v 3 - 18v 18. 4g 4 + I2q
20 . lOx 2 + 15x 3 21 . 4a 2 - 8a 5
19. 3x - 9x 2
22 . 241 5 + 6 1 3
23. 15X 3 — 5x 2 — lOx 24. 4a 5 + 8a 3 — 2a 2
25. 18 d 6 — 6 d 2 + 3d
Student HcCp
^ .
► Homework Help
Example 1: Exs. 17-25
Example 2: Exs. 36-44
Example 3: Exs. 26-31
Example 4: Exs. 32-35
Example 5: Exs. 32-35
Example 6: Exs. 59-61
FACTOR BY GROUPING Factor the expression.
26. x 2 + 2x + xy + 2y
28. 2x 3 — 3x 2 — 4x + 6
30. 8x 2 - 3x - 8x + 3
27. a 2 + 3a + ab + 3b
29. 10x 2 — 15x + 2x — 3
31. 10x 2 — lx — lOx + 7
SUM AND DIFFERENCE OF TWO CUBES Factor the expression.
32. m 3 +1 33. c 3 - 8 34. r 3 + 64 35. m 3 - 125
Chapter 10 Polynomials and Factoring
FACTORING COMPLETELY Factor the expression completely.
36. 24x 3 + 18x 2 37. 2v 3 - 1Ov 2 - I2y 38. 5.v 3 + 30s 2 + 40s
39. 4f 3 — 144? 40. -12z 3 + 3z 2 41. c 4 + c 3 - 12c-12
42. x 3 — 3x 2 + x — 3 43. 3x 3 + 3000 44. 2x 3 — 6750
SOLVING EQUATIONS Solve the equation. Tell which method you used.
45. y 2 + ly + 12 = 0 46. x 2 - 3x - 4 = 0
47. 27 + 6w - w 2 = 0 48. 5x 4 - 80x 2 = 0
49. — 16x 3 + 4x = 0 50. 10x 3 - 290x 2 - 620x = 0
! Student HeCp
p Look Back
For help with finding
roots, see p. 534.
I ___>
FINDING ROOTS OF POLYNOMIALS Use the quadratic formula or
factoring to find the roots of the polynomial. Write your solutions in
simplest form.
51 _ 4x 2 — 9x — 9 = 0 52, 5x 2 + 2x — 3 = 0 53. 2x 2 + 5x + 1 = 0
54. 3x 2 - 4x + 1 = 0 55. 6X 2 - 2x - 7 = 0 56. 3X 2 + 8x - 2 = 0
Science Link/ In Exercises 57 and 58, use the vertical motion models,
where h is the height (in feet), v is the initial upward velocity
(in feet per second), s is the initial height (in feet), and t is the time
(in seconds) the object spends aloft.
Vertical motion model for Earth: h — —16 1 2 + vt + s
16 o
Vertical motion model for the moon: h = —— t + vt + s
o
Note: the two equations are different because the acceleration due to gravity on
the moon’s surface is about one-sixth that of Earth.
PACKAGE DESIGNERS
consider the function of a
package to determine the
appropriate size, shape,
weight, color and materials
to use.
57. EARTH On Earth, you toss a tennis ball from a height of 96 feet with an
initial upward velocity of 16 feet per second. How long will it take the tennis
ball to reach the ground?
58. MOON On the moon, you toss a tennis ball from a height of 96 feet with an
initial upward velocity of 16 feet per second. How long will it take the tennis
ball to reach the surface of the moon?
PACKAGING In Exercises 59-61, use the following information. Refer to
the diagram of the box.
The length f of a box is 3 inches less than the
height h. The width w is 9 inches less than the
height. The box has a volume of 324 cubic inches.
59. Copy and complete the diagram by labeling
the dimensions.
60. Write a model that you can solve to find
the length, height, and width of the box.
61 . What are the dimensions of the box?
10.8 Factoring Cubic Polynomials
Standardized Test 62. MULTIPLE CHOICE Which of the following is the complete factorization of
Practice x 3 — 5x 2 + 4x — 20 ?
(A) (x + 2)(x + 2)(x -5) CD (x + 2)(x - 2)(x - 5)
CD (x 2 + 4)(x - 5) CD (x ~ 4)(x - l)(x - 20)
63. MULTIPLE CHOICE Solve x 3 - 4x = 0.
CD 0 and 2 CD 0, 2, and—2 CD 2 and—2 CD - 2and0
Mixed Review SOLVING INEQUALITIES Solve the inequality. (Lesson 6.3)
64. 7 + x < -9 65. -3 > 2x - 5 66. -x + 6 < 12
SOLVING ABSOLUTE-VALUE EQUATIONS Solve the equation. (Lesson 6.6)
67. | x | = 3 68. | x - 5 | = 7 69. | x + 6 | = 13 70. | 4x + 3 | = 9
GRAPHING INEQUALITIES Graph the inequality. (Lesson 6.8)
71. x + y < 9 72. _y — 3x > 2 73.y-4x<10
Maintaining Skills
RECIPROCALS
74. 18
Find the reciprocal. (Skills Review p. 763)
75. -7 76. | 77. l|
79. -2- c
80. 9
10
Quiz 3 -
Factor the expression. Tell which special product factoring pattern you
used. (Lesson 10.7)
1 . 49x 2 - 64 2. 121 - 9x 2 3. 4f 2 + 20f + 25
4. 72 - 50/ 5. 9y 2 + 42y + 49 6 . 3 n 2 - 36 n + 108
Solve the equation by factoring. (Lesson 10.7)
7. x 2 — 8x + 16 = 0 8- 4x 2 + 32x + 64 = 0 9- x 3 + 9x 2 — 36x = 0
Find the greatest common factor and factor it out of the expression.
(Lesson 10.8)
10. 3x 3 + 12x 2 11.6x 2 + 3x 12. 18x 4 - 9x 3 13. 8 X 5 + 4X 2 - 2x
Factor the expression completely. (Lesson 10.8)
14, 2x 3 — 6x 2 + 4x 15- x 3 + 3x 2 + 4x + 12 16- 4x 3 — 500
Solve the equation by factoring. (Lesson 10.8)
17. 108_y 3 — 75 y = 0 18. 3x 3 — 6x 2 + 5x = 10
i ~~
Chapter Summary
and Review
• monomial, p. 568
• degree of a monomial, p. 568
• polynomial, p. 569
• binomial, p. 569
• trinomial, p. 569
\ _
• standard form, p. 569
• degree of a polynomial
in one variable, p. 569
• FOIL pattern, p. 576
• factored form, p. 588
• zero-product property, p. 588
• factor a trinomial, p. 595
• perfect square trinomial,
p. 609
• prime polynomial, p. 617
• factor a polynomial
completely, p. 617
Adding and Subtracting Polynomials
Examples on
pp. 568-570
To add or subtract polynomials, add or subtract like terms.
HORIZONTAL FORMAT VERTICAL FORMAT
(4x 3 + 6x — 8) — (—x 2 + lx — 2) —2x 3 — 4x 2 — x + 5
= 4x 3 + 6x — 8 + x 2 — lx + 2 3x 3 + 2x 2 — 4x + 9
— 6 + —x 3 + 5x 2 — x — 1
4x 3 + x 2
3x 2 — 6x + 13
Use a vertical format or a horizontal format to add or subtract.
1. (5x - 12) - (2x - 7) 2. (24m - 13) - (18m + 7) + (6m - 4)
3- (—x 2 + x + 2) + (3x 2 + 4x + 5) 4- (x 2 + 3x — 1) — (4x 2 — 5x + 6)
5- (x 3 + 5x 2 — 4x) — (3x 2 — 6x + 2) 6- (4x 3 + x 2 — 1) + (2 — x — x 2 )
Multiplying Polynomials
Examples on
pp. 575-577
To multiply polynomials, use the distributive property or FOIL pattern,
a. (3x + 2)(5x 2 - 4x + 1) = 5x 2 (3x + 2) + (-4x)(3x + 2) + l(3x + 2)
= 15x 3 + 10x 2 — 12x 2 — 8x + 3x + 2
= 15x 3 — 2x 2 — 5x + 2
First Outer Inner Last
b. (Ax + 5)(—3x - 6) = -\2x 2 - 2Ax - \Sx - 30
= — 12X 2 — 39x — 30 Combine like terms.
Chapter Summary and Review
Chapter Summary and Review continued
Find the product.
7. 3a(2a 2 — 5a + 1) 8. —4x 3 (x 2 + 2x — 7) 9. (a — 5 )(a + 8)
10. (4x - l)(5x + 2) 11. (d + 2)(d 2 -3d - 10) 12. (2b - 1)(3 b 2 + 5b + 4)
10.3
Special Products of Polynomials
Examples on
pp. 581-584
Use special product patterns to multiply some polynomials.
- b ) = sf- - b 1 (a + b ) 2 = sf- + lab + b 1
(3x + l)(3x - 7) = (3x) 2 - l 2 (5 1 + 4) 2 = (5 1) 2 + 2(50(4) + 4 2
= 9x 2 - 49 = 251 2 + 40? + 16
(a + b)(a
In Exercises 13-16, find the product.
13. (x + 15)(x - 15) 14. (5x - 2)(5x + 2) 15. (x + 2) 2 16. (7m - 6) 2
17, Write two expressions for the area of the figure
at the right. Describe the special product pattern
that is represented.
10.4 Solving Quadratic Equations in Factored Form
Examples on
pp. 588-590
Solve the equation (x + l)(x — 5) = 0.
Solution
(x + l)(v — 5) = 0 Write original equation.
x+l=0 or x — 5 = 0 Use the zero product property.
x = — 1 I x = 5 Solve for x.
ANSWER ► The solutions are — 1 and 5. Check these in the original equation.
20. (y - 7) 2 = 0
23. n(n + 9)(n - 12) = 0
26. 2c(4c + 3) 2 = 0
H ~~
Solve the equation.
18. (x + l)(x + 10) = 0 19. (x — 3)(x — 2) = 0
21.b(5b - 3) = 1 22 . 6(5 a - 1)(3 a + 1) = 0
24. (c + 5)(2c - l)(3c + 2) = 0 25. (3x + l)(x - 4) 2 = 0
Chapter Summary and Review continue of
Factoring x 2 4- bx +- c
Examples on
pp. 595-598
Factor x 2 — 6x + 8.
The first term of each binomial factor is x. For this trinomial, b = — 6 and c = 8.
Because c is positive, you need to find numbers p and q with the same sign. Find
numbers p and q whose sum is —6 and whose product is 8.
p and q
p + q
-1,-8
-9
^1"
1
<N
1
-6
The numbers you need are —2 and —4.
ANSWER ^ x 2 — 6x + 8 = (x — 2)(x — 4). Check your answer by multiplying.
Factor the trinomial.
27.x 2 + lOx + 24 28. a 2 — 6a — 16 29. m 2 — 8 m — 20
Solve the equation by factoring.
30. b 2 - lib + 28 = 0 31. y 2 + 4y — 32 = 0 32. a 2 - 6a - 40 = 0
Factoring ax 2 4-bx4-c
Examples on
pp. 603-605
Factor 3x 2 + 5x — 2.
For this trinomial, a = 3, b = 5 and c = — 2. Because c is negative, you need to
find numbers p and q with different signs.
Q Write the numbers m and n whose
product is 3 and the numbers p
and q whose product is —2.
0 Use these numbers to write trial
factors. Then use the Outer and
Inner products of FOIL to
check the middle term.
mand n
1,3
Trial Factors
(x - l)(3x + 2)
(x + l)(3x - 2)
(x + 2)(3x - 1)
p and q
- 1,2
1.-2
Middle Term
2x — 3x = —x
—2x + 3x = x
—x + 6x = 5x
ANSWER ► 3x 2 + 5x — 2 = (x + 2)(3x - 1).
Factor the trinomial.
33. 12x 2 + lx + 1 34. 3x 2 - 8x + 4 35. Ar 2 + 5r - 6 36. 5c 2 - 33c - 14
Solve the equation by factoring.
37. 2 p 2 ~ p — 1 = 0 38. 4x 2 — 3x — 1 = 0 39. 2 a 2 + la = 4
Chapter Summary and Review
Chapter Summary and Review continued
Factoring Special Products
Examples on
pp. 609-612
Factor using the special product patterns to solve the equations.
a 2 - b 2 = (a + b)(a - b)
x 2 _ 64 = o
x 2 - 8 2 = 0
(x + 8)(x - 8) = 0
x + 8 = 0 or x — 8 = 0
x = — 8
x = 8
ANSWER ^ The solutions are —8 and 8.
a 2 - lab + b* = (a - b) 2
x 2 — 4x + 4 = 0
x 2 - 2(x)(2) + 2 2 = 0
(x - 2) 2 = 0
x — 2 = 0
x = 2
ANSWER ► The solution is 2.
Use factoring to solve the equation.
40. b 2 - 49 = 0 41. 16a 2 -1=0
43. m 2 - 100 = 0 44. 4b 2 - 12b + 9 = 0
42. 9 d 2 — 6d + 1 = 0
45. 25x 2 + 20x + 4 = 0
Factoring Cubic Polynomials
Examples on
pp. 616-619
Factor using the distributive property or the special product patterns.
a 3 + b 3 = (a + b)(a 2 - ab + A 2 )
X 3 + 125 = X 3 + 5 3
= (x + 5XX 2 - 5x + 25)
a 3 - 6 s = (a - AHa 2 + ab+
c 3 - 216 = c 3 - 6 3
= (c — 6 )(c 2 + 6c + 36)
Factor by Grouping
- 4x 2 - 4x + 16
= (x 3 - 4x 2 ) + (—4x + 16)
= x 2 (x — 4) + (—4)(x — 4)
= (x — 4)(x 2 — 4)
= (x — 4)(x + 2)(x — 2)
Factor the expression completely.
46. -2X 3 + 6x 2 - 14x 47. 5v 4 - 20v 3 + 10v 2
49. 3v 3 - 4y 2 - 6v + 8 50. x 3 - 64
48. x 3 + 3x 2 — 4x — 12
51. 27 b 3 + 1
Solve the equation.
52.x 2 — 6x + 5 = 0 53. 2x 2 — 50 = 0 54. 8x 3 + 25x = 30x 2
Chapter 10 Polynomials and Factoring
u.
Iiapi^r
Chapter Test
Use a vertical format or a horizontal format to add or subtract.
1. (x 2 + 4x - 1) + (5x 2 + 2) 2 . (5 1 2 - 9t + 1) - (8? + 13)
3. ( In 3 + 2n 2 - n - 4) - (4n 3 - 3 n 2 + 8) 4. (x 4 + 6x 2 + 7) + (2x 4 - 3x 2 + 1)
Find the product.
5. (.x + 3)(2x + 3)
6. (3x - l)(5x + 1)
7. (w — 6)(4w 2 + w — 7)
8. (5 1 + 2)(4 1 2 + St-7)
9. (3z 3 - 5z 2 + 8)(z + 2)
10. (4x + l)(4x — 3)
11.(x- 12) 2
12. (7x + 2) 2
13. (8x + 3)(8x — 3)
Use the zero-product property to solve the equation.
14. (6x — 5)(x + 2) = 0
15. (x + 8) 2 = 0
16. (x + 3)(v — l)(3v + 2) =
Find the x-intercepts and the vertex of the graph of the function. Then
sketch the graph.
17.y = (x + l)(x — 5)
18. y = (x- 4)(x + 4)
1 9. y = (x + 2)(x + 6)
Solve the equation by factoring.
20.x 2 + 13x 4- 30 — 0
21.x 2 - 19x + 84 = 0
22. x 2 — 34x — 240 = 0
23. 2x 2 + 15x - 108 = 0
24. 9x 2 - 9x = 28
25. 18x 2 - 57x = -35
Factor the expression.
26.x 2 - 196
27. 16x 2 - 36
28. 128 - 50x 2
29.x 2 - 6x + 9
30. 4x 2 + 44x + 121
31. —6X 3 — 3X 2 + 45x
32. 9 1 2 - 54
33. x 3 + 2x 2 - 16x - 32
34. 2x 3 - 162x
Solve the equation by a
method of your choice.
35.x 2 - 60 = -11
36. 2x 2 + 15x - 8 = 0
37. x 2 - 13x = -40
38. x(x — 16) = 0
39. 12x 2 + 3x = 0
40. x 4 + 7X 3 — 8x — 56 = 0
41. 5x 3 — 605x = 0
42. 4x 3 + 24x 2 + 36x = 0
43. 16x 2 - 34x - 15 = 0
44. ROOM DIMENSIONS A room’s length is 3 feet less than twice its width.
The area of the room is 135 square feet. What are the room’s dimensions?
2
45. RUG SIZE A mg 4 meters by 5 meters covers — of the floor area in a room.
The mg touches two walls, leaving a strip of uniform width around the other
two walls. How wide is the strip?
Chapter Test
Chapter Standardized Test
Tip
<32^>C^>ClD
Some questions involve more than one step. Read each
question carefully to avoid missing preliminary steps.
1. Classify 3x 2 — 7 + 4x 3 — 5x by degree
and by the number of terms.
CD quadratic trinomial
CD cubic polynomial
CD quartic polynomial
CD quadratic polynomial
CD None of these
2. Which of the following is equal to
( -x 2 - 5x + 7) + {—lx 2 + 5x - 2)?
CD — 8x 2 + 5
Cp — 8x 2 + lOx + 5
CD 6x 2 + 5
Cp — 8x 2 — lOx + 5
3. Which of the following is equal to
(5x 3 + 3x 2 — v + 1) — (2x 3 + v — 5)?
CD lx 3 + 3x 2 — 2x + 6
Cp 3x 3 + 3x 2 — 2x — 4
Cp 3x 3 + 3x 2 — 2x — 6
Cp 3x 3 + 3x 2 — 2x + 6
4. Which of the following is equal to
(4x - l)(5x ~ 2)7
Cp 20x 2 — 5x + 2
CD 20x 2 - 13jc + 2
Cp 20x 2 — 3x — 2
Cp 20x 2 — 8x — 2
5. Which of the following is equal to
(2x - 9) 2 ?
CD 4x 2 + 81
CD 4x 2 - 18 jc + 81
Cp 4x 2 + 36x + 81
Cp 4x 2 — 36x + 81
6. Which of the following is one of the
solutions of the equation x 2 — 2x = 120?
CD -12 CD -10
CD 20 CD 60
7. Which of the following is a correct
factorization of — 45x 2 + 150x — 125?
CD — 5(3x + 5) 2
CD -5(3jc + 5)(3jc - 5)
CD — 5(3x - 5) 2
CD — 5(9x + 25)
8. Which of the following is equal to the
expression x 3 — 2x 2 — 1 lx + 22?
CD 0 - 2)(jc - 11) <3D(x~ 2)(x 2 + 11)
CD 0 - 2)(jc + 11) CD (x ~ 2)(x 2 - 11)
9. Which of the following is equal to x 3 + 64?
Cp x(x + 4)(x — 4)
CD (x + 4)(x 2 — 4x + 16)
CD (x ~ 4)(x 2 + 4x + 16)
Cp (x + 8)(x 2 — 8x + 16)
Chapter 10 Polynomials and Factoring
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
1 The Least Common Denominator
3 2 5
Write the numbers —, —, and — in order from least to greatest.
Solution The LCD of the fractions is 24.
3 _ 3 • 6 _ 18 2 _ 2 • 8 _ 16 5 _ 5 » 3 _ 15
4 4-6 24 3 3-8 24 8 8-3 24
5 2 3
Compare the numerators: 15 < 16< 18, so — < — <~.
5 2 3
ANSWER In order from least to greatest, the fractions are — , —, and —
L
Try These
Write the numbers in order from least to greatest.
1 .
I 2
4’ 5
5 A 1 XL
10’ 4’ 20
2 .
6 .
4 3
,15 1
3’ 6 ’ 2
3 i i
4’ 6 ’ 2
7’ 8
7 5 7
7 5. A
3’ 4’ 6
8 . 2 ± if.
5
8 ’ 4’ 24
6
2 Operations with Fractions
Add f+ f.
Solution
1 ,3 = 20 _9_
6 + 8 24 + 24
= 20 + 9
24
29 t 5
= 24- or >24
Rewrite fractions using the LCD.
Add numerators.
Simplify.
Student HeCp
► Extra Examples
More examples
anc j p ract j ce
exercises are available at
www.mcdougallittell.com
Try These
Add or subtract. Write the answer as a fraction or
mixed number in simplest form.
_7_ 1
10
14 L _|_ 11
9 12
’5. if + 3|
_5_
12
16. 2 ^ -
XL
20
Maintaining Skills
Rational Expressions
and Equations
How do scale models fit into the
design process?
Application: Scale Models
A floor plan is a smaller diagram of a room or
a building drawn as if seen from above. Two- and
three-dimensional scale models are used by architects,
builders, and city planners in the design process.
Think & Discuss
In the floor plan below, 1 inch represents 14 feet.
In the floor plan, Bedroom 2 is 1 inch by y inch.
1. What is the actual length of Bedroom 2?
2 . What is the actual width of Bedroom 2?
3 x
3. Solve the equation y = y^ to find the actual
length x of the whole floor of the house.
Learn More About It
You will write and use a proportion for a problem
about a scale model in Exercise 37 on page 637.
-• LJ
APPLICATION LINK More about scale models is available
at www.mcdougallittell.com
f
nipTtr
PREVIEW
PREPARE
STUDY TIP
Study Guide
What’s the chapter about?
Recognizing direct variation and inverse variation models
Simplifying, adding, subtracting, multiplying, and dividing rational expressions
Solving proportions and rational equations
r Key Words
• proportion, p. 633
• inverse variation, p. 639
• least common
• extremes, p. 633
• rational number, p. 646
denominator (LCD), p. 663
• means, p. 633
< _
• rational expression, p. 646
• rational equation, p. 670
_ >
Chapter Readiness Quiz
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer top. 132)
1. Which of the following are equivalent equations?
(A) y = x 2 + 3 CM) y = (x — 5)(x + 1)
x 2 + y = 3 y — x 2 = —4x — 5
CM) y = (2x + l)(x + 4) CM) y = (x — 5) 2
y = 2x 2 + 8x + 4 y = x 2 -25
Skill Check (refer to pp. 462, 605)
2. Simplify the expression 49x 2 '■
-lx
(A)
343x 3
CD ~21x
CD —21x 2
CD 21x
3. Solve the equation 4x 2 — lOx + 6 = 0 by factoring.
(A) x = - 1 CD * = 3
CD x = -3 CD x = 1
Preview and Review
Before studying the chapter, list
what you know about each topic.
After studying the chapter, go back
to each topic and list what you know
about each topic. Compare the two
sets of notes and see what you
have learned.
&
Chapter 11 Preview
Proportions-, deal with fractions, ratios
Direct and Inverse Variation:
direct variation equation, y = kx
graph of direct variation is linear
Chapter 11 Rational Expressions and Equations
Proportions
Goal
Solve proportions.
Key Words
• ratio
• proportion
• extremes
• means
• reciprocal property
• cross product property
• cross multiplying
How many clay warriors were buried in the tomb?
Many real-life quantities are
proportional to each other.
In Example 5 you will use
a proportion to estimate the
number of clay warriors
buried in Emperor Qin Shi
Huang’s tomb.
Student HeCp
► Reading Algebra
The proportion p ) = ^
is read as “a is to b as
c is to d"
\ _ >
An equation that states that two ratios are equal is a proportion,
^ where a , b, c, d A 0.
When the ratios are written in this order, a and d are the extremes of the
proportion and b and c are the means of the proportion. If two nonzero numbers
are equal, then their reciprocals are equal. This property carries over to ratios.
RECIPROCAL PROPERTY OF PROPORTIONS
If two ratios are equal, then their reciprocals are also equal.
■ t a c .. b d „ .24 3 6
lf ~b = d' then a = c- Example: 3 = 6 2 = 4
Student HeCp
►Vocabulary Tip
Solving for a variable
in a proportion is
called solving the
proportion.
V _ J
Use the Reciprocal Property
Solve the proportion ^ = ~ using the reciprocal property.
Q Write the original proportion.
© Use the reciprocal property.
© Multiply each side of the equation by 60 24 = x
to clear the equation of fractions.
ANSWER ► The solution is x = 24. Check this in the original equation.
5 = 60
2 x
2 _ x
5 ” 60
CROSS PRODUCT PROPERTY By writing both fractions in the proportion
a_
b
c ad be
= over a common denominator bd, the proportion becomes ^ This
observation is the basis for the cross product property, shown on the next page.
11.1 Proportions
CROSS PRODUCT PROPERTY OF PROPORTIONS
The product of the extremes equals the product of the means.
If then ad = be. Example: ^ ^ 2 • 6 = 3 • 4
2 Use the Cross Product Property
3 5
Solve the proportion — = — using the cross product property.
3 5
Q Write the original proportion. — = ^
y °
© Use the cross product property. 3 • 8 = y • 5
© Simplify the equation. 24 = 5 y
24
© So/ve by dividing each side by 5. — = y
CHECK y Substituting fory, -jj becomes 3 • which simplifies to
T
Student HeCp
► Study Tip
Remember to check
your solution in the
original proportion.
Since Example 3 has
two solutions, you need
to check both of them,
v_ /
3 Use the Cross Product Property
3 x+ 1
Solve the proportion ^ ^ .
© Write the original proportion.
0 Use the cross product property.
© Multiply.
© Collect terms on one side.
© Factor the right-hand side.
© Solve the equation.
3 _ x + 1
x ~ 4
(3X4) = (x)(x + 1)
12 = x 2 + x
0 = x 2 + x
12
0 = (jc - 3)(jc + 4)
x = 3 or —4
ANSWER ^ The solutions are x = 3 and x = —4. Check both solutions.
Use the Cross Product Property
Solve the proportion. Check your solutions.
25
n
4
x + 6
Consider the equation — = - where p, q , r and s are polynomials and g and s are
restricted so that they do not equal zero. Writing both fractions with a common
ps qr
denominator leads to — = —, and then to ps = qr. This reasoning is the basis f
qs qs
cross multiplying , a method of solving equations used in Example 4.
Chapter 11 Rational Expressions and Equations
J j 4 Cross Multiply and Check Solutions
y 2 -9 y - 3
Solve the equation - + - = —-—.
Solution
y 2 -9 y - 3
y + 3 '
(j 2 - 9)2 = (>■ + 3 )(y
2y 2 - 18 = y 2 - 9
O Write the original equation.
0 Cross multiply.
0 Multiply.
0 Isolate the variable term.
0 Solve by taking the square root of each side. y = ±3
The solutions appear to be y = 3 and y = — 3. However, you must discard
y = — 3, since the denominator of the left-hand side would become zero.
ANSWER ^ The solution is y = 3. Check this in the original equation.
3 )
y 2 = 9
Link
ArcfiaeoCogy
CLAY WARRIORS In 1974,
archaeologists excavated the
tomb of Emperor Qin Shi
Huang (259-210 B.C.) in China.
Buried close to the tomb was
an entire army of life-sized
clay warriors.
More about this
r excavation at
www.mcdougallittell.com
EXCLUDE ZERO DENOMINATORS Because division by zero is undefined, when
dealing with proportions, you must check your answer to make sure that any
values of a variable that result in a zero denominator are excluded from the final
answer, as shown in Example 4.
5 Write and Use a Proportion
CLAY WARRIORS Pit 1 of the tomb of Emperor Qin Shi Huang, shown below,
consists of two end sites, containing a total of 450 warriors, and a central
region. The site (shown in red) in the central region contains 282 warriors.
This 10-meter-wide site is thought to be representative of the 200-meter central
region. Estimate the total number of warriors in Pit 1.
Not drawn to scale
Solution Let n represent the number of warriors in the 200-meter central
region. You can find the value of n by solving a proportion.
Number of warriors found _ Number of meters excavated
Total number of warriors Total number of meters
282 = 10
n 200
ANSWER ^ The solution is n = 5640, indicating that there are about 5640
warriors in the central region. With the 450 warriors at the ends,
that makes a total of about 6090 warriors in Pit 1.
11.1 Proportions
I Exercises
Guided Practice
Vocabulary Check
1. Identify the extremes and the means of the proportion.
a.
_9_
12
b.
12
3
4*
Skill Check Solve the proportion. Check your solution.
2 _ — =
16
40
-72 x
96 4
2
1
x + 1
6 .
2x + 1
2
3
Determine whether the equation follows from ^
S. ad — be 9. ba = dc 10- ■§ = * 11. — = —
d c a c
Practice and Applications
RECIPROCAL PROPERTY Solve the proportion using the reciprocal
property. Check your solution.
N
X |U>
II
13 — = —
1J ' 4 3c
II
CROSS PRODUCT PROPERTY Solve the proportion using the cross
product property. Check your solution.
15 — = —
8 56
x _ 7
3 3
17. —
4 z
42 3
18.^ = -
28 x
5 8
19. - = Q
4 7
20. x- = ^
2w 3
21 — = —
3d 3
__ 14 7b
22- -y = ~2
oo 3 1
23 ' 10 10 a
P Student HeCp
Homework Help
Example 1: Exs. 12-14
Example 2: Exs. 15-23
Example 3: Exs. 24-35
Example 4: Exs. 24-35
Example 5: Ex. 36
x. _ j
CHECKING SOLUTIONS Solve the equation. Check your solutions.
27 x ~ 2 = x+ 10
4 10
t ~ 1
t
33.
-2
a-1
a_
5
28.
r + 4
3
L
5
5
x + 3
x
x + 6
x 4- 6 _ x — 5
7
- 3
3
x
35.
9-x
x + 4
2x
Chapter 11 Rational Expressions and Equations
36. CLAY POTS Assume that a 15-meter-wide site is representative of a larger
60-meter-wide site. If an archaeologist excavates the 15-meter-wide site and
finds 30 clay pots, estimate the number of clay pots in the larger 60-meter¬
wide site. Assume that both sites are the same length.
Lin
History
JOHN WESLEY DOBBS
(1882-1961) was a prominent
community and civil rights
leader in Atlanta, Georgia.
The sculpture pictured above,
entitled "Through His Eyes,"
is a memorial to Dobbs.
J Scale Models
You want to make a scale model of one of the clay horses found in Emperor
Qin Shi Huang’s tomb. The clay horse is 1.5 meters tall and 2 meters long. Your
scale model will be 18 inches long. How tall should it be?
Solution Let h represent the height of the model.
Height of actual statue Height of model
Length of actual statue
1.5
2
Length of model
h
18
(1.5)(18) = 2 h
27 = 2 h
13.5 = h
Write verbal model.
Write proportion.
Use cross product property.
Multiply.
Divide by 2.
,1 •
ANSWER ► Your scale model should be 13— inches tall.
37. H istory Link / The ratio of the sculpture of John Wesley Dobbs’ head to
actual size is about 10 to 1. Suppose that his head was 9 inches high and
6^- inches wide. Estimate the height and width of the sculpture. Write the
answer in feet.
MURAL PROJECT In Exercises 38 and 39, use the following information.
Refer to the example above if necessary.
Art is the Heart of the City is a
fence mural project in Charlotte,
North Carolina. Artists Cordelia
Williams and Paul Rousso along
with 22 high school students
created drawings of the mural.
Then slides of the drawings were
made and projected to fit onto
4-foot-wide by 8-foot-long sheets
of plywood used for the fence
panels. Students traced and later
painted the enlarged images.
38. If the paper used for the original drawings was 11 inches wide, how long did
it need to be?
39. Suppose the height of a flower on the panel shown is 2j feet. Use
Exercise 38 to find the height of the flower in the student’s drawing.
11.1 Proportions
Standardized Test
Practice
Mixed Review
Maintaining Skills
40. CHALLENGE A scale model uses a scale of yy inch to represent 1 foot.
Explain how you can use a proportion and the cross product property to show
that a scale of tz in. to 1 ft is the same as a scale of 1 in. to 192 in.
16
1 x
41. MULTIPLE CHOICE What are the extremes of the proportion — = yy?
X 1
What are the extremes of — = —?
to J
(3)1,3;*, 18 (D x, 18; 1,3 CD x, 3; 1, 18 CD 1,18;*, 3
42. MULTIPLE CHOICE Solve = ^4
x + 5 x + 2
CD 1 CD —2 and —5 CED 2 and 5 CD No solution
43. MULTIPLE CHOICE Solve —^-7 = —^ 777 .
c - 4 c - 10
(A) 0 CD 2 and 16 CD — 18 and 32 CD No solution
POINT-SLOPE FORM Write in point-slope form the equation of the line
that passes through the given point and has the given slope. (Lesson 5.2)
44. (-1, -2),m = 2 45. (5,-3), m = -4 46. (-8, 8), m = - 1
STANDARD FORM Write in standard form the equation of the line that
passes through the given point and has the given slope. (Lesson 5.4)
47. (10, 6), m = —2 48. (-7, -7), m = y 49. (1, 8), m = |
50. (0,5), m = 3 51.(6, 12), m= -12 52. (6,-1), m = 0
FINDING SQUARE ROOTS Evaluate the expression. Check the results by
squaring the answer. (Lesson 9. V
53. V64 54. -V9 55. V10,000 56. ±Vl69
SIMPLIFYING RADICALS Simplify the radical expression. (Lesson 9.3)
57. Vl8 58. V20 59. V80 60. Vl62
61.9V36 62. 63. yV28 64 ‘/f
65. FRACTIONS, DECIMALS, AND PERCENTS Copy and complete the
table. Write the fractions in simplest form. (Skills Review pp. 767-769)
Decimal
?
0.2
?
0.073
?
?
Percent
78%
?
?
?
3%
?
Fraction
?
?
2
3
?
?
12
25
Chapter 11 Rational Expressions and Equations
Direct and Inverse Variation
Goal
Use direct and inverse
variation.
Key Words
• direct variation
• inverse variation
• constant of variation
How are banking angle and turning radius related?
In Lesson 4.6 you studied direct
variation. In Example 4 you will use
a different kind of variation to relate
the banking angle of a bicycle to its
turning radius.
In this lesson you will review direct variation and learn about inverse variation,
where the product of two variables is a constant.
Student HeCp
->
► Study Tip
Direct and inverse
variation are
sometimes called
direct and inverse
proportions.
I J
MODELS FOR DIRECT AND INVERSE VARIATION
Direct Variation
Uy
The variables x and y vary directly if for a
/ y—kx
constant k
X *>o
/
— = k, or y = kx, where k ¥= 0.
X
X
Inverse Variation r
4
The variables x and y vary inversely if for a
II
*1
constant k
y k> o
k
xy = k, or y = where k ± 0
X
X
The number k is the constant of variation.
l , _____
J i Use Direct Variation
Find an equation that relates x and y such that x and y vary directly, and y — 4
when x — 2.
Solution
0 Write the direct variation model.
© Substitute 2 for x and 4 for y.
© Simplify the left-hand side.
y
ANSWER The direct variation that relates x and y is — = 2, or y = 2x.
2 = k
11.2 Direct and Inverse Variation
Iezebbi 2 Use Inverse Variation
Find an equation that relates x and y such that x and y vary inversely, and y = 4
when x = 2.
O Write the inverse variation model. xy = k
0 Substitute 2 for x and 4 fory. (2)(4) = k
© Simplify the left-hand side. 8 = k
g
ANSWER ► The inverse variation that relates x and y is xy = 8, or y =
Student UeCp
■ ^
► Study Tip
Direct and inverse
variation models
represent functions
because for each
value of xthere is
exactly one value of y.
For inverse variation,
the domain excludes 0.
V _>
Student HeCp
^
►Vocabulary Tip
A hyperbola is a curve
with two branches.
You will learn more
about hyperbolas in
later math courses.
■afMUM Jl 3 Compare Direct and Inverse Variation
Compare the direct variation model and the inverse variation model you found
in Examples 1 and 2 using x = —4, —3, —2, — 1, 1, 2, 3, and 4.
a. numerically b. graphically
Solution
g
a. Use the models y = 2x and y = — to make a table.
y y x
x-value
-4
-3
-2
-1
1
2
3
4
Direct, y=2x
-8
-6
-4
-2
2
4
6
8
Inverse, y = -
7 X
-2
1
LU | 00
-4
-8
8
4
8
3
2
direct variation: Because k is positive, y increases as x increases.
As x increases by 1, y increases by 2.
inverse variation: Because k is positive, y decreases as x increases.
b. Use the table of values to graph each model.
direct variation: The graph for this model
is a line passing through the origin.
inverse variation: The graph for this
model is a hyperbola. Since neither x nor y
can equal 0, the graph does not intersect
either axis.
Compare Direct and Inverse Variation
1. Suppose y = 6 when x = 2. Find an equation that relates x and y such that:
a. x and y vary directly. b_ x and y vary inversely.
2_ Compare the direct and inverse variation models in Checkpoint 1
numerically and graphically using x = —4, —3, —2, — 1, 1, 2, 3, and 4.
grr
Chapter 11 Rational Expressions and Equations
Write and Use a Model
BICYCLE BANKING
ANGLE A bicyclist tips the
bicycle when making a turn.
The angle B of the bicycle
from the vertical direction is
called the banking angle.
BICYCLE BANKING ANGLE Assume that the graph below shows an inverse
relationship between the banking angle B and the turning radius r for a bicycle
traveling at a particular speed.
Bi
40
i
(A
03
03
h.
03
03
an
» ( 3 . 5 , 32 )
•D
JU
"5)
e
20
CO
e>
c
10
c
(C
GO
0
(
)
]
L
2
1
£
X
i
(
5 r
Turning radius (fee
t)
a. Find an inverse variation model that relates B and r.
b. Use the model to find the banking angle for a turning radius of 5 feet.
c. How does the banking angle change as the turning radius gets smaller?
Student HeCp
^ More Examples
More examples
are available at
www.mcdougallittell.com
Solution
a. From the graph, you can see that B = 32° when r = 3.5 feet.
Q Write the inverse variation model. B = —
r
© Substitute 32 for B and 3.5 for r. 32 =
© Multiply each side by 3.5. 112 = k
112
ANSWER The model is B = ——, where B is in degrees and r is in feet.
112
b. Substitute 5 for r in the model found in part (a). B = —- = 22.4°
c. As the turning radius gets smaller, the banking angle becomes greater.
From the graph, you can see that the increase in the banking angle is about
10° for a 1-foot decrease in banking angle from 4 to 3 feet, but the increase
in banking angle is only about 4° for a 1-foot decrease from 6 feet to 5 feet.
Write and Use a Model
Use the inverse variation model B = -.
r
3. What is the bicycle banking angle when the turning radius is 8 feet?
4. Does this model apply when r = 1? Explain.
11.2 Direct and Inverse Variation
aeg r jgs- dsr •
■IBa Exercises
Guided Practice
Vocabulary Check
1. What does it mean for two quantities to vary directly?
2 . What does it mean for two quantities to vary inversely?
Skill Check
Does the graph model direct variation , inverse variation , or neither ? Explain.
Does the equation model direct variation, inverse variation , or neither ?
6. x = y 7. y = lx — 2 8. x = 1 2y 9. xy = 9
Suppose y = 6 when x = 4. For the given type of variation, find an
equation that relates x and y.
10 .x and y vary directly. 11 .x and y vary inversely.
Practice and Applications
DIRECT VARIATION EQUATIONS The variables x and y vary directly. Use
the given values to write an equation that relates x and y.
12. x = 3, y = 9 13. x = 2, y = 8 14. x = 18, y = 6
15. x = 8, y = 24 16. x = 36, y = 12 17. x = 27, y = 3
Student HeCp
►Homework Help
Example 1: Exs. 12-17
Example 2: Exs. 18-26
Example 3: Exs. 27-34
Example 4: Exs. 35-42
INVERSE VARIATION EQUATIONS The variables x and y vary inversely.
Use the given values to write an equation that relates x and y.
18. x = 2, y = 5 19. x = 3, y = 1 20. x = 16, y = 1
21. x = 11, y = 2 22.x = ^,y = 8 23. x = 5 ,y = -j~
24. x = 1.5, y = 50 25. x = 45, y = 0.6 26. x = 10.5, y = 1
DIRECT OR INVERSE VARIATION Make a table of values for x = -4, -3,
-2, -1, 1, 2, 3, and 4. Use the table to sketch the graph. State whether x
and y vary directly or inversely.
27. y = — 28. y = 29. y = 3x 30. y = —
y x y 2 y y x
Chapter 11 Rational Expressions and Equations
Student HeCp
► Homework Help
Extra help with
w* problem solving in
Exs. 31-33 is available at
www.mcdougallittell.com
J
Link to
Showsfioes
SNOWSHOES distribute a
person's weight over a large
area, allowing a person to
walk over deep snow without
sinking. Native Americans
were among the first people
to use snowshoes.
VARIATION MODELS IN CONTEXT In Exercises 31-33, state whether the
variables model direct variation , inverse variation , or neither.
31 - BASE AND HEIGHT The area B of the base and the height h of a prism with
a volume of 10 cubic units are related by the equation Bh = 10.
32. MASS AND VOLUME The mass m and the volume V of a substance are
related by the equation 2V = m , where 2 is the density of the substance.
33. HOURS AND PAY RATE The number of hours h that you must work to earn
$480 and your hourly rate of pay p are related by the equation ph = 480.
34. MODELING WITH GRAPHS Which graph models direct variation where the
constant of variation is 3?
5
1
-
1 ,
r 1
3 x
SNOWSHOES In Exercises 35-37, use the following information.
When a person walks, the pressure on each boot sole varies inversely with the
area of the sole. Denise is walking through deep snow, wearing boots that have a
sole area of 29 square inches each. The pressure on the sole is 4 pounds per
square inch when she stands on one foot.
35. Use unit analysis to explain why the constant of variation is Denise’s weight.
How much does she weigh? /
36. Using the constant of variation from Exercise 35, write an equation that
relates area of the sole A and pressure P.
37. If Denise wears snowshoes, each with an area of 319 square inches, what is
the pressure on the snowshoe when she stands on one foot?
OCEAN TEMPERATURES In Exercises 38 and 39, use the graph and
the following information.
The graph at the right shows water
temperatures for part of the Pacific
Ocean. At depths greater than
900 meters, the temperature of ocean
water (in degrees Celsius) varies
inversely with depth (in meters).
38. Find a model that relates
the temperature T and the
depth d.
39. Find the temperature at a depth
of 2000 meters. Round to the
nearest tenth.
Pacific Ocean Temperatures
L
_ f
370
0,12)
I
l
0 1000 2000 3000 4000 5000 d
Depth (meters)
11.2 Direct and Inverse Variation
Standardized Test
Practice
Mixed Review
Maintaining Skills
CHALLENGE You are taking a trip on a highway in a car that gets a gas
mileage of 26 miles per gallon for highway driving. You start with a full
tank of 12 gallons of gasoline.
40. Find your rate of gas consumption (gallons of gas used to drive 1 mile).
41. Use your results from Exercise 40 to write an equation relating the number of
gallons of gas g in your tank and the number of miles m you have driven.
42. Do the variables g and m vary directly , inversely , or neither ? Explain.
43. MULTIPLE CHOICE Assuming y — 14 when x = 6, find an equation that
relates x and y such that x and y vary directly.
(a) xy = 84 CD y = \x <D y = jx CD *y = \
44. MULTIPLE CHOICE Assuming y — 9 when x = 10, find an equation that
relates x and y such that x and y vary inversely.
9 10 9
CD xy = 90 <D y = "iq* (E) y = y x ® x;y = To
USING PERCENTS Evaluate. (Lesson 3.9)
45. 45% of 10 46. 30% of 42
48. 150% of 300 49. 11% of 50
47. ±% of 200
50. 99% of 10,000
CHECKING SOLUTIONS Decide whether the ordered pair is a solution of
the inequality. (Lesson 9.8)
51. y < x 2 + 6x+ 12; (-1,4) 52.y<x 2 - 7x + 9; (-1,2)
53. _y > x 2 - 25; (5, 5)
54. _y > x 2 — 2x + 5; (1, —7)
FACTORING EXPRESSIONS Completely factor the expression.
(Lesson 10.6)
55. x 2 + 5x — 14 56. 7X 2 + 8 x + 1 57. 5x 2 — 51x +54
58. 36X 3 - 9x 59. 15x 4 - 50x 3 - 40x 2 60. 6 x 2 + 16x
61. POPULATION The population P of Texas (in thousands), as projected
through 2025, is modeled by P = 18,870(1.0124)*, where t = 0 represents
1995. Find the ratio of the population in 2025 to the population in 2000.
► Source: U.S. Bureau of the Census (Lesson 8.3)
SUBTRACTING FRACTIONS Subtract. Write the answer as a whole
number, fraction, or mixed number in simplest form. (Skills Review p. 765)
62 --S
7
8
63 - f
<*-'!
4
3
66 . f
1
2
67 'f
-4
68 . 12| - y
69.it
-4
Chapter 11 Rational Expressions and Equations
USING A GRAPHING CALCULATOR
IIIHHiBIWMi l
For use with
Lesson 11.2
20
1.06771
19
1.11276
18
1.17583
17
1.24341
16
1.32450
15
1.40559
14
1.51371
13
1.62183
12
1.74347
11
1.90566
Student HeCp
► Keystroke Help
See keystrokes for
several models of
graphing calculators at
www.mcdougallittell.com
Use a graphing calculator to develop an inverse or direct variation model.
Sampl*
During a chemistry experiment, the volume of a fixed mass of air was decreased
and the pressure at different volumes was recorded. The data are shown at the
left, where x is the volume (in cubic centimeters) and y is the pressure (in
atmospheres). Use a graphing calculator to determine if a direct or an inverse
variation model is appropriate. Then make a scatter plot to check your model.
Solution
Since y increases as x decreases, direct variation can be ruled out.
Let L x represent the volume x and L 2 represent the pressure y. Use the Stat
Edit feature to enter the ordered pairs from the table. Then create lists L 3 and
and
L, using
Notice that the values in L 3 are all
different. However, the values in L A
are all about 21.1. Thus, x and y can
be modeled using inverse variation.
r~
L 3
1.0677
.05339
21.354
1.1128
.05857
21.142
1.1758
.06532
21.165
1 .2434
.07314
21.138
1 .3245
.08278
21 .192 |
l_4= (21 .
. 3542,21 . . . .
© Choose the constant of variation k using the List Math feature to calculate
mea n ( L 4 ). Use the rounded value, 21.12, to write an inverse variation model
in the form y = —
y x
Q Set the viewing rectangle so that
11 < x < 20 and 1 < y < 2. Use the
Stat Plot feature to make a scatter
plot of Lj and L r Then graph
21.12
y = -
on the same screen.
TtyThts*
Decide whether the data might vary directly or inversely. Then choose the
constant of variation and write a model for the data.
1. (10, 8.25), (9, 7.425), (8, 6.6), (7, 5.775), (6, 4.95), (5, 4.125), (4, 3.3)
2 . (18, 1.389), (17, 1.471), (16, 1.563), (15, 1.667), (14, 1.786), (13, 1.923)
11.2 Using a Graphing Calculator
Goal
Simplify rational
expressions.
Simplifying Rational
Expressions
What is the air pressure at 36,000 feet?
Key Words
• rational number
• rational expression
• simplest form of a
rational expression
In Exercise 47 you will
simplify an expression that
models the relationship
between air pressure and
altitude. Then you will use the
simplified expression to
determine the air pressure on
Breitling Orbiter 3 , the first
manned balloon to circle Earth.
A rational number is a number that can be written as the quotient of two
14 7
integers, such as —, —, and A fraction whose numerator and denominator are
nonzero polynomials is a rational expression. Here are some examples.
3 2x 3x + 1 2x 2 + x - 2
x + 4 x 2 - 9 x 2 + 1
Simplifying rational expressions is similar to simplifying fractions because the
variables in a rational expression represent real numbers. To simplify a rational
expression, we factor the numerator and denominator and then divide out any
common factors. (Exercise 48, page 650, shows the reasoning used.) A rational
expression is in simplest form if its numerator and denominator have no factors
in common other than ±1.
SIMPLIFYING RATIONAL EXPRESSIONS
Let a, b r and c be nonzero polynomials.
ac _ a • £ _ _a
be b • jef b
1 Simplify Rational Expressions
Simplify the rational expression if possible.
I4x _ 2 • x _ ^ . 6x _ 2 _ 2
7 X ~ lX 9X 2 ~ X-3 -X- x~ 3x
Chapter 11 Rational Expressions and Equations
2 Write in Simplest Form
Simplify the expression if possible.
Student Hedp
► Study Tip
When you simplify
rational expressions,
you can divide out only
factors , not terms.
For example,
4 . , . x + 4
-= 4, but-
cannot be simplified.
2x
a ' 2(x + 5)
Solution
2x _ Z_ • x
2(x + 5) ~ Z‘ (x + 5)
x + 5
x(x 2 + 6) _ /« (i 2 + 6)
b. 2
X X
= X 2 + 6
X
X + 4
c. -
X
x(x 2 + 6) X +4
x 2 x
Divide out the common factor 2.
Simplify.
Divide out the common factor x.
Simplify.
Already in simplest form.
Write in Simplest Form
Simplify the expression. If not possible, write already in simplest form.
3X 3 _ 3 m _ * 2 (* + 3)
6x 2
3m
3 (m - 4)
n + 5
Student Hedp
► More Examples
^ ore exam P' es
are available at
www.mcdougallittell.com
3 factor Numerator and Denominator
2x 2 — f)x
Simplify —
Solution
0 Write the original expression.
Q Factor the numerator and denominator.
© Divide out the common factors 2 and x.
0 Simplify the expression.
2x 2 - 6x
6x 2
2x(x - 3)
2 • 3 • x • x
Mix - 3)
• 3 • x
x - 3
3x
Factor Numerator and Denominator
Simplify the expression.
5.
2x — 6
4
6 .
5x
10x 2 — 5x
7.
4m 3
2m 3 + 8m 2
11.3 Simplifying Rational Expressions
Student HeQp
► Study Tip
Rational expressions in
3 — b
the form t -are
b - a
equal to -1 because
of the following.
a - b _ ~(~3 + A _
b - a b - a
-Sb^ai
— r —= -1
| Recognize Opposite Factors
4 -x 2
Simplify >^ 2 -
Solution
© Write the original expression.
0 Factor the numerator and denominator.
© Factor — 1 from (2 — x).
© Divide out the common factor (x — 2).
© Simplify the expression.
4 — x 2
X 2 -
x - 2
(2-
x)(2 + x)
(x -
2)(x + 1)
-(x
- 2)(2 +x)
(x -
- 2)(x + 1)
— Cx^=^2f(x + 2)
U^2J(x + 1)
x + 2
x + 1
Recognize Opposite Factors
Simplify the expression.
3(4 -m)
3(m — 4)
2x — 5
10 .
12 .
13.
3 — x
x 2 - 9
y 2 + 3y- 28
11 .
14.
20 8x 16 — y 2
5 Divide a Polynomial by a Binomial
4(1 — in)
m 2 — 2m + 1
10x~5
1 — 2x
Divide (x 2 — 2x — 3) by (x — 3).
Solution
x 2 - 2x - 3
0 Rewrite the problem as a rational expression. x - 3
© Factor the numerator.
(.x - 3)(x + 1)
x — 3
© Divide out the common factor (x — 3).
© Simplify the expression.
fr^5)(jc + 1)
v + 1
Divide a Polynomial by a Binomial
Find the quotient.
15. Divide (x 2 — 4) by (x + 2).
17. (m 2 — 4m + 3) -r- (m — 1)
Chapter 11 Rational Expressions and Equations
16. Divide (2rc 2 - 8rc + 8) by (n - 2).
18. (x 2 — 2x — 8) -5- (jc — 4)
1WM Exercises
Guided Practice
Vocabulary Check
1. Define rational number. Which of the following are rational numbers?
2 _1 -, V3, 1.45, 0, K
5,
3’
2 . Define rational expression. Give an example of a rational expression.
3. Define the simplest form of a rational expression. Give an example of a
rational expression in simplest form.
Skill Check Simplify the expression. If not possible, write already in simplest form.
28 y
16
128c
10 .
4
18
8.
t\t + 2)
2x + 4
11 .
y ~ y
Find the quotient.
13. Divide (3 y 2 + 22 y + 7) by (y + 7).
14. Divide (* 2 + 5* + 6) by (x + 3).
15. Divide (2x 2 — 5* — 7) by (2* — 7).
6 .
12r 2
6*
8n 3
12 .
12n 4 + 40n 2
1 — m
m 2 — 49
Practice and Applications
Student HeCp
► Homework Help
Example 1: Exs. 16-21
Example 2: Exs. 22-24
Example 3: Exs. 25-42
Example 4: Exs. 25-42
Example 5: Exs. 43-46
SIMPLIFYING EXPRESSIONS Simplify the expression. If not possible,
write already in simplest form.
— 18v 2
16.
19.
22 .
4x
20
14x 2
5 Ox 4
x 14
17.
20 .
23.
45x
15
10x 5
16v 3
t\t + 2)
18.
21 .
24.
12x
36x
27x
10(r ~ 6)
lOr
FACTORING AND SIMPLIFYING Simplify the expression. If not possible,
write already in simplest form.
25.
28.
31.
lx
26.
3x 2 — 18*
27.
12x + x 2
-9x 2
x 2 + 25
29.
2(5 - d)
30.
2x + 10
2 (d - 5)
x 2 + * — 20
32.
x 3 + 9x 2 + 14x
33.
x 2 + 2x — 15
x 2 - 4
42x — 6x 3
36*
x 2 + 8* + 16
3* + 12
3
XT ~ X
x 3 + 5x 2 — 6x
11.3 Simplifying Rational Expressions
BALLOONING On March 20,
1999, Dr. Bertrand Piccard
(pictured above) and Brian
Jones became the first
balloonists to circle the globe
nonstop. The 29,000 mile trip
at an altitude of 36,000 feet
took them 19 days, 21 hours,
and 55 minutes.
Standardized Test
Practice
SIMPLIFYING EXPRESSIONS Simplify the expression if possible.
34.
Os
1
(N
X
35.
2x 2 + 1 lx — 6
36.
121 - x 2
x 2 — 5x — 6
x + 6
x 2 + 15x + 44
37.
1 - X
38.
12 - 5x
39.
1
(N
oo
x 2 — X
1 Ox 2 - 24.V
14y 2 - 1 6 v 3
40.
5 — x
41.
9 - 2y
42.
3x — 5
x 2 - 8x + 15
2 y 1 - 3y - 27
25 - 30x + 9x
DIVIDING POLYNOMIALS Find the quotient.
43. Divide ( a 2 — 3a + 2) by ( a — 1). 44. Divide (5 g 2 + 13 g — 6) by (g + 3).
45. Divide (.x 2 — 6x — 16) by (x + 2). 46. Divide (—5m 2 + 25m) by 5m.
47. Science The air pressure at sea
level is about 14.7 pounds per square
inch. As the altitude increases, the air
pressure decreases. For altitudes between
0 and 60,000 feet, a model that relates
air pressure to altitude is
P =
2952x - 44x 2
200x + 5x 2
where P is
measured in pounds per square inch
and x is measured in thousands of feet.
Simplify this rational expression. Suppose
you are in Breitling Orbiter 3 at 36,000
feet. What is the pressure at that altitude?
Air Pressure
p
0 20 40 x
Altitude
(thousands of ft)
48. LOGICAL REASONING Copy and complete the proof to show why you can
divide out common factors.
statement
ac __ a
be ~~ b
P 5 i,
—b
9 =-
— b
Explanation
Apply the rule for Multiplying rational expressions.
Any nonzero number divided by itself is 1.
Any nonzero number multiplied by 1 is itself.
49. MULTIPLE CHOICE Simplify the expression
6 + 2x
x 2 + 5x + 6 ’
(A)
x + 2
QD
x + 3
CD
x + 5
CD
2x
x 2 + 5x
50. MULTIPLE CHOICE Simplify the expression
1 ^1 ^ -1
3 — x
x 2 — 5x + 6
CjD
x + 2
CD
x — 2
CH)
x + 2
x — 2
Chapter 11 Rational Expressions and Equations
Mixed Review PRODUCTS AND QUOTIENTS Simplify. (Lessons 2.5, 2.8)
Maintaining Skills
51.
■m
52. ( 15)( ~
2 14
53. — h ——
7 24
54. J - (-36)
57. • 6m 2
55.
58.
36
45a
■m
~9a
56.
59. 18c
(-5 TO)
-27c
-4
60. Geometry Link/ The area
/
V
of the triangle is 192 square
5x /
meters. What is the value
4x \
of x? What is the perimeter?
/ r
\
(Lesson 9.2)
3x
3x
SKETCHING GRAPHS Sketch the graph of the function. (Lesson 9.4)
61. y = x 2 62. _y = 4 — x 2
64. y = 5x 2 + 4x - 5 65. y = 4x 2 - x + 6 66. y = ~3x 2 -x + 1
_ _ 1 2
63. y = /c
ADDING DECIMALS Find the sum. (Skills Review p. 759)
67. 0.987 + 1.4 68. 0.009 + 9 69. 75.6 + 35.8
70. 1.23 + 0.45
71. 0.01 + 0.01
72. 100.02 + 10
Quiz 7
Solve the proportion. Check your solutions. (Lesson 11.1)
. x _ 4 9 — — — 3 x _ 2_ 6x + 4 _ 2
1- 10 “ 5 Zm x~9 3 '4x-8“x 5 “x
The variables x and y vary directly. Use the given values to write an
equation that relates x and y. (Lesson 11.2)
5. x = 8, y = 32 6. x = 5, y = 3 7. x = 10, v = 15
The variables x and y vary inversely. Use the given values to write an
equation that relates xand y. (Lesson 11.2)
8. x = 12, y = 2 9. x = 4, y = 4 10. x = 3, y = 2.5
Simplify the expression if possible. (Lesson 11.3)
11 .
15x 2
lOx
12 .
x 2 — 7x + 12
x 2 + 3x — 18
13.
3 — x
x 2 + x — 12
14.
5x
1 lx + x 2
Find the quotient. (Lesson 11.3)
15. Divide (x 2 — 3x — 28) by (x — 7).
16. Divide (6x 2 + llx + 3) by (3x + 1).
11.3 Simplifying Rational Expressions
Multiplying and Dividing
Rational Expressions
Goal
Multiply and divide
rational expressions.
What is the ratio of two prairie dog populations?
Key Words
• rational expression
• reciprocal
• divisor
In Exercise 47 you will use
the rules for dividing rational
expressions to compare the growth
of two prairie dog populations.
Because the variables in a rational expression represent real numbers, the rules
for multiplying and dividing rational expressions are the same as the rules for
multiplying and dividing fractions that you learned in previous courses.
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS
Let a, b, c, and d be nonzero polynomials.
3 C 3C
to multiply, multiply numerators and denominators. ~b * ~d = ~bd
to divide, multiply by the reciprocal of the divisor. ^ -r- ^
B Student HeCp
>
► Study Tip
In Step 3, you do not
need to write the prime
factorizations of 24 and
60 if you recognize 12
as their greatest
common factor.
j
Bsai Multiply Rational Expressions
C . yr 3x 3 8x
Simphfy 4^--^.
3 x 3
Q Write the original expression. ^ 4
© Multiply the numerators
and denominators.
24jc 4
60x 5
© Factor and divide out
the common factors.
2 * 2*2 *2 * 2 * 2 * 2 * 2
2*2*2* 5 *X*2*2*2*x
© Simplify the expression.
2 _
5x
Chapter 11 Rational Expressions and Equations
Student HeGp
► Study Tip
In some cases it saves
work to simplify the
rational expressions
before multiplying,
k _ J
2 Multiply Rational Expressions
Simplify • — ~r -—-—-
3x 2 — 9x 2x 2 + x — 3
0 Write the original expression.
Q Factor the numerators and denominators.
© Multiply the numerators and denominators.
x x — 3
3x 2 — 9x 2x 2 + x — 3
x x — 3
3x{x - 3) # (jc - l)(2x + 3)
_ x(x — 3)_
3x(x - 3)(jc - l)(2x + 3)
© Divide out the common factors.
0 Simplify the expression.
__
3/&^3)(x ~ l)(2x + 3)
1
3(x - l)(2x + 3)
Multiply Rational Expressions
Write the product in simplest form.
„ ^ 4 >’ 2 „ 5x + 10 x 2 - 9
1 -•- 2 -•-
2y 2 6 x - 3 5
_ 4x 2 x 2 + 3x + 2
3.-T •-;-
(x + 2) 2 x 2 + 1
3 Multiply by a Polynomial
Simplify ———-• (x + 4).
F J x 2 + 5x + 4
Solution
lx
X 2 + 5x + 4
(x + 4) =
lx
X 2 + 5x + 4
x + 4
1
_ lx_ _ # x + 4
(x + l)(x + 4) 1
lx(x + 4)
(x + l)(x + 4)
lx{xjp^r)
{x + l)(x>4)
lx
X + 1
Write x + 4 as * + 4 .
Factor.
Multiply numerators and
denominators.
Divide out common factor.
Write in simplest form.
Multiply by a Polynomial
Write the product in simplest form.
4. • (2x + 2) 5. ^ + 4 • (x 2 + 2x) 6. (x - 3) •
11.4 Multiplying and Dividing Rational Expressions
Student HeCp
► More Examples
More examples
are available at
www.mcdougallittell.com
EsEfi
Simplify
9 4 Divide Rational Expressions
4 n . n — 9
n + 5 n + 5
O Write the original problem.
©Multiply by the reciprocal.
4 n n — 9
n + 5 + 5
4n n + 5
ft + 5 ft — 9
© Multiply the numerators and denominators.
4ft(ft + 5)
(ft + 5)(ft - 9)
© Divide out the common factor (n + 5).
© Simplify the expression.
4/?(ftjh-5)
(ftj>5)(ft — 9)
4 ft
ft — 9
Divide Rational Expressions
Write the quotient in simplest form.
4 3 g x + 3 2x + 6
7 " x + 2 " x + 2 4 " 3~~
g 3x ^ 6x 2
2x - 4 ' x - 2
Em 5 Divide by a Polynomial
Simplify h- (x — 3).
Solution
* 2 ~ 9 _ ox _ a 2 ~ 9 1
4-x 2 X 4x 2 x - 3
= (x + 3)(x - 3) . 1
4x 2 x ~ 3
_ (x + 3)(x — 3)
4x 2 (x - 3)
_ (x + 3)(x—"^”3)
4x 2 (x-=^T)
_ x + 3
4x 2
Multiply by reciprocal.
Factor.
Multiply numerators and
denominators.
Divide out common factor.
Write in simplest form.
Divide by a Polynomial
Write the quotient in simplest form.
x + 2
10.444 (2x + 2)
x + 2
11 .
1
-i- (x z + 2x)
r 2 - 4
12 . - (4jc - 8)
x + 2
Chapter 11 Rational Expressions and Equations
1L4 Exercises
Guided Practice
Vocabulary Check
Skill Check
In Exercises 1 and 2, complete the sentence.
1. To multiply rational expressions, multiply the ? and ? .
2 - To divide rational expressions, multiply by the ? of the ? .
Simplify the expression.
3x Ax 3
3.
6 .
8x 2 3x 4
3x
4.
x 2 - 1
2x
(x + 3) 7.
x
X
3x - 3
2x
x 2 - 2x - 15 ^ '' 8 - 2x ' 4 - x
9- ERROR ANALYSIS Find and correct the error.
5.
8.
x x - 5
x 2 - 25 x + 5
4x 2 - 25
4x
- (2x - 5)
Practice and Applications
Student He dp
p Homework Help
Example 1: Exs. 10-15
Example 2: Exs. 16-24
Example 3: Exs. 25-30
Example 4: Exs. 31-39
Example 5: Exs. 40-45
MULTIPLYING RATIONAL EXPRESSIONS Write the product in simplest
form.
10 .
13.
16.
19.
22 .
4x 1
3 * x
6x '
14 * 5x 5
_3x_ x — 6
x 2 — 2x — 24 6x 2
3 a a 2 + 5a + 4
a + 4 a 2 + a
45x 3 - 9x 2 2
11
14.
9 x 2 8
4 * 18x
12 .
Id 2 12 d 2
6 d 2d
6(x — 5)
17.
20 .
23.
y 4/
15.
-3
x - 4
16 * y 2
x — 4 *
1
<N
z 2 + 8z + 7 z 2
18.
5 — 2x
24
10z z 2 - 49
6
# 10 - 4x
3x 2 — 6x ' 4x + 2
21.
X
x 2 - 3x + 2
2x + 1 x - 2
x - 2
x - 1
c 2 — 64 c
24.
3
x — 3
4c 3 c 2 + 9c + 8
x 2 - 5x + 6 x - 2
MULTIPLYING BY POLYNOMIALS Write the product in simplest form.
25.
3x
x + 4
(3x + 12)
27. {y - 3) 2 •
2y - 2
y 2 - 4y + 3
29. + 3 — • (x 2 - 9)
2x 2 — 3x — 9
26 --irfTTfr-^ +1 »
28. (x 2 + 2x + 1) •
30. 3z 2 + lOz + 3 •
X + 2
x 2 + 3x + 2
z + 3
3 z 2 + 4z + 1
11.4 Multiplying and Dividing Rational Expressions
Student HeCp
► Homework Help
Extra help with
probIem solving in
Exs. 31-39 is available at
www.mcdougallittell.com
DIVIDING RATIONAL EXPRESSIONS Write the quotient in simplest form.
31
34.
25x 2
lOx
37.
5x
lOx
x ^ x + 5
x + 2 x + 2
x . x + 3
X + 6 ' X 2 - 36
32.
35.
38.
16.v 2
8x
4x 2
16x
2(x + 2) ^ 4(x - 2)
5(x - 3) * 5x - 15
3x + 12 . x + 4
4x
2x
33.
36.
39.
3x 2 . 9X 3
10
25
2x - 2
x — 2 x 2 — 3x + 2
2x 2 + 3x + 1 . x 2 — 1
12x - 12
6x
DIVIDING BY POLYNOMIALS Write the quotient in simplest form.
x + 5 / 2
40.
42.
2 + 3x
x 2 + 19x - 20
- (x 2 - 25)
H- (x 2 - 1)
„„ 3x 2 + 2x - 8 . /0
44. -^-- (3x - 4)
43.
45.
-5x 2
y ~ 12
2y + 3
4x + 3 -
X — 1
-H (y 2 - 14 y + 24)
-h (4x 2 + x — 3)
PRAIRIE DOGS In Exercises 46-49, use the following information.
Scientists are monitoring two distinct prairie dog populations, P { and P 2 ,
modeled as follows.
n 100x 2 , n 100x 2 ,
P ! = ——r and = —— where x is time in years,
i x + 1 2 x + 3 J
46. Copy and complete the table below. Round to the nearest whole number.
2
3
4
5
6
7
8
9
10
?
?
?
?
?
?
?
?
?
?
LL
?
?
?
?
?
?
?
?
?
47. Find the ratio in simplest form of Population 1 to Population 2, that is —.
48. Add another row to your table labeled and evaluate for each value of x.
49. Describe the pattern in the ratios you found in Exercise 48. If the value of x
P x
gets very large, what value does — approach? Explain.
50. Geo metry Link / Write the ratio in simplest form comparing the area of
the smaller rectangle to the area of the larger rectangle.
x —3
x 2 —3x—4
x 2 —16
x 2 —x— 6
x+4
2x
x +1
2x 2
Chapter 11 Rational Expressions and Equations
Standardized Test
Practice
Mixed Review
Maintaining Skills
51. CHALLENGE Write the expression in simplest form.
x 2 + llx + 18 . I4x 3 x 2x — 1 2x 2 + 9x — 5
x 2 - 25 ^ x 2 - x - 20 * x + 4 " 6x * x 2 + 3x + 2
52. MULTIPLE CHOICE Which of the following represents the expression
X 2 - 3x (x - 2) 2 . . ! , r o
—-• A ~—— in simplest form?
x 2 - 5x + 6 2x
-T(-T 3) ^ 4.1 + 4 / ^-^'\ X 2
CD X _ 2 CD — ®2
X 2 “1“ X
53. MULTIPLE CHOICE Which product represents ( 2x + 2) h--—?
CD
2x + 2
4
CD
2x + 2
x 2 + X
2x + 2
x 2 + X
1
4
CD
1
4
CD
2x + 2
4
2x + 2
x 2 + X
1
x 2 + X
FUNCTIONS In Exercises 54-56, use the function y = x + 9, where
2 < x < 6. (Lesson 1.8)
54. Calculate the output y for several inputs x.
55. Make an input-output table.
56. State the domain and range of the function.
ABSOLUTE-VALUE INEQUALITIES Solve the absolute-value inequality.
(Lesson 6.7)
57. U + 7 | < 12
60. | 3jc — 10 | < 4
58. |2x - 15 | < 15
61. U + 5 | > 17
59. | jc + 13 | > 33
62. 15x — 1 | < 0
QUADRATIC EQUATIONS Solve the quadratic equation. (Lesson 9.6)
63. 2x 2 + 12x — 6 = 0 64. x 2 — 6x + 7 = 0 65. 3x 2 + 1 lx + 10 = 0
66. 6x 2 = 130 67. 5 = 6x 2 + lx 68. 2x 2 + 4x = 7
POLYNOMIALS Add or subtract the polynomials. (Lesson 10.1)
69. (—5x 2 + 2x — 12) — (6 — 9x — lx 2 ) 70. (a 4 — 12 a) + (4 a 3 + lla — l)
71. (16 p 3 -p 2 + 24) + (12 p 2 -8 p- 16) 72. (4 1 2 + 5t + 2) - (t 2 - 3t - 8)
ADDING FRACTIONS AND DECIMALS Add. Write the answer as a
decimal. (Skills Review pp. 759 , 767)
73. 0.35 + | 74. 0.58 + | 75. 0.99 + | 76. 0.06 + 4
2 J 4 O
77. J + 0.25 78. | + 0.4 79. + 0.12 80. + 0.45
o D 12 10
11.4 Multiplying and Dividing Rational Expressions
Adding and Subtracting
with Like Denominators
Goal
Add and subtract rational
expressions with like
denominators.
Key Words
• rational expression
• common denominator
What happens when you hit a tennis ball?
In Exercises 42-45 you will
subtract rational expressions with
like denominators to analyze the
effects of hitting a tennis ball with
a racket.
As with fractions, to add or subtract rational expressions with like, or the same,
denominators, combine their numerators and write the result over the common
denominator.
ADDING OR SUBTRACTING WITH LIKE DENOMINATORS
Let a, b, and c be polynomials, with c =£ 0.
to add, add the numerators. — + — = a + b
c c c
to subtract, subtract the numerators. — -SI— = -—-
c c c
1 Add Rational Expressions
Simplify £ +
Solution
O Write the original expression.
© Add the numerators.
© Combine like terms.
0 Simplify the expression.
5 x — 5
2x 2x
5 + (x - 5)
2x
x
2x
1
2
Chapter 11 Rational Expressions and Equations
Student HeCp
| ►Study Tip
When you are
subtracting rational
expressions, make
sure the quantity that
you are subtracting is
in parentheses so that
you remember to
distribute the negative.
2 Subtract Rational Expressions
o- rf 4 x + 4
Simplify, + 2 , + 2 -
Solution
Q l/l/r/fe the original expression.
4 x + 4
x + 2 x + 2
0 Subtract the numerators.
4 — (x + 4)
v + 2
0 Distribute the negative.
4 - x - 4
x + 2
0 Simplify the numerator.
X
x + 2
Add and Subtract Rational Expressions
Simplify the expression.
x + 2 3x - 2 - x + 2 3x + 2
I ■ "T~ m ^ ^
X X x 2 + 5 x 2 + 5
3.
3x - 4
x - 4
2x
x-4
Student HeCp
► More Examples
More examples
-^07 gre ayajiabie a t
www.mcdougallittell.com
ms Simplify after Subtracting
Simplify
4x
x — 2
3x 2 — x — 2 3x 2 — x — 2
Solution
4x _ x — 2 _ 4x — (x — 2)
3x 2 — x — 2 3x 2 — x — 2 3x 2 - x - 2
_ 3x + 2
3x 2 — x — 2
_ 3x + 2
” (3x + 2)(x - 1)
_ 3xj>-2
~ (3x> 2 )(x - 1 )
1
x — l
Subtract numerators.
Simplify.
Factor.
Divide out common factor.
Write simplest form.
Simplify after Adding or Subtracting
Write the sum or difference in simplest form.
4 2x 2 2x - 4 x — 1
" x 2 + 2x + 1 x 2 + 2x + 1 " x 2 + 3x x 2 + 3x
11.5 Adding and Subtracting with Like Denominators
iW3l Exercises
Guided Practice
Vocabulary Check 1 . Complete: To add or subtract rational expressions with like denominators,
add or subtract their numerators, and write the result over the ? .
Skill Check Add. Simplify your answer.
2 ± +
3x 3x
8y 10 ~3y
y + 3 y + 3
4.
+
3x + 1
x z — 9 x z
9
Subtract. Simplify your answer.
1
6 .
12k 3k + 7
c + 1
c + 6
3 r 3r ~ k 2 k 2
Add or subtract, then factor and simplify.
9.
c 2 - 4 c z -4
0 5x , 20
O. , +
■12y
10 .
x + 4 4 + x
2_y + 3 -y + 15
+
84
y 2 - 4y y 2 - 4y
Practice and Applications
11 .
y 2 -9y + 14 / - 9y + 14
10 _ —2r
r 2 + 9r + 20 r 2 + 9r + 20
ADDING RATIONAL EXPRESSIONS Simplify the expression.
12 2 -+ ±+ 2 .
2x 2x
15.
4 2x — 2
x + 1
X + 1
13.
2
+
5
14.
x + 7
x + 7
16.
a + 1
15(3
+
2(3—1
15(3
17.
4t- 1 2f + 3
1 - 4f 1 - 4f
2 x
+
4x + 6 4x + 6
SUBTRACTING RATIONAL EXPRESSIONS Simplify the expression.
lx 6x ^8 + 6( 5^-6 ^ 2x 2x + 1
3
18.
21
x x
2
3x — 1 3x — 1
5x
19.
8 + 6 1
L/l
1
On
20.
2 x
31
3t
x + 2
22.
4x
16
23.
4m
2 x + 6
2 x + 6
m — 2
x + 2
2m + 4
m — 2
FACTORING AFTER ADDING OR SUBTRACTING Simplify the expression.
8 ~ 2 - 2 . 4a - 3
Student ttcCp
► Homework Help
Example 1: Exs. 12-17
Example 2: Exs. 18-23
Example 3: Exs. 24-31
24.
26.
28.
30.
x 2 + 5x - 24 x 2 + 5x - 24
2 x 8
x 2 + 5x + 4 x 2 + 5x + 4
2 x x
x 2 + 5x x 2 + 5x
y 2 ~ 2y 9(y - 2)
a 2 — 25 a z — 25
x 1 — 10 3x
27. —;-+
y 2 -ly - 18 y 2 - 7y - 18
29.
31.
x 2 4 x 2 - 4
2x(x + 4) 3x — 3
(x + l) 2
„2
(x + l) 2
12-y
y 2 -3y- 28 y 2 - 3y - 28
Chapter 11 Rational Equations and Functions
ERROR ANALYSIS In Exercises 32 and 33, find and correct the error.
32.
3h 3 - 36h
3 ^- 12 ) __
"7 f« - 12) (n - 12) m - 12
I
Student Hedp
► Homework Help
Extra help with
problem solving in
Exs. 34-39 is available at
www.mcdougallittell.com
COMBINING OPERATIONS In Exercises 34-39, simplify the expression.
1 lx — 5 llx + 12 3x - 100 „ 4 + x 6 + x _ 1 - x
2x + 5 2x + 5 + 2x + 5 x - 9 + x - 9 x - 9
36.
38.
c ~ 15 2c
2 c + 6
2 c + 6
+
12
2 c + 6
37.
2 x
x 2 — 9
4x + 2 _ 4
x 2 - 9 x 2 - 9
39.
3x - 5
6 x - 8
Link to
Science
KINETIC ENERGY Every
moving object has kinetic
energy. The tennis player and
tennis racket have kinetic
energy as he swings the
racket at the ball. The ball
has kinetic energy as it flies
through the air.
Find expressions for the perimeter of the rectangle and
triangle. Simplify your answer.
Science Li In Exercises 42-45, use the following information.
When a tennis player hits a ball that is already moving, the work done by the
racket is the change in the ball’s kinetic energy. The total work done on an object
is given by the formula
w = k 2 -k {
where W represents work, K x represents the initial kinetic energy of an object,
and K 2 represents the final kinetic energy of an object. Work and energy are
measured in joules. (A joule is the amount of work done when a force of
1 newton acts on an object that moves 1 meter.)
Copy and complete the table, by computing the work done on the tennis ball.
42.
43.
44.
45 .
K 2 /r, Work
9
X
7
X
? ?
5
a
5 — a
a
? ?
2 1
t + 4
? ?
t- 1
t- 1
x 2 - 7
1
§
1
-0
?
X 2 - 100
I X 2 - 100
11.5 Adding and Subtracting with Like Denominators
Standardized Test
Practice
Mixed Review
Maintaining Skills
46. MULTIPLE CHOICE Which of the following expressions can be simplified
to x + 3?
(A)
x + 3 x + 3
x — 6 _ x + 9
x — 3 x — 3
CD
4x + 21
x - 7 x - 7
Co) None of these
47. MULTIPLE CHOICE Simplify
25
x + 5 x + 5 *
CD
x — 5
(G)
x 2 - 25
x + 5
(H) x - 5
CD
x — 5
x + 5
24 y 2 + 24 73y
48. MULTIPLE CHOICE Simplify
(A)
CD
(8 y + 3)(3y + 8)
8y - 3
3y - 8
8y - 3
8y — 3 8y — 3‘
CD ^ ——
® (8y - 3) 2
CD 3y - 8
SIMPLIFYING EXPRESSIONS Rewrite the expression with positive
exponents. (Lesson 8.2)
-6 -- -1 1 - 3
49. x 5 y~
53. (- 6 c)~
50. 8x _1 y“ 3
51. —)—7
52.
2x‘y~
4
54. (-yA
55.4
C
56.
10r 3 r _1
_ 1 _
(-7m) -3
SIMPLIFYING EXPRESSIONS Simplify the expression. The simplified
expression should have no negative exponents. (Lesson 8.4)
57.
58.x 5
1
61.
m 8 • m 10
m
62.
(a 3 ) 4
(a 3 ) 8
59.
63.
-l
-2u 2 v
uv
60.
64.
y 5 \ 2
7
42a 3 fr~
6 ab
EVALUATING EXPRESSIONS Perform the indicated operation. Write the
result in scientific notation. (Lesson 8.5)
q X 10 -3 1 A v in_1
65
5 X 10“
66 .
1.4 X 10
3.5 X 10
-4
68 . 2 X 10 + 3 X 10 69. (2.5 X 10)
-2
67. (3 X 10“ 2 )
70. 3.2 X 10 + 5.8 X 10
PATTERNS List the next three numbers suggested by the sequence.
(Skills Review pp. 781)
71. 1,3, 5, 7, ?, ?, ?
73. 60, 57, 53,48, ?, ?, ?
7 13
75 9 — 5 — 779
72. 1,3,6, 10, ?, ?, ?
12 3 4
74 — — — D 9 9 7
2’ 3’ 4’ 5’ ’’ ’’ '
76. 100,81,64, 49, ?, ?, ?
Chapter 11 Rational Equations and Functions
Adding and
with Unlike
Subtracting
Denominators
Goal
Add and subtract rational
expressions with unlike
denominators.
Key Words
• least common
denominator (LCD)
How should you plan a 300-mile car trip?
In Example 6 you will add
rational expressions with
unlike denominators to
analyze the total time
needed for a 300-mile
car trip.
As with fractions, to add or subtract rational expressions with unlike
denominators, you first rewrite the expressions so that they have like
denominators. The like denominator that you usually use is the least
common multiple of the original denominators, called the least common
denominator or LCD.
Student MeCp
► Skills Review
For practice on finding
the LCD of numerical
fractions, see p. 762.
\_ j
i| Find the LCD of Rational Expressions
Find the least common denominator of
Solution
0 Factor the denominators.
0 Find the highest power of each factor
that appears in either denominator.
© Multiply these to find the LCD.
ANSWER ^ The LCD is 120x 4 .
12x
and
2 + x
40x 4 '
12x = 2 2
40x 4 = 2 3 • 5 •
2 3 , 3, 5, x 4
2 3 • 3 • 5 • x 4 = 120*
Find the LCD of Rational Expressions
Find the least common denominator.
^ v + 1 2x ^ 1 3x + 1
1b 5 ’ ~6 Z '36? 9x 5
5*+ 9 7
16x 3 24x 2
A X 2x 3
x - 5’ x + 7
5.
12 x
X + 1 ’ X — 1
6 .
2x + 3 1
30* 5 ’ 8x
11.6 Adding and Subtracting with Unlike Denominators
2 Rewrite Rational Expressions
Find the missing numerator.
2 = ?
a ' 3y lSy
Solution
2 _ ?
a - 3y 15y
2 _ 10
3y 15y
3x — 1 _ ?
4x 2 “ 36x 5
Multiply 3y by 5 to get 15y.
Therefore, multiply 2 by 5 to get 10.
3x - 7 = (3x - 7) * 9x 3
4x 2 36x 5
3x — 7 _ 27x 4 — 63x 3
4x 2 36x 5
Multiply 4x 2 by 9x 3 to get 36x 5 .
Therefore, multiply (3x - 7) by 9x 3 to get
(3x - 7) • 9x 3 .
Simplify.
Rewrite Rational Expressions
Find the missing numerator.
9 ? y ~ 1
7 -^=^ 8 -—
13 y 2
9.
c
c + 1
?
(c + l)(c - 3)
Student HeCp
► Study Tip
2
To rewrite - with a
x
denominator of x 2 ,
multiply the numerator
and denominator by x.*
1 x lx
3 Add with Unlike Denominators
Simplify | + 1
Solution
O Find the LCD.
© Write the original expression.
© Rewrite the expression using the LCD.
O Add.
The LCD is x 2 .
2 1 ~ 2x
x x 1
2x 1 — 2x
2 ' 2
X X
2x + (1 - 2x)
0 Simplify the expression.
L_
Add with Unlike Denominators
Write the sum in simplest form.
_ 1 , 2 2 ,3 — 2 m
10. H— 11.7-h 7
X 3m nr
Chapter 11 Rational Expressions and Equations
12 .
3
15x 2
Student HeCp
p More Examples
More examples
are available at
www.mcdougallittell.com
J 4 Subtract with Unlike Denominators
X + 1
8X 2 ‘
Simplify
Solution
The LCD is 24x 2 .
1_ _ x + 1 = J_'4x_ (x + 1)
6x 8x 2 6x 4.v 8x 2
_ 28x 3x + 3
~~ 24x 2 24x 2
_ 28x — (3x + 3)
24x 2
= 25x - 3
24x 2
l_
Rewrite using LCD.
Simplify numerators and denominators.
Subtract.
Simplify.
5 Add with Unlike Binomial Denominators
0 . ™ x + 2 12
Simplify—f +
Solution
Neither denominator can be factored. The least common denominator is the
product (x — l)(x + 6) because it must contain both of these factors.
x±2 + _L2_
x — 1 x + 6
(x + 2)(x + 6) . 12(x — 1)
(x — l)(x + 6) (x — l)(x + 6)
x 2 + 8x + 12 12x — 12
(x — l)(x + 6) (x — l)(x + 6)
jc 2 + 8x + 12 + (12 jc ~ 12)
(x — l)(x + 6)
x 2 + 20x
(x — l)(x + 6)
x(x + 20)
(x — l)(x + 6)
Write original expression.
Rewrite using LCD.
Simplify numerators.
Add.
Combine like terms.
Factor.
Add or Subtract with Unlike Denominators
Simplify the expression.
3 2
13.
x 2 3x
14.
16. — 7—r + 1
X + 1 X — l
17.
x — 6
1 - 10 p
1 * 3 + 4x
1
5 P 2
15 ‘ 4x 3
Kk 2
, 1
x — 5
x + 2
+ -
18. , , -
x + 5
x — 2
11.6 Adding and Subtracting with Untike Denominators
Write and Use a Model
PLANNING A TRIP You are planning a 300-mile car trip. You can make the
trip using a combination of two roads: a highway on which you can drive
60 mi/h and a country road on which you can drive 40 mi/h. Write an
expression for the total time the trip will take driving on both roads.
Solution
Algebraic T
Model
x 300 — x
40 + 60
3x 2(300 - x)
120 + 120
3x + 600 — 2x
120
x + 600
120
Write algebraic model.
Rewrite using LCD.
Add.
Simplify.
Write and Use a Model
19. Evaluate the expression for the total time at 60 mile intervals by completing
the table. The table can help you decide how many miles to drive on each
road.
Distance (country), x
0
60
120
180
240
300
Total time, T
?
?
?
?
?
?
Chapter 11 Rational Expressions and Equations
L ' / - I •
m Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Explain what is meant by the least common denominator of two rational
expressions.
2. ERROR ANALYSIS Find and correct the error.
_2 _ 3x _ Z(x — 1)
X —X X(X — 1) X(X “4
In Exercises 3-6, simplify the expression.
3 —+ -
^ 12 + 4
10c
1 x + 6 _ 4
4x 2 x + 1 2x + 3
6 .
x - 2 x + 3
2jc - 10
5 2
7. You can use v — 3 as the LCD when finding the sum _ + _ . What
number can you multiply the numerator and the denominator of the second
fraction by to get an equivalent fraction with x — 3 as the new denominator?
Practice and Applications
f
Student HeCp
► Homework Help
Example 1: Exs. 8-15
Example 2: Exs. 16-21
Example 3: Exs. 22-27
Example 4: Exs. 28-33
Example 5: Exs. 34-42
Example 6: Exs. 43-48
FINDING THE LCD Find the least common denominator of the pair of
rational expressions.
1 1
8.
12 .
3*’ 9x 3
10 10
13v 7 ’ 3v 5
9.
4x 3x 2
10.
17/
8 z
ii.4#!
15’ 5
z 2 ’
3 y
c 3 ’ 7c 5
6b -5
5 ’ b
14.
X — 1
x-T
x — 3
’ x - 4
.x + 1 25
15 ' 15x ’ 18x 3
REWRITING RATIONAL EXPRESSIONS Find the missing numerator.
16.
19.
11 _ ?
3* 12x 3
3^+1 ?
17.
20 .
8 _ ?
5 15y 2
x 9
9 a 5 63 a 11 w 2x + 3 x(2x + 3)
ADDING Write the sum in simplest form.
18.
21 .
x — 3 _
2
2a-3
35 a 2
7
28v
140<+
3 , 1
22 . — + -
2 z z
_ 2x + 3 . x + 1
25. —--h
11 ± 2
23 ' 6x + 13*
24. T- +
26.
3 m + 1
4x —5x
01 - 4 - — + 1
15 30n
2 12m 3 4m 3
11.6 Adding and Subtracting with Unlike Denominators
SUBTRACTING Write the difference in simplest form.
28.
31.
2x
x + 1
29 -%-
2
30.
5
4
(N
X
x - 1
2
«■§-
2 + c
33.
6x 2 3x
25 c
3
6 b 2
4 b
ADDING OR SUBTRACTING Simplify the expression.
34.
x + 1
+
37.#^r +
x — 2
x + 3
40.
3x — 1
4x
5x —
x + 1
2x
5x + 1
35.
38.
+
x + 4
x — 10 ' x + 6
4
41.
x + 4
2x
1 _
5x
lx
X
1 X + 4
36.
39.
42.
2x - 1
1
3x
" 11
x - 3
x + 3 +
x + 9
x — 3
2x + 1
x + 4
3x — 1
x — 2
3x + 10
X
lx- A
4x + 3
TRAVEL BY BIKE In Exercises 43-45, use the following information.
You are riding your bike to a pond that is 8 miles away. You have a choice to
ride in the woods, on the road, or both. In the woods, you can ride at a speed of
10 mi/h. On the road, you can ride at a speed of 20 mi/h.
43. Write an expression for your total time.
44. Write your answer to Exercise 43 in simplest form.
45. Evaluate the expression for total time at 2 mile intervals.
Link
Transportation
TRAVEL BY BOAT Many
cities with harbors offer a
variety of types of water
transportation. These include
regular commuter service,
sight-seeing tours, and
recreational trips.
TRAVEL BY BOAT In Exercises 46-48, use the following information.
A boat moves through still water at x kilometers per hour (km/h). It travels 24 km
upstream against a current of 2 km/h and then returns to the starting point with
the current. The rate upstream is x — 2 because the boat moves against the current,
and the rate downstream is x + 2 because the boat moves with the current.
46. Write an algebraic model for the total time for the round trip.
47. Write your answer to Exercise 46 as a single rational expression.
48. Use your answer to Exercise 47 to find how long the round trip will take if
the boat travels 10 km/h through still water.
Geometry Unk ^ In Exercises 49-51, use the diagram of the rectangle.
1
2x — 1
2x+ 1
49. Find an expression for the perimeter of the rectangle.
50. What is the perimeter of the rectangle when x = 3?
51. What is the area of the rectangle when x = 3?
Chapter 11 Rational Expressions and Equations
Standardized Test
Practice
Mixed Review
Maintaining Skills
15 9
52. MULTIPLE CHOICE Find the LCD of — j and — 7 .
3t 6 2 1 4
(A)
6f 6
CM) 6t 2
(c) 6t 6
CM) 6t 10
53- MULTIPLE CHOICE Find the missing numerator
5x + 6
8x 2
48x
.3 •
CD 6x
CD 41jc
(JD 30x 2 + 36x CD llx + 6
1
54- MULTIPLE CHOICE What is the difference of _ and in simplest
XL AX \ L
form?
(A)
CM)
x — 1
(x - l)(2x + 1)
2X 2 + 1
(x - l)(2x + 1)
CD -
CD
X — 1
2x 2 - 1
(jc - l)(2x + 1)
POINT-SLOPE FORM Write in point-slope form the equation of the line
that passes through the given point and has the given slope. (Lesson 5.2)
55- (—3, —2), m = 2
58- (5, 5), m = 5
56- (0, 5), m = — 1
59. (7, 0), m - 3
57. (-3, 6), m = j
60. (14, -3), m
SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 8.4)
„2 1 4 42x 4 v 3
61.
65
lOx
12x
144x 2
62.
66 .
Am 2
6m
2x 2 y 3 z 4
63.
67.
16x 4
32x 8
33 p 4
64.
68 .
6x 3 y 9
15w 2
9w 5
5x 4 y 3 z 2 " 44/? 2 ^
STANDARD FORM Write the equation in standard form. (Lesson 9.6)
69. 6x 2 = 5x-l 70. 9 - 6x = 2x 2 71. -4 + 3y 2 = y
72. 12x = x 2 + 25 73. 7 - 12x 2 = 5x 74. 8 = 5x 2 - 4x
75. GAME SHOW A contestant on a television game show must guess the price
of a trip within $1000 of the actual price in order to win. The actual price of
the trip is $8500. Write an absolute-value inequality that shows the range of
possible guesses that will win the trip. (Lesson 6.7)
FRACTIONS AND DECIMALS Write the fraction as a decimal rounded to
the nearest thousandth. (Skills Review p. 767)
47
76 ‘ 99
77
200
32
78 ‘ T 55
79
199
80.
144
8’.-§
“■-i
79
83. ——
145
23
84
° 25
<*4
86. f
87
87 “ 35
11.6 Adding and Subtracting with Untike Denominators
Rational Equations
Goal
Solve rational equations.
Key Words
• rational equation
• cross product property
• least common
denominator (LCD)
How long does it take to shovel a driveway?
If you can shovel a snowy driveway in
3 hours and your friend can do it in 2
hours, how long would it take to shovel
it together? In Example 4 you will write
and solve a rational equation to answer
this question.
A rational equation is an equation that contains rational expressions. Example 1
and Example 2 show the two basic strategies for solving a rational equation.
Student tieCp
► Study Tip
When you solve
rational equations, be
sure to check your
answers. Remember,
values of the variable
that make any
denominator equal to 0
are excluded.
\ _ /
i Cross Multiply
Solve _ = ,.
y + 2 3
5 _ y
y + 2 3
0 Write the original equation.
© Cross multiply.
5(3) = y(y + 2)
© Simplify each side of the equation.
15 = j 2 +
0 Write the equation in standard form.
0 = j 2 + — 15
© Factor the right-hand side.
0 = (y + 5)(y - 3)
ANSWER ^ The solutions arey = — 5 andy = 3.
CHECK y Neither —5 nor 3 results in a zero denominator. Substitute y =
and y = 3 into the original equation.
- 5 :
3 :
(-5)+ 2
5 z 3
3 + 2 3
Since
-3
y = —5 is a solution. S
Since y = 3 is a solution. S
Cross Multiply
Solve the equation. Check your solutions.
_ x _ x + 2 _ 3 _ m + 1
"26 " 2m 4m
6
y + 7
Chapter 11 Rational Expressions and Equations
TWO METHODS Cross multiplying is appropriate for solving equations in which
each side is a single rational expression. A second method, multiplying by the
LCD, works for any rational equation.
■flcisidf 2 Multiply by the LCD
2 1 4
Solve - + ^ = -.
x 3 x
Student HeCp
► Study Tip
In Example 2, you
must check that the
solution x = 6 does
not result in a zero
denominator in the
original equation.
0 Find the LCD.
0 Write the original equation.
© Multiply each side by the LCD 3x.
0 Simplify each side of the equation.
© Solve by subtracting 6 from each side.
The LCD is 3x.
2 1 4
3jc • - + 3jc • ~ = 3x • -
x 3 x
6 + v = 12
x = 6
Student HeCp
►Study Tip
To find the LCD in
Example 3, look at the
equation in factored
form. The product of
the highest powers of
the factors in either
denominator is (x + 3) 2 .
L J
3 Factor First, then Multiply by the LCD
Solve ■
+
x + 3 x 2 + 6x + 9
Solution
■x
+
= 1 .
* + 3 x 2 + 6x + 9
3 + 4
x + 3
(x + 3) 2 +
x + 3 (x + 3) 2
4 / ■ ox2
(x + 3) 2
Write original equation.
Factor denominator.
• (x + 3f = 1 • (jc + 3) 2
3(jc + 3) + 4 = (jc + 3) 2
3x + 13 = x 2 + 6x + 9
0 = x 2 + 3x — 4
0 = (jc + 4)(jc - 1)
x = —4 andx = 1
Multiply by
LCD (x + 3) 2 .
Simplify.
Simplify each side.
Write in standard
form.
Factor.
Solve.
ANSWER ► The solutions are x = —4 and x = 1. Check both values.
L
Multiply by the LCD
Solve the equation. Check your solutions.
4. - + j = - 5. —J-r + - =
x 4 x n + 1 n n 2 + n
6 .
+
x + 3
= 1
11.7 Rational Equations
WORK PROBLEMS Writing and solving rational equations can help to solve
problems such as finding out how long it would take you and a friend to clear
snow off of a driveway.
Student HeCp
► More Examples
More examples
are ava j| a kie a t
www.mcdougallittell.com
4 Solve a Work Problem
SHOVELING SNOW Alone, you can shovel your driveway in 3 hours. Your
friend Amy can shovel the driveway in 2 hours. How long will it take you and
Amy to shovel your driveway, working together?
Solution
Student ftedp
>
► Study Tip
To find how many
minutes are in \ hour,
0
do the following
calculation.
4 • 60 minutes =
b
12 minutes. Therefore
11 hours equals
1 hour 12 minutes.
^ _ J
Verbal
Model
Algebraic
Model
Part you
Time you
Part Amy
Time Amy
do in
•
spend
+
does in
•
spends
1 hour
shoveling
1 hour
shoveling
part you do
+
part Amy does
= 1
whole
job
Labels
Part of the job you can do in 1 hour = —
(no units)
Time you spend shoveling = t
(hours)
Part of the job Amy can do in 1 hour = —
(no units)
Time Amy spends shoveling = t
(hours)
— t H- t — 1
1 + 1=1
3 2
6 + 2 * 6
1 • 6
2t + 3t — 6
5t = 6
t
5 5
Write algebraic model.
Simplify equation.
Multiply by LCD 6.
Simplify.
Combine like terms.
Divide each side by 5.
ANSWER ► The time that it will take you and Amy to shovel the driveway
is 1-| hours, or 1 hour 12 minutes.
Solve a Work Problem
7. You can clean your house in 4 hours. Your sister can clean it in 6 hours.
How long will it take you to clean the house, working together?
8. A roofing contractor estimates that he can shingle a house in 20 hours and
that his assistant can do it in 30 hours. How long will it take them to shingle
the house, working together?
MIXTURE PROBLEMS Mixture problems —problems that involve combining
two or more items—occur in many different settings. Example 5 discusses
mixing roasted nuts and raisins. The exercise set presents mixture problems from
other fields, such as chemistry.
5 Solve a Mixture Problem
RAISINS AND NUTS A store sells a mixture of raisins and roasted nuts.
Student HeCp
^ -
► Study Tip
Because the number
of kilograms of the
mixture is 20 and the
number of kilograms of
raisins is x, the number
of kilograms of nuts is
20 - x. ...
Raisins cost $3.50 per kilogram and nuts cost $4.75 per kilogram. How many
kilograms of each should be mixed to make 20 kilograms of this snack worth
$4.00 per kilogram?
Solution
When you solve a mixture problem, it is helpful to make a chart.
Let x = Number of kilograms of raisins.
Then 20^- x = Number of kilograms of nuts.
Use the information from the problem to complete the chart below. Then write
and solve an equation that relates the cost of the raisins, the cost of nuts, and
the cost of mixture.
Number of kg x Price per kg = Cost
Raisins
X
3.50
3.5x
Nuts
H
1
O
<N
4.75
4.75(20 - x)
Mixture
20
4.00
80
Cost of raisins + Cost of nuts = Cost of mixture Write verbal model.
3.5x + 4.75(20 - x) = 80
350x + 475(20 - x) = 8000
350x + 9500 - 475x = 8000
9500 — 125x = 8000
— 125x = -1500
x =12
Therefore, 20 — x = 8
Write algebraic model.
Multiply each side by
100 to clear equation
of decimals.
Use distributive property.
Combine like terms.
Subtract 9500 from
each side.
Divide each side by -125.
ANSWER ► 12 kilograms of raisins and 8 kilograms of nuts are needed.
Solve a Mixture Problem
9. You make a mixture of dried apples costing $6.00 per kilogram and dried
apricots costing $8.00 per kilogram. How many kilograms of each do you
need to make 10 kilograms of a mixture worth $7.20 per kilogram? Make a
chart to help you solve the problem.
11.7 Rational Equations
JL7 Exercises
Guided Practice
Vocabulary Check 1. What are two methods of solving rational equations?
2. Which method is limited to solving equations in which each side is a single
rational expression?
Skill Check Find the least common denominator.
1 £|_ /L — LI 5 2 1
x 3’3* 4x’6v 2 ’ 8x 2 x 9 3x 2 ’x 3
Solve the equation using the cross product property. Remember to check
your solutions.
G 3 _ x x = 3 3 _ 1
x 12 x + 2 x — 2 u + 2 u- 2
Solve the equation by multiplying by the least common denominator.
Check your solutions.
9.
1
5
2 _
5x
1
x
X
x + 2
= 1
Practice and Applications
CROSS MULTIPLYING Solve the equation by cross multiplying. Check
your solutions.
15.
7
x + 1
2
J-3
13 2L = 1±
10 5
16.
19.
x + 2 4
3Q 2 + 1)
6t 2 - t - 1
2
14.
17.
12
x 5(x + 2)
5 5
x + 4 3(x +1)
2 o.^±i2=i
(X - 3 ) 2
Student HeCp
->
► Homework Help
Example 1: Exs. 12-20
Example 2: Exs. 21-32
Example 3: Exs. 33-38
Example 4: Exs. 48-50
Example 5: Exs. 51-53
MULTIPLYING BY THE LCD Solve the equation by multiplying each side
by the least common denominator. Check your solutions.
21 .- + 2
x
24.
3 1
x
4
_ 2
" 3
27 . - + = i
s s + 2
2 w
30 - M = 5 _ 2
22 .
25.
28.
31.
9
x + 9 x + 9
+ 4
x(x +1) X
2 2
—--h 2 = -
3x + 1 3
X + 1 X + 1
12
x
23.
26.
29.
3x
x - 1
x
5
+
1
x + 3 x — 3
5 3
2r + 1 2r - 1
= 1
= 0
5 , 250
32. - + —
3 9r
Chapter 11 Rational Expressions and Equations
Student HeCp
► Homework Help
Extra help with
"<^0^ problem solving in
Exs. 33-38 is available at
www.mcdougallittell.com
FACTOR FIRST Factor first, then solve the equation. Check your solutions.
4 _ 3 1 1
„ 2 , 1
33.-- +
y - 2 y + 2 y 2 - 4
35.
37.
+
10
x - l x 2 - 2x + 1
2 x 6
34.
36.
x + 1 x — 2 x 2 x 2
1 4
+
x - l x + 3 x 2 + 2x - 3
38.
x + 3 x - l x 2 + 2x-3
1 2 2
y
16 y + 4 y - 4
CHOOSING A METHOD Solve the equation. Check your solutions.
„ 1 , 4 1
39. 7 + - = -
4 x x
42.
45.
x _
9 x ” 9
-3 = 2
x + 7 x + 2
40.
43.
46.
— 3x
-2
x + 1 x — 1
x + 42
x
2
x
+ i = a
x + 3 x 3x
41.
44.
47.
1
x
2 _
x 2
6
3
4
l
" 9
48. MOWING THE LAWN With your new lawn mower, you can mow a lawn
in 4 hours. With an older mower, your friend can mow the same lawn in
5 hours. How long will it take you to mow the lawn, working together?
SPORTS REPORTER
Sports reporters gather
statistics, such as a baseball
players batting average, and
prepare stories that cover all
aspects of sports from local
sporting events to
international competitions.
More about sports
reporters is available
at www.mcdougallittell.com
49. HIGHWAY PAVING The county’s new asphalt paving machine can surface
one mile of highway in 10 hours. A much older machine can surface one
mile in 18 hours. How long will it take them to surface 1 mile of highway,
working together? How long will it take them to surface 20 miles?
50. CAR WASHING Arthur can wash a car in 30 minutes, Bonnie can wash a
car in 40 minutes, and Claire can wash a car in 60 minutes. How will it take
them to wash a car, working together?
51. NOODLE MIXTURE A grocer mixes 5 pounds of egg noodles costing
$.80 per pound with 2 pounds of spinach noodles costing $1.50 per pound.
What is the cost per pound of the mixture?
52. JUICE MIXTURE A farm stand owner mixes apple juice and cranberry juice.
How much should he charge if he mixes 8 liters of apple juice selling for
$0.45 per liter with 10 liters of cranberry juice selling for $1.08 per liter?
53. COINS You have 12 coins worth $1.95. If you only have dimes and quarters,
how many of each do you have?
54. BATTING AVERAGE You have 35 hits in 140 times at bat. Your batting
35
average is yyy = 0.250. How many consecutive hits must you get to increase
your batting average to 0.300? Use the following verbal model to answer
the question.
Desired Batting average
_Past hits + Future hits_
Past times at bat + Future times at bat
11.7 Rational Equations
Standardized Test
Practice
Mixed Review
55. CHALLENGE How many liters of water must be added to 50 liters of a 30%
acid solution in order to produce a 20% acid solution? Copy and complete
the chart to help you solve the problem.
Number of liters x % acid = Liters of acid
Original Solution
?
?
?
Water Added
X
?
?
New Solution
?
?
?
56. IVIULTIPLE CHOICE What is the LCD of iy, and
(A) 56x 4 CD 28a- 2 CD 28a- CD 7x 2
57. MULTIPLE CHOICE What is the solution of the equation
CD 8 CD | CD 10 GD ^
58. IVIULTIPLE CHOICE What is the solution of the equation 2- — ^ = 0?
(A) 6, —6 CD 6 CD 36 CD None of these
59. IVIULTIPLE CHOICE Solve the equation —^-r + — = —%
X ~r l x z — 1 X 1
CD 1 CD 0 CD§ CDf
FUNCTION VALUES Evaluate the function when x = 0, 1, 2, 3, and 4.
(Lesson 4.8)
60. f(x) = 4a-
63. f(x) = -x 2
61. f(x) = — x + 9
64. fix) = x 2 - 1
62. fix) = 3x + 1
65. fix) = j
EVALUATING EXPRESSIONS Evaluate the expression.
(Lessons 8.1, 8.2)
66. 2 4 • 2 3 67. 6 3 • 6 _1 68 . (3 3 ) 2
69. (4 5 ) 0 70. 12“ 5 • 12 3 71. 5 2 • 5 1
RADICAL EXPRESSIONS Simplify the radical expression. (Lesson 9.3)
72. V50
73. V72
74. |vTT2
75. |V52
76. Vf28
77. ^V90
78. 3V63
79. JVl53
O
80. |Vl8
81. V27
82. |V500
83. |Vl47
Chapter 11 Rational Expressions and Equations
Maintaining Skills
OPERATIONS WITH FRACTIONS Evaluate the expression. Write
the answer as a fraction or mixed number in simplest form.
(Skills Review p. 764)
M 2 , 1 1
84. — + — — —
3 6 3
o* 3 5 _ 1
85 ‘ 4 + 8 2
2 3 1
“■5 + 8-i
2 14
87 - 9-3 + 5
8 o_L + 1_A + 2
88 ' 10 + 5 10 + 5
89 i+2_3 4
89 ‘ 4 + 4 4 + 4
__ 3 3,1
9 °' 17 34 + 2
1 3.5 7
91 ' 2-4 + 6"8
12 , 7 1
92 -T3 + 26 “ 2
QO 103 1 _ \__
93 “ 202 + 2 101
„ 7 , 1 2
94 ‘3 + 5 _ I5
5 4,3
95 -TT-5 + 4
Quiz 2
Multiply or divide. Simplify the expression. (Lesson 11.4)
5x 2 14x 2
lm 2x * lOx
2 .
10 + 4x
(20 + 8x)
3.
3x + 12
4x
x + 4
2 x
4.
5x 2 — 30x + 45
x + 2
- (5jc - 15)
Add or subtract. Simplify the expression. (Lessons 11.5 , 7 7.^)
5.---+-^- 6. 4x ~ 1 -- ^ -
x 2 — 2x — 35 x 2 — 2x — 35 3x 2 + 8 x + 5 3x 2 + 8x + 5
6 lx 3x 2 2x
' x 2 - 1 x + 1 " 3x — 9 x 2 -x-6
Solve the equation. Check your solution. (Lesson 11.7)
r*3_9
9 ' JC 2(JC + 2) 10 ' 2 + r t
11 ,
1
x — 5
+
1
x + 5
x + 3
x 2 — 25
16
x - 2
3
4
CANOEING In Exercises 13-15, use the following information.
You are on a canoe trip. You can paddle your canoe at a rate of x + 2 miles per
hour downstream and x — 2 miles per hour upstream. You travel 15 miles
downstream and 15 miles back upstream. (Lesson 11.6)
13. Write an expression for the travel time downstream and an expression for the
travel time upstream.
14. Write and simplify an expression for the total travel time.
15. Find the total travel time if x = 3.
16. RAKING LEAVES You can rake your neighbor’s yard in 3 hours. Your
neighbor can rake his yard in 4 hours. How long will it take you if you rake
the yard together? (Lesson 11.7)
11.7 Rational Equations
Goal
Perform operations on
rational functions.
Key Words
• rational function
Rational Functions
The inverse variation models you graphed in Lesson 11.2 are a type of rational
function. A rational function is a function that can be written as a quotient
of polynomials.
fix) =
polynomial
polynomial
In this lesson you will perform arithmetic operations on rational functions using
properties to combine and simplify functions.
Student HeCp
-N
► Reading Algebra
The function notation
f(x) is read “f of x."
I j
OPERATIONS ON FUNCTIONS
Let f and g be two functions. Each function listed below is defined
for all values of x in the domain of both fand g.
Sum of functions f and g
(f+ g)(x)
= f(x) + gix)
Difference of functions f and g
(f - g)(x)
II
—K
><.
1
><.
Product of functions f and g
if' g){x)
II
-K
><.
•
&
Quotient of functions f and g, if g(x) = 0
if+ g)(x)
II
><.
•1-
><.
mi
Add Rational Functions
Let fix) =
1
2 x + 2
and g(x) =
2 x + 2
. Find a rule for the function (/ + g)(x).
Solution
Q Write the rule for the sum of functions.
(/+ g)(x) = fix) + gix)
Q Substitute
1
2 x + 2
for /(x) and
2 x + 2
for g(x).
1
+
© Add.
© Factor the denominator.
© Divide out common factors and simplify.
ANSWER) (/+ g)(x) = ~
2 x + 2 2x + 2
1 + x
2x + 2
(1 +*)
2(x + 1)
iq+^) = i
2(lJ^x) 2
L_
Add Rational Functions
Find a rule for the function (f + g)(x).
1 -fix) = qA gix) = J;
2 . fix)
1
X —
3> =
1
x + 3
Chapter 11 Rational Equations and Functions
Student HeCp
^
► Study Tip
In Step 6, you must
factor the numerator
to determine whether
the numerator and
denominator have any
common factors.
v j
2 Subtract Rational Functions
Let f{x) = x + 2 anc * #(■*) = Fiud a m ^ e f° r function (/ — g)(x).
0 Write the rule for the difference (/ — g)(x) = f(x) — g(x)
of functions.
0 Substitute — ~ - for /(x) and ^ for g(x).
© Rewrite the expressions using the
LCD x(x + 2).
0 Simplify {he numerators.
0 Subtract
0 Factor the numerator.
answer t (/- g)(x) = 1}
X 1
x + 2 x
x(x)
l(x + 2)
x(x + 2)
x(x + 2)
x 2
x + 2
x(x + 2)
x(x + 2)
x 2 - x - 2
x(x + 2)
(x - 2)(x + 1)
x(x + 2)
Subtract Rational Functions
Find a rule for the function (f - g)(x).
3. /(*) = g(x) = \ 4 .f{x) = g(x) =
3 Multiply and Divide Rational Functions
Let f(x) = ——— and g(x) = - - j . Find a rule for the function.
a. (/ • g)(x) = fix) • g(x)
x - 4 x + 4
= - • -
x x — 4
_(x^4Xx + 4)
xipcjzJ^-
= x + 4
X
b- if ^ g)(x) = f( x ) + g(x)
_ x - 4 _ x + 4
x x — 4
x - 4 x - 4
= - • -
x x + 4
(x - 4) 2
x(x + 4)
Multiply and Divide Rational Functions
v 9 x -h 14
5. Let/(x) = and g(x) = - 5 - . Find a rule for the function (/ • g)(x).
X ~r / o
1 _ ^ _ 1
6. Let/(x) = —-— and g(x) = • Find a rule for the function (/ 4- g)(x).
Rational Functions
Exercises
SUMS Find a rule for the function (f + g)(x).
i-/(*) = yt!h)>$(*)= 9
3./(x) = ~~ff~ ’gO)
x — 9
6x
1
x
5. fix) = y, g(x)
5x
x ~\~ 1
2. fix) = ~ 2 - gix) = ~Y^~
x 2 - 25
4./(x)
x — 3
»g<»
X 2 - 25
7
6 - /(x) = %T’ g(x)
3 — x
X +4
15x
DIFFERENCES Find a rule for the function (f- g)(x).
7- fix) = 3jc 4 * g(x) = * 5
3x+ 7
9-/W = y, g(x) = 2X ^ 3
11 ■/(*)
1
x + 9
, gix)
x - 9
8-fix) =
x 2 - 36
, g(x) =
10 .fix)
X + 4
2 x
»g(*)
X 2 - 36
4
12-/(x) = g(x)
x — 2
3
2x — 6
PRODUCTS Find a rule for the function (f • g)(x).
13 -^ (x) = ^ (x) = 7TT5
14./(x) = T^T, g(x) = x+2
15 .f(x)
17. f(x)
5x + 6
2x
»g(*)
x 2 + 3x — 10
x + 2
, g(x)
3x - 6
x — 3
= X 2 - 4
x + 5
16 .f(x)
18. fix)
3x + 6
x + 2
x 2
,g(*)
X
8x
4X 2 - 16
x 2 — 3x + 2
x 2 + 3x + 2
, gix)
8x + 8
4x+ 8
QUOTIENTS Find a rule for the function (f 4- g)(x).
x+3 /x X + 1
19./(x) = ——, gix) =
21 . fix)
23. fix)
2 x +
y, gix)
X
2x + 1
20 . fix) = j- x , gix) = ±
x 2 + 3x — 10
x + 2
, gix)
X 2 - 4
x + 5
22 . fix)
24. fix)
2 x
x 3 - 5X 2
x 2 — x — 20
5x - 25
, gix) =
, gix)
10
x 2 — 5x
_ X — 1
x 2 — 25
Student HeCp
► Homework Help
Example 1: Exs. 1-6
Example 2: Exs. 7-12
Example 3: Exs. 13-24
*_ )
GRAPHING Graph the function by making a table of values, plotting the
points, and then connecting them with two smooth curves.
2.5. m =
26 ./ W = y -3
27. gix)
X
2x + 3
28. gix) = yy^y 1
Chapter 11 Rational Equations and Functions
Chaf/rif
Chapter Summary
and Review
/-
-\
• proportion, p. 633
• inverse variation, p. 639
• least common denominator
• extremes, p. 633
• rational number, p. 646
(LCD), p . 663
• means, p. 633
• rational expression, p. 646
• rational equation, p. 670
Proportions
Examples on
pp. 633-635
12 5
Solve the proportion — = — using the cross product property.
12 = 5
7 x
O l/Vr/fe the original proportion.
© Use the cross product property. 12 • x = 7 • 5
© Divide each side by 12.
35
12
Solve the proportion. Check your solutions.
v 4
1 .
2 7
o _Z_ _ 9 + v
10 “ v
3.
x 2 - 16 = x - 4
x + 4 3
v — 6
v + 6
Direct and Inverse Variation
Examples on
pp. 639-641
Assuming y = 4 when x = 8, find an equation that relates x and y in each case.
a. x and y vary directly.
b. x and y vary inversely.
Solution
y
a. — = k Write direct variation model.
JT
3
II
Write inverse variation model.
4
7T = k Substitute 8 for x and 4 for y.
o
(8)(4) = k
Substitute 8 for x and 4 for y.
2 — k Simplify.
32 = k
Simplify.
ANSWER t ^ i or y = jk.
ANSWER ^ xy
= 32 or y = — .
y X
Chapter Summary and Review
Chapter Summary and Review continued
Find an equation such that xand / vary directly.
5. y = 50 when x = 10 6. y = 6 when x = 24
8. y = 20 when x = 2 9. y = 7 when x = ^
Find an equation such that xand / vary inversely.
11 - y = 3 when x = 12 12. y = 10 when x = 20
2 11
14. v = 3 when x = — 15.); = — when x = 4
7. y = 36 when x = 45
10. y = 132 whenx = 66
13. y = 5 when x = 90
16. y = when x = 24
11.3
Simplifying Rational Expressions
Examples on
pp. 646-648
Simplify iXfrf-
To simplify a rational expression, look for common factors.
Solution
© Write the original expression.
© Factor the numerator and denominator.
© Divide out the common factor (x + 2).
© Simplify the expression.
2x 2 + 3x - 2
2x 2 + 5x + 2
(2x ~ l)(x + 2)
(2x + l)(x + 2)
(2x - 1 )C*^Z)
(2x + l)Cx-4''2)
2x - 1
2 x + 1
In Exercises 17-25, simplify the expression.
3x _ 6 x 2
17 .
20 .
23.
9x 2 + 3
5x 2 + 21x + 4
25x + 100
2x 2 + 17x + 21
2x 2 + x — 3
18.
21 .
24.
12x 4 + 18X 2
+ 4x + 4
x 2 + 9x + 14
13x 2 - 39x
3x 2 — 8x — 3
19.
22 .
25.
7x 3 - 28x
3x 2 + 8x + 4
6X 2 - 19x + 10
2X 2 — 5x
y 1 -2y- 48
2v 2 + 9y - 18
26. Find the ratio of the area of the smaller rectangle to the area of the larger
rectangle. Simplify the expression.
4(x + 3)
2(x +1)
Chapter 11 Rational Expressions and Equations
Chapter Summary and Review continued
Multiplying and Dividing Rational Expressions
Examples on
pp. 652-654
Simplify
6x 2 + x - 1 . 9x — 3
2 x + 1 x + 1
To divide rational expressions, multiply by the reciprocal.
6x 2 + x - 1 ^ 9x - 3 _ 6x 2 + x - 1 # x + 1
2x+l x + 1 2x+l 9x — 3
_ (2x + l)(3x — 1) # x + 1
2x + 1 3(3x — 1)
= (2x +J^(3x ^l)<x + 1)
3Gx ^
_ x + 1
” 3
Multiply by reciprocal.
Factor numerators and denominators.
Multiply and divide out common
factors.
Write in simplest form.
Simplify the expression.
27.
30.
12x 2 25x 4
5x 3
6y 2
3x
9y
y + 3 (y + 3) z
28.
31.
a 2 — la — 18 12
4 a 2 + 8 a a 2 — 81
9x 3 x - 8
X 3 - X 2 ' X 2 - 9x + 8
29.
32.
2x 2 + 9x + 7 16x 2
2x
3
X J — X
x 2 + 3x + 2 x 2 + 5x + 4
x 2 + lx + 12 x 2 + 5x + 6
Adding and Subtracting With Luce Denominators
Examples on
pp. 658-659
Simplify
2x + 6
x 2 + 2x — 8 x 2 + 2x — 8
_ 3x — 6
x 2 + 2x - 8
= 3 (^- 2 )
(x^2)(x + 4)
= 3
x + 4
Subtract numerators.
Simplify numerator.
Factor and divide out common factor (x - 2)
Write in simplest form.
5x
2x + 6
x 2 + 2x — 8 x 2 + 2x — 8 ’
5x - (2x + 6)
x 2 + 2x — 8
In Exercises 33-36, simplify the expression.
__ 2x + 1 . x + 5
33. —^-+
3x
3x
34.
-2b ~ 5
+
35.
6x _ 5x — 4
x + 4 x + 4
x(x ± 1)
36 - r.v - 3 ) 2
12
(x - 3) 2
37. Find an expression in simplest form for the perimeter of a rectangle whose
side lengths are X ^ 1 and
Chapter Summary and Review
Chapter Summary and Review continued
Adding and Subtracting With Unlike Denominators
Examples on
pp. 663-666
■JMgfiMI Sim P lif y 7^5-jh-
The LCD is (x — 5)(x + 2).
x _ 2 _ x(x + 2) _ 2(x — 5)
x — 5 x + 2 (x — 5)(x + 2) (x - 5)(x + 2)
x 2 + 2x _ 2x — 10
(x - 5)(x + 2) (x - 5)(x + 2)
_ (pc 2 + 2x) — ( 2x — 10)
(x — 5)(x + 2)
= x 2 + 10
(x — 5)(x + 2)
Rewrite fractions using LCD.
Simplify numerators.
Subtract fractions.
Simplify.
In Exercises 38-41, simplify the expression.
x +
3x — 1
+
39.
-5x ~ 10
x 2 - 4
+
4x
x — 2
P
P+ 1
41.
x 4
2x
x — 6
3x
42. Find an expression in simplest form for the perimeter of a rectangle whose
side lengths are
x + 3
x — 2
and
6
x + 4’
Rational Equations
Examples on
pp. 670-673
Solve the equation
2x
1 1
The LCD is 9x.
2x 1 1
9x • ^ - 9x • - = 9x • 4
9x3
2x 2 — 9 = 3x
2x 2 - 3x - 9 = 0
(2x + 3 )(jc - 3) = 0
Multiply each side of original equation by LCD 9x.
Simplify equation.
Write equation in standard form.
Factor left side of equation.
ANSWER ► When you set each factor equal to 0, you find that the solutions are x
Check your solutions back into the original equation.
and 3.
Solve the equation. Check your solutions.
* + 2 -4 „„ 1 _L J = I
43
44.i + ^
s s + 2
45.
1 + 1
1 x + 2 x 2 + x- 2
Chapter 11 Rational Expressions and Equations
Solve the proportion. Check your solutions.
1 .
17
5
_ x _ x + 8
Zm 4 ~ x
3.
10
4.
x 2 + 4
_4_
5x
Make a table of values for x = 1,2, 3, and 4. Use the table to sketch the
graph. State whether x and y vary directly or inversely.
5- y = 4x
6 -y
50
x
7.y
9
8.y
15
2x
Simplify the expression.
5 6x 6
9.
12 .
4a 4
2a - 14
3a 2 — 21a
10 .
13.
5 x 2 — 15x
15 a 4
A 2 - 1
2a 2 + a — 1
Write the product or quotient in simplest form.
15.
18.
6a 2 . ~4a 3
8a 2a 2
3a 2 + 6a . 15
16.
A 3 + A 2
A + 4
A 2 - 16 3a 4 + a 3 - 2a 2
4a
8a 2
19.
x 4- 3
x 2 -9
t 3 — x 2 — 6x
11 .
14.
x 2 — x — 6
x 2 -4
2x 2 + 12x + 18
x 2 — x — 12
17.
3x 2 — 6x
x 2 - 6x + 9
x 2 — x — 6
A 2 - 4
20 .
x 2 4- x — 2
Write the sum or difference in simplest form.
\2x — 4
21 . , +
24.
x — 1
4
+
4x
x — l
3x
x 4- 3 x — 2
22 .
25.
6(2); + 1) 2{5y ~ 7)
r
5x
100
4_
A 2
r
100
23.^j + ^~
2a 2 3a
26.
5x 4- 1
2x
x — 3 x — 1
Solve the equation. Check your solutions.
27.
30.
4x — 9
3
x_
3
5 2 3
28. f + -f = ~
9 9x x
1
u + 2 u — 2
31.
6
x
3
x
29.
32.
6
x
x
+
x + 1 x - 2
33. LENGTH AND WIDTH The length i and width w of a rectangle with an area
of 60 square units are related by the equation f w = 60. Does this model
represent direct variation , inverse variation , or neither ?
34. STREET SWEEPERS A town’s old street sweeper can clean the streets in
60 hours. The new street sweeper can clean the streets in 20 hours. How
long would it take the old sweeper and the new sweeper to clean the
streets together?
Chapter Test
Chapter Standardized Test
Tip
Think positively during a test. This will help you keep up
your confidence and enable you to focus on each question.
1, Which of the following is the solution of
4 6
the proportion = —-?
(A) -82 CD -41
CD 7 CD 41
2. The variables x and y vary inversely. When
x is 9, y is 36. If x is 3, what is y?
(A) 12
CD 108
CD 36
CD 324
3. What is the simplest form of the
. x 3 - lOx 2 + 9x„
expression ^ + _ 6 ?
V - 9
(A)
CD
x + 6
CD
x + 6
x(x - 9)
(x — l)(x + 6) x + 6
4. What is the simplest form of the product
9 x 2 16x 3 .
- • c '
4x x 5
(A) 36x
64
CD 36x 3
x 2 - 64
by (x - 8).
5. Divide
x + 8
x — 8
® 3x 2
® 3x 2
x + 8
x 3 - 512
® 3 jc
® 3x 2 (x - 8)
6 _ What is the simplest form of the sum
x + 2 3
X 2 - 25 + X 2 - 25 '
x + 5
(A)
CD
x 2 - 25
3x + 6
x — 5
CD
CD
x + 5
(x 2 - 25) 2
1
x — 5
7. What is the simplest form of the difference
2x + 9 x — 4 r
-?
x + 5 x — 2 *
x 2 + 6x + 2
(A)
(x + 5)(x - 2)
CD
x 2 + 6x — 38
(x + 5)(x - 2)
.— N x 2 + 4x - 38 .—. x 2 + 4x + 2
CD , - ... —^7 CD
(x + 5)(x — 2)
8 . Solve the equation
(x + 5)(x — 2)
—-b - = 1
x + 2 x
(A) -1
CD -2
CD 1, -6
CD -1,6
CD None of these
9, What is the ratio in simplest form of the
area of the red rectangle to the area of the
blue rectangle?
CDtt CD
1
6
x(x + 7)
6x(x + 7)
X
x + 7
3x+21
Chapter 11 Rational Expressions and Equations
Maintaining Skills
The basic skills you’ll
review on this page will
help prepare you for
the next chapter.
Simplify the expression
1 Simplify Radicals
63
100
using the quotient property.
Solution
[63
V 100
V63
Vioo
V9 ♦ 7
V100
V9 • V7
VToo
3V7
10
l_
Write original expression.
Use quotient property.
Factor using perfect square factors.
Use product property.
Simplify.
Try These
Simplify the expression using the quotient property.
2 Factor Perfect Squares
Factor x 2 + 16x + 64.
Solution Recall from Chapter 10 the pattern for factoring a perfect square
trinomial: a 2 + lab + b 2 = (a + b ) 2 or a 2 — lab — b 2 = (a — b ) 2 .
x 2 + 16x + 64 = x 2 + 2(x)(8) + 8 2 Write as o 2 + lab + b 2 .
= (x + 8) 2 Factor using pattern.
Student HeCp
► Extra Examples
More examples
and practice
exercises are available at
www.mcdougallittell.com
l J
Try These
Factor the trinomial.
9- a 2 — 18 a + 81
12 . 169 + 16m + m 2
15. 4x 2 + 20x + 25
10 . x 2 + 6x + 9
13. 225 + 30 r + r 2
16. 9b 2 - 6a + 1
11 . y 2 - 22y + 121
14. 100 - lOt + t 2
17. 16 — 56x + 49x 2
Maintaining Skills
j
Radicals and More
Connections to Geometry
I How are passengers kept in place on
a spinning amusement ride?
Learn More About It
You will calculate the centripetal force exerted on a
rider in Example 5 on p. 706.
APPLICATION LINK More about amusement park rides is
available atwww.mcdougallittell.com
APPLICATION: Spinning Rides
Some amusement park rides spin so fast that
the riders "stick" to the walls of the ride. The force
exerted by the wall on the rider is called centripetal
force. You'll learn more about calculating centripetal
force in Chapter 12.
Think & Discuss
When designing spinning rides, engineers must calculate
the dimensions of the ride as well as how many times
per minute it will spin. The table shows the height and
revolutions per minute for four spinning rides.
Ride name
Height
(feet)
Revolutions
per minute
Football Ride
34.4
15
Chaos
36
12
Centrox
44.3
17.5
Galactica
44.3
17
Based on the numbers in the table, is revolutions
per minute a function of height? Explain.
You are designing a spinning ride that is 40 feet
high. Use the information in the table to decide on
a reasonable range for how many revolutions per
minute the ride would make.
Study Guide
PREVIEW
What’s the chapter about?
• Solving radical equations and graphing radical functions
• Applying the Pythagorean theorem
• Proving theorems by using algebraic properties and logical reasoning
Key Words
• square root function,
p. 692
• extraneous solution,
p . 70S
• rational exponent,
p. 711
• completing the square,
p. 716
• theorem, p. 724
• Pythagorean theorem,
p. 724
• hypotenuse, p. 724
• legs of a right triangle,
p. 724
• converse, p. 726
• distance formula, p. 730
• midpoint, p. 736
• midpoint formula,
p. 736
• postulate, p. 740
• axiom, p. 740
• conjecture, p. 741
• indirect proof, p. 742
PREPARE
Chapter Readiness Quiz
STUDY TIP
Take this quick quiz. If you are unsure of an answer, look back at the
reference pages for help.
Vocabulary Check (refer to p. 512)
1 _ Which is the simplest form of the radical expression
V36
V9 •
(a) —f CD V2
CD 2
Skill Check ( refer to pp. 511, 596)
2. Which is the simplest form of Vl40?
Ca) 2V35 CD 4V35 CD 10 V 7 CD 14V5
3. Which of the following is the correct factorization of the trinomial
x 2 — 3x — 18?
(A) (x + 3)(x + 6)
C© (x ~ 3)(x - 6)
Cb) (x + 3)(x — 6)
Co) (x — 3)(x + 6)
Draw Diagrams
Including a diagram or
another visual aid when you
take notes can be helpful.
Chapter 12 Radicals and More Connections to Geometry
DEVELOPING CONCEPTS
For use with
Lesson 12.1
Goal
Use a function's graph to
determine its domain and
range.
Materials
• graph paper
• pencil
Question How do you determine the domain and range of functions
^ ~~~~ with radicals?
A function’s graph can provide a representation of the domain and range. Recall
that when a function is given by a formula, its domain is all possible input values.
The range of a function is the set of output values.
Explore
© Copy and complete the table of values for the function y = Vx.
Round to the nearest tenth.
For what values of x is Vx not defined?
© For those values of x for which Vx is
defined, plot the points from the table
on a piece of graph paper and
connect them with a smooth curve.
© The table of values suggests that the domain of the function is the set of all
nonnegative real numbers. You can verify this observation as follows: (1) The
square root of a negative number is not defined. (2) The square root of any
nonnegative real number is defined. The range is the set of all nonnegative
real numbers because every nonnegative real number is the square root of
its square.
Think About It
Use the formula for y to identify the
Explain your reasoning.
1 .y = Vx + 1
domain and range of the function.
Developing Concepts
Functions Involving Square
Roots
Goal
Evaluate and graph a
function involving
square roots.
Key Words
• square root function
• domain
• range
How fast can a dinosaur walk?
The maximum walking speed of a
dinosaur is a function of the length
of its leg. In Exercise 57 you will use
a function involving a square root to
compare the maximum walking
speeds for two species of dinosaurs.
The square root function is defined by the equation
y = V£.
Its domain is all nonnegative numbers, and its range is all nonnegative numbers.
Understanding the square root function will help you work with other functions
involving square roots.
Student HeCp
► Study Tip
Recall that the square
root of a negative
number is undefined.
Vx can be evaluated
only when x> 0.
v j
ESSU *1 Evaluate Functions Involving Square Roots
Find the domain of y = 2Vx. Use several values in the domain to make
a table of values for the function.
Solution
A square root is defined only when
the radicand is nonnegative. Therefore
the domain of y = 2 Vx consists of
all nonnegative numbers. A table of
values for x = 0, 1, 2, 3, 4, and 5 is
shown at the right.
0
o
II
<N
II
1
<N
II
■>
<N
II
2
y = 2V2 = 2.8
3
y = 2 V 3 « 3.5
4
y = 2V4 = 4
5
y = 2V5 = 4.5
Evaluate Functions Involving Square Roots
Find the domain of the function. Then use several values in the domain
to make a table of values for the function.
1 . y = Vx 2. y = 3\/x 3 - y = \flx 4 .y — \fx — 1
Chapter 12 Radicals and More Connections to Geometry
It is a good idea to find the domain of a function before you make a table of
values. This will help you choose appropriate values of x for the table.
Student HeCp
► More Examples
More examples
are available at
www.mcdougallittell.com
2 Graph y = 2 Vx
Sketch the graph of y = 2Vx. Then find its range.
Solution
From Example 1, you know
the domain is all nonnegative
real numbers. Use the table
of values from Example 1.
Then plot the points and
connect them with a smooth
curve. The range is all
nonnegative real numbers.
Student HeCp
► Study Tip
When you make a
table of values to
sketch the graph of a
function, choose
several values to see
the shape of the curve.
V _ )
3 Graph y = Vx -H
Find the domain of y = Vx + 1. Then sketch its graph and find the range.
Solution
The domain is the values of x for which the radicand is nonnegative, so the
domain consists of all nonnegative real numbers. Make a table of values, plot
the points, and connect them with a smooth curve.
0
y = Vo +1 = 1
1
y = V1+1=2
2
y = V 2 + 1 = 2.4
3
y = V 3 + 1 = 2.7
4
y = V4 + 1 = 3
5
y = V5 + 1 =3.2
The range is all real numbers that are greater than or equal to 1.
Graph Functions Involving Square Roots
Find the domain of the function. Then sketch its graph and find the range.
5. y = — 3Vx 6- y = —2 Vx 7. y = Vx + 2
8. y = Vx — 2 9. y = 3 — Vx 10- y = 2Vx + 1
12.1 Functions Involving Square Roots
Forked Lunate
Truncate Rounded
FISH TAILFINS, also called
caudal fins, help fish swim
and steer. The speed at
which a fish moves through
the water is affected by the
size of the fish tailfin.
— 1 < Graphy = Vx-^3
Find the domain of y = Vx — 3. Then sketch its graph.
Solution
To find the domain, find the values of x for which the radicand is nonnegative,
x — 3 > 0 Write an inequality for the domain,
x > 3 Add 3 to each side.
The domain is all numbers that are greater than or equal to 3. Make a table of
values, plot the points, and connect them with a smooth curve.
= V3 - 3 = 0
= V4 - 3 = 1
= V5 - 3 = 1.4
= V6 - 3 = 1.7
= V7 - 3 = 2
= V8 - 3 = 2.2
5 Use a Square Root Model
FISH TAILFINS The tailfin height h of a tuna can be modeled by h = V7.5A
where A is the surface area of the tailfin. Sketch the graph of the model.
Solution
The domain is all nonnegative numbers. Make a table of values, plot the points,
and connect them with a smooth curve.
0
h = V7.5 -0=0
1
h = V7.5 -1 = 2.7
2
h = V7.5 • 2 = 3.9
3
h = V7.5 -3 = 4.7
4
h = V7.5 -4 = 5.5
Graph Functions Involving Square Roots
11. The tailfin height h of a bottom-dwelling fish can be modeled by h = V0.6A,
where A is the surface area of the tailfin. Sketch the graph of the model.
Chapter 12 Radicals and More Connections to Geometry
VIA Exercises
Guided Practice
Vocabulary Check
1. Describe the square root function.
Skill Check
2. Complete: Finding the ? of a square root function helps you choose
appropriate input values of x for a table of values.
Evaluate the function for x = 0, 1, 2, 3, and 4. Round your answers to the
nearest tenth.
3 .y = 4Vx
6. y = 6Vx — 3
4. y = — Vjc
7. y = Vx + 2
5. y = 3Vx + 4
8.y = V4x - 1
Find the domain and the range of the function.
9. y = 5Vx 10- y — Vx
12. y = Vx + 6 13. y = Vx + 5
11 . y = Vx - 10
14. y = Vx ~ 10
Find the domain of the function. Then sketch its graph.
15. y = 4Vx 16. y = Vx + 5 17. y = 3 Vx + 1
FIRE HOSES In Exercises 18 and 19, use the following information. For a
particular fire hose, the flow rate /(in gallons per minute) can be modeled by
/ = 120Vp, where p is the nozzle pressure in pounds per square inch.
18. Find the domain of the flow rate model. Then sketch its graph.
19. If the nozzle pressure is 100 pounds per square inch, what is the flow rate?
Practice and Applications
Student HeCp
► Homework Help
Example 1: Exs. 20-39
Example 2: Exs. 40-55
Example 3: Exs. 40-55
Example 4: Exs. 40-55
Example 5: Exs. 56-59
L _/
EVALUATING FUNCTIONS Evaluate the function for the given value of x.
20. y = TVx\ 9 21. y = —2 Vx; 25 22. y = V32x; 2
23. y = V3x; 12 24. y = Vx + 4; 4 25. y = 10 - Vx; 16
26. y = Vx — 7; 56 27. y = V3x — 5; 7 28. y = V21 — 2x ; —2
FINDING THE DOMAIN Find the domain of the function. Then use several
values in the domain to make a table of values for the function.
29. y = 6Vx 30. y = Vx — 17 31. y = V3x — 10
32. y = Vx + 1 33. y = 4 + Vx 34. y = Vx - 3
35. y = Vx + 9 36. y = 2V4x 37. y = xVx
12.1 Functions Involving Square Roots
INVESTIGATING ACCIDENTS In Exercises 38 and 39, use the following
information. When a car skids to a stop, its sp eed S (in miles per hour) before
the skid can be modeled by the equation S = V30 df, where d is the length of the
tires’ skid marks (in feet) and/is the coefficient of friction for the road.
38. In an accident, a car makes skid marks that are 120 feet long. The coefficient
of friction is 1.0. What can you say about the speed the car was traveling
before the accident?
39. In an accident, a car makes skid marks that are 147 feet long. The coefficient
of friction is 0.4. A witness says that the driver was traveling under the speed
limit of 35 miles per hour. Can the witness’s statement be correct? Explain
your reasoning.
Link t o
Careers
PALEONTOLOGISTS Study
fossils of animals and plants
to better understand the
history of life on Earth.
More about
' paleontologists at
www.mcdougallittell.com
GRAPHING FUNCTIONS Find the domain of the function. Then sketch its
graph and find the range.
40. y = 7Vx
41. y =
4Vx
42. y= 5Vx
43. y= 6Vx
44. y = V3x
45. y =
—\flx
46. y = Vx + 4
47. y = Vx — 3
48. y = 5 — Vx
49. y =
6 — Vx
50. y = 2Vx + 3
51. y = 5Vx — 2
52. y = Vx - 4
53. y =
Vx + 1
54. y = V3x + 1
55. y = 2V4x + 10
DINOSAURS In Exercises 56 and 57, use the following information.
In a natural history museum you see leg bones for two species of dinosaurs
and want to know how fast they walked. The maximum walking speed S
(in feet per second) of a dinosaur can be modeled by the equation below,
where L is the length (in feet) of the dinosaur’s leg. ►Source: Discover
Walking speed model: S = V32L
56. Find the domain of the walking speed model. Then sketch its graph.
57. For one dinosaur the length of the leg is 1 foot. For the other dinosaur the
length of the leg is 4 feet. How much faster does the taller dinosaur walk than
the shorter dinosaur?
CHALLENGE In Exercises 58 and 59, use the
following information. The lateral surface area S
of a cone whose base has radius r can be
found using the formula
S = 7T • rVr 2 + h 2
where h is the height of the cone.
58. For r = 14 and h > 0, sketch the graph
of the function.
59. Find the lateral surface area of a cone that has a height of 30 centimeters
and whose base has a radius of 14 centimeters.
Chapter 12 Radicals and More Connections to Geometry
Standardized Test
Practice
Mixed Review
Maintaining Skills
3
60. CRITICAL THINKING Find the domain of y = 77=—-
J \x - 2
61. MULTIPLE CHOICE Which function best represents the graph?
(A) y — 2Vx — 3
® y = V2x — 3
eg) y = V2x — 3
CD) = 2Vx" — 3
Ce) None of these
SIMPLIFYING
62. V24
Simplify the radical expression. (Lesson 9.3)
63. V60 64. Vl75 65. V360
67. |V80 68. ^=- 69.4
SOLVING EQUATIONS Use the quadratic formula to solve the equation.
If the solution involves radicals, round to the nearest hundredth.
(Lesson 9.6)
70. x 2 + 4x - 8 = 0 71. x 2 - 2x - 4 = 0 72. x 2 - 6x + 1 = 0
73. x 2 + 3x - 1 = 0 74. 2X 2 + x - 3 = 0 75. 4X 2 - 6x + 1 = 0
MULTIPLYING EXPRESSIONS Find the product. (Lesson 10.2)
76. (x - 2)(x + 11) 77. (x + 4)(3x - 7) 78. (x - 5)(x - 4)
79. (2x — 3)(5x — 9) 80. (6x + 2)(x 2 — x — l) 81. (2x — l)(x 2 + x + l)
82. MOUNT RUSHMORE Carved on Mount Rushmore are the faces of four
Presidents of the United States: Washington, Jefferson, Roosevelt, and
Lincoln. The ratio of each face on the cliff to a scale model is 12 to 1. How
tall is Washington’s face on Mount Rushmore if the scale model is 5 feet tall?
(Lesson 11.1)
MULTIPLYING RATIONAL EXPRESSIONS Write the product in simplest
form. (Lesson 11.4)
8v 1
— •
3
9
16x
85.
x
x + 6
x + 6
x + 1
AREA Find the area of a triangle with the given base and height.
(Skills Review p. 774)
86. b = 4, h = 9
89.b = 6,h = S
92. b = 0.75, h = 4
87. b = 1, h = 1
90.b = S,h = 3
93. b = 0.85, h = 0.62
88. b = 12, h = 9
91. b = 10, h = 7
94. b = 0.25, h = 1.75
12.1 Functions Involving Square Roots
Operations with Radical
Expressions
Goal
Add, subtract, multiply,
and divide radical
expressions.
Key Words
• simplest form of a
radical expression
How far can you see to the horizon?
The distance you can see to Earth’s
horizon depends on your eye-level
height. In Example 4 you will compare
the distance you can see to the distance
a friend can see when you are at
different heights on a schooner’s mast.
You can use the distributive property to simplify sums and differences of radical
expressions when the expressions have the same radicand.
SUM: V2 + 3V2 = (1 + 3) V2 = 4V2
DIFFERENCE: V2 — 3 V2 = (1 — 3) V2 = — 2^2
In part (b) of Example 1, the first step is to identify a perfect square factor in the
radicand, as you learned on page 511.
Student MeCp
p Look Back
For help simplifying
radical expressions,
see pp. 511-512.
L _/
^^3331 1 Add and Subtract Radicals
Simplify the radical expression.
a. 2V2 + V5 — 6V2 = ( 2 V 2 — 6 V 2 ) + V5 Group radicals having the
same radicand.
= -4V2 + V5 Subtract.
b. 4\/3 — V27 = 4V3 — V9 • 3
= 4V3 - V9 • V3
= 4V3 - 3V3
= V3
Factor using perfect square factor.
Use product property.
Simplify.
Subtract.
Add and Subtract Radicals
Simplify the radical expression.
1.V3 + 2V3 2.3V5-2V5
4. V8 - V2 5. Vl8 + V2
3. V7 + V2 + 3V7
6 . 5V3 - Vl2
Chapter 12 Radicals and More Connections to Geometry
Student HeCp
► Study Tip
As you can see in part
(c) of Example 2, the
product of two radical
expressions having the
sum and difference
pattern has no radical.
In general,
(a + Vb)(a - Vb) =
a 2 - b.
\ _/
J 2 Multiply Radicals
Simplify the radical expression.
a. V2 • V8 = Vl6 = 4
b. V2(5 — V3) = V2 • 5 — V2 • V3
= 5V2-V6
c. (2 + V3)(2 - VJ) = 2 2 - (V3) 2
=4-3=1
Use product property and simplify.
Use distributive property.
Use product property.
Use sum and difference pattern.
Evaluate powers and simplify.
Multiply Radicals
Simplify the radical expression.
7. V3 • Vl2 8 . V5(V2 + l)
9. (V2 + l)(V2 - l)
To simplify expressions with radicals in the denominator, you may be able to
rewrite the denominator as a rational number without changing the value of the
expression, as you learned on page 512.
Student MeGp
► Study Tip
Multiplying the
fractions in Example 3
V5 , 2 +V3 .
by vT and 7TW ls
justified since both are
equivalent to 1.
v _ J
B2ZEH3I 3 Radicals
Simplify the radical expression.
I
V 5
V5 V5
Multiply by
3V5
V5 • V5
Multiply fractions.
3V5
5
Simplify perfect square.
b.
1
2 -13
1 2 + V 3
2-V3 *
Use the fact that the product (o + Vb)(o - Vb)
2 + Vb
does not involve radicals: multiply by ——
_ 2 + V3
(2 - V3)(2 + V3)
2 + V3
_ 2 2 - (V3) 2
= 2 + V3
Multiply fractions.
Use sum and difference pattern.
Evaluate powers and simplify.
Simplify Radicals
Simplify the radical expression.
10 .
1
V2
11 .
Vl8
V2
12 .
3 - V2
13.
11
5 + V3
12.2 Operations with Radicai Expressions
TTy
4 Use a Radical Model
SAILING You and a friend are working on a schooner. The distance d (in miles)
you can see to the horizon can be modeled by the equation
where h is your eye-level height (in feet) above the water. Your eye-level height
is 32 feet and your friend’s eye-level height is 18 feet. Write an expression that
shows how much farther you can see than your friend. Simplify the expression.
Not drawn to scale
Solution
Verbal
Difference
Your
Your friend’s
Model
in distances
distance
distance
Labels
I
Algebraic
Model
Difference in distances = D
Your distance =
Your friend’s distance
3 ( 18 )
(miles)
(miles)
(miles)
D=f- f>
3 ( 18 )
D
= V48 -
- V 27
D
= Vl6 •
3 - V9 • 3
D
= 4V3 -
- 3V3
D
= V3 «
1.7
You can
see about 1.7 miles i
Write algebraic model.
Simplify.
Factor using perfect square factors.
Use product property and simplify.
Subtract like radicals.
(§2l33 U se ° Radical Model
14. Your eye-level height is 16 feet and your friend’s eye-level height is 20 feet.
Write an expression showing how much farther your friend can see than you.
Chapter 12 Radicals and More Connections to Geometry
fefei Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Complete: In the expression “3 Vt\ 2 is called the ? .
4
2. Which of the following is the simplest form of the radical expression ^T?
D.
Vn
3
Simplify the expression.
3. 4 + V5 + 5V5
6 . V3 • V8
9 - w
4. 3V7 - 2V7
7. (3 + V7) 2
5. 3V6 + V24
8 . V3(5V3 - 2V6)
11 .
VTo
12, SAILING In Example 4 on page 700, suppose your eye-level height is
24 feet and your friend’s is 12 feet. Write an expression that shows how
much farther you can see than your friend. Simplify the expression.
Practice and Applications
ADDING AND SUBTRACTING RADICALS Simplify the expression.
13. 5V7 + 2V7
16. 2 V6 - V6
19. V32 + V2
22 . V72 - Vl8
14. V3 + 5V3
17. 4V5 + V3 + V5
20 . V75 + V3
15. llV3 - 12V3
18. 3VTT - V5 + VTT
21 . V80 - V45
23. 4V5 + Vl25 + V45 24. V24 - V% + V6
Student MeCp
► Homework Help
Example 1: Exs. 13-24,
53
Example 2: Exs. 25-40
Example 3: Exs. 41-52,
54
Example 4: Exs. 55, 56
v _ J
MULTIPLYING RADICALS Simplify the expression.
25. V3 • V75 26. Vl6 • V4 27. Vl8 • V5
28. V5 • V8 29. V6(V6 - l) 30. V6(7V3 + 6)
31.V5(4 + V5) 32. V2(V8-4) 33. V3(5V2 + V3)
MULTIPLYING RADICALS Simplify the expression using the sum and
difference pattern.
34. (V2 + 6)(V2 - 6) 35. (l + Vl3)(l - Vl3)
36. (V2 + V3)(V2 - V3) 37. (V7 + V2)(V7 - V 2 )
12.2 Operations with Radicai Expressions
38.
Find the area. (See the Table of Formulas on page 798.
Student HeCp
► Homework Help
Extra help with
1 ^ 0 ^ problem solving in
Exs. 41-52 is available at
www.mcdougallittell.com
V17
V68
SIMPLEST FORM Simplify the radical expression.
41.
45.
49.
V7
VTo
V3
1
2 + V2
42.
46.
50.
V2
43 ‘ V48
44.
Vl3
V3
48.
9
V7
47 ‘ 6 + V3
5 - V7
6
51 ^
52.
V3
10 + V2
51 -3 —V5
V3 - 1
ERROR ANALYSIS In Exercises 53 and 54, find and correct the error.
53.
54.
55. POLE-VAULTING A pole-vaulter’s approach velocity v (in feet per second)
and height reached h (in feet) are related by the following equation.
Pole-vaulter model: v = 8 Vh
Suppose you are a pole-vaulter and reach a height of 20 feet and your opponent
reaches a height of 16 feet. Write an expression that shows how much faster
you ran than your opponent. Simplify the expression and round your answer to
the nearest hundredth.
56. Science Ltnk % Many birds drop clams or
other shellfish in order to break the shell and
get the food inside. The time t (in seconds) it
takes such an object to fall a certain distance
d (in feet) is given by the following equation.
Vd
1 4
A gull drops a clam from a height of 50 feet.
A second gull drops a clam from a height
of 32 feet. Write an expression that shows
the difference in the time that it takes for
the two clams to reach the ground. Simplify
the expression.
Chapter 12 Radicals and More Connections to Geometry
Standardized Test
Practice
Mixed Review
Maintaining Skills
57. MULTIPLE CHOICE Simplify V5(6 + V5).
(A) V30 + 5 CD5V6 + 5 (© 6V5 + 5 CD llV5
58. MULTIPLE CHOICE Which of the following is equal to the difference
V3 - 5V9?
C© V3 - 15 (© -4V3 CED V3 — 3 GD 3 + 2V5
59. MULTIPLE CHOICE Simplify
5 — V2 ’
(A)
15 + 3V2
23
CD
15 + 3V2
25
CD
15 + V6
23
CD
15 + V6
25
PERCENTS Solve the percent problem. (Lesson 3.9)
60- What is 30% of 160? 61 - 105 is what percent of 240?
62. 203 is what percent of 406? 63. What is 70% of 210?
SOLVING QUADRATIC EQUATIONS Solve the equation by factoring.
(Lesson 10.5)
64. x 2 — 25 = 0 65. x 2 + 2x — 15 = 0 66 . x 2 — I3x = —42
67. x 2 — 26 = llx 68 . — 9x + 4 = —lx 2 69. 2 + 3x 2 = —5x
CROSS PRODUCT PROPERTY Solve the equation using the cross product
property. Check your solutions. (Lesson 11.7)
„ 2 1 __ 6 7 „ 7 2
70.—— =- 7 71-- =-f 72 .——- = - 7
x + 3 x — 6 x x — 5 x + A x — 6
FINDING THE DOMAIN Find the domain of the function. Then use several
values in the domain to make a table of values for the function.
(Lesson 12.1)
73. y = Vx - 3 74. y = Vx + 4 75. y = 6\Tx
76. y = 11 Vx 77. y = Vx + 3 78. y = Vx - 8
COMPARING PERCENTS AND DECIMALS Complete the statement using
<, >, or =. (Skills Review pp. 768, 770)
79.40% ? 0.35
82.0.22 ?) 20%
85.0.3 ? 33%
88.5% ? 0.5
91.101% ?H.l
94.2.25 ? 250%
80.110% ? 110
83.200% ?>1.0
86.0.75 ? 85%
89. 1.5 ? 150%
92.20% ? 0.25
95.80% ? 1.8
81. 1.8 % 180%
84. 12% ? 1
87. 1% ? 0.1
90.0.9 ? 89%
93.0.66 ? 60%
96. 100% ? 1.0
12.2 Operations with Radicai Expressions
Solving Radical Equations
Goal
Solve a radical equation.
Key Words
• radical
• extraneous solution
What is the nozzle pressure of a de-icing hose?
The nozzle pressure of a hose is a
function of the flow rate of the hose
and the diameter of the nozzle. In
Exercises 37 and 38 you will use an
equation involving radicals to find
the nozzle pressure of a hose used
to de-ice an airplane.
In solving an equation involving radicals, the following property can be useful.
SQUARING BOTH SIDES OF AN EQUATION
If a = b, then a 2 = b 2 , where a and b are algebraic expressions.
Example: Vx + 1 = 5, so x + 1 = 25.
) 1 Solve a Radical Equation
a. Solve Vx — 7 = 0.
Solution
a. Vx - 7 = 0
Vx = 1
(Vx) 2 = 7 2
x = 49
b. Solve 3Vx + 4 =15.
Write original equation.
Isolate the radical expression on one side of the equation.
Square each side.
Simplify.
ANSWER ^ The solution is 49. Check the solution in the original equation.
b. 3Vx + 4 = 15
Vx + 4 = 5
(Vx + 4 ) 2 = 5 2
x + 4 = 25
x = 21
Write original equation.
Divide each side by 3.
Square each side.
Simplify.
Subtract 4 from each side.
ANSWER ► The solution is 21. Check the solution in the original equation.
Chapter 12 Radicals and More Connections to Geometry
2 Solve a Radical Equation
3+4 = 5, you need to isolate the radical
To solve the equation v2x
expression first.
Q Write the original equation.
© Subtract 4 from each side of the equation.
© Square each side of the equation.
© Simplify the equation.
© Add 3 to each side of the equation.
o Divide each side of the equation by 2.
V2x - 3 +4 = 5
(Vz
V2x - 3 = 1
3) 2 = l 2
2x - 3 = 1
2x = 4
x = 2
ANSWER ► The solution is 2. Check the solution in the original equation.
Solve a Radical Equation
Solve the equation.
1- Vx = 3
4. Vn + 1 = 1
2 . Vm" —4 = 0
5. Vx - 4 + 5 = 11
3. Vx — 6 = 4
6 . V3n + 1 -3 = 1
EXTRANEOUS SOLUTIONS Squaring both sides of an equation can introduce a
solution to the squared equation that does not satisfy the original equation. Such a
solution is called an extraneous solution. When you solve by squaring both sides
of an equation, check each solution in the original equation.
Student HeCp
3 Check for Extraneous Solutions
► More Examples
More examples
W* are available at
www.mcdougallittell.com
Vx + 2 = x
(Vx + 2) 2 = x 2
x + 2 = x 2
Solve Vx + 2 =x and check for extraneous solutions.
Solution
Q Write the original equation.
0 Square each side of the equation.
© Simplify the equation.
© Write the equation in standard form.
© Factor the equation. (x — 2)(x + 1) = 0
© Use the zero-product property to solve for x. x = 2 or x = — 1
CHECK / Substitute 2 and — 1 in the original equation.
V2 + 2 1 2 V-l + 21-1
2 = 2 / 1 # -1
ANSWER ► The only solution is 2, because x = — 1 does not satisfy the
original equation.
12.3 Solving Radical Equations
4 Check for Extraneous Solutions
Solve Vx + 13 = 0 and check for extraneous solutions.
Solution
Vx + 13 = 0
Vx = -13
Write original equation.
Subtract 13 from each side.
(Vx) 2 = (— 13) 2 Square each side.
x = 169 Simplify.
ANSWER ^ Vl69 + 13 ^ 0, so x = 169 is not a solution. The equation has no
solution because Vx > 0 for all values of x.
Check for Extraneous Solutions
Solve the equation. Check for extraneous solutions.
7. Vx+ 6 = x 8- x = V8 — 2x 9- Vn + 4 = 0
Link to
Science
CENTRIPETAL FORCE
keeps you spinning in a circle
on an amusement park ride.
Forces can be measured
in newtons. A force of one
newton will accelerate a
mass of one kilogram at one
meter per second per second.
5 Use a Radical Model
CENTRIPETAL FORCE The centripetal force F exerted on a passenger by a
spinning amusement park ride and the number of seconds t the ride takes to
complete one revolution are related by the following equation.
t
162Q7T 2
F
Find the centripetal force experienced by this person if t = 10.
Solution
t =
10 2 =
' 16207T 2
100 =
46207V
162071 2
F
16207T 2
100
160
Write model for centripetal force.
Substitute 10 for tand square each side.
Simplify.
Solve for F.
ANSWER ► The person experiences a centripetal force of about 160 newtons.
Use a Radical Model
10. Find the centripetal force exerted on the passenger in Example 5 if the
amusement park ride takes 11 seconds to complete one revolution.
Chapter 12 Radicals and More Connections to Geometry
3 Exercises
Guided Practice
Vocabulary Check
Skill Check
3 . 8 = Vx
6 . Vx = — 7
9 . Vx + 6 = 0
12 . x = Vx + 12
a radical equation is.
an extraneous solution is.
ion. Check for extraneous solutions.
4 . Vx = 11
II
in
7 . 6 = Vx
8 . Vx = 1
10 . Vx - 20 = 0
11 . Vix -1=3
13 . -5 + Vx = 0
14 . x — V5x + 24
= 12 16 . VAx + 5 = x
17 . \/x + 6 = jc
Practice and Applications
SOLVING RADICAL EQUATIONS Solve the equation.
18 . Vx - 9= 0
21 . Vx - 10 = 0
24 . V6x — 13 = 23
19 . Vx — 1=0
22 . Vx - 15 = 0
20 . Vx - 5 = 0
23 . Vx - 16 = 0
25 . V4x + 1 + 5 = 10 26 . V9 - x - 10 = 14
27 . V5x + 1 + 2 = 6 28 . V6x - 2 - 3 = 7 29 . 4 = 7- V33x - 2
30 . 4V3x + 3 = 24
31 . V2x + 4+1 = 11 32 . 8Vx + 3 = 64
ERROR ANALYSIS In Exercises 33 and 34, find and correct the error.
34 .
Student HeCp
► Homework Help
Example 1: Exs. 18-36
Example 2: Exs. 18-36
Example 3: Exs. 40-54
Example 4: Exs. 40-54
Example 5: Exs. 37, 38,
55, 56
Geometry Linkfr Find the value of x.
35 . Perimeter = 30 36 . Area = 88
6 /
/ r
_ c
V5x-2
Vx + 6
12.3 Solving Radical Equations
Student HeCp
► Homework Help
Extra help with
-^0y p ro | 3 | em solving in
Exs. 37-38 is available at
www.mcdougallittell.com
PLANE DE-ICING In Exercises 37 and 38, use the following information.
You work for a commercial airline and remove ice from planes. The relationship
among the flow rate r (in gallons per minute) of the antifreeze for de-icing, the
nozzle diameter d (in inches), and the nozzle pressure P (in pounds per square
inch) is shown in the diagram. You want a flow rate of 250 gallons per minute.
37. Find the nozzle pressure
P for a nozzle whose
diameter is 1.25 inches.
38. Find the nozzle pressure
P for a nozzle whose
diameter is 1.75 inches.
Nozzle diameter d
r=30rf 2 V/>
__ f
Flow rate r
l
Nozzle pressure P
t
39. MATHEMATICAL REASONING Write a radical equation that has a
solution of 18.
CHECKING SOLUTIONS Solve the equation. Check for extraneous
solutions.
40.
43.
46.
49.
52.
^l"
II
m
1
41. Vx — 6 = 0
42. Vx + 5 = 1
6 + V3x = -3
44. Vx + 5 = 7
45. V5x + 10 = -5
Vx +11 = 1
47. x = Vx + 42
48. Vx - 5 = 20
1
II
O
1
50. 3Vx = -21
51.x = V2x + 3
00
T
II
<N
53. x = V —X + 12
54. 2 Vx + 7 = 19
HAMMER THROWING is
a sports event in which
athletes throw a metal ball,
called a hammer, as far as
possible. A hammer weighs
about 16 pounds and is
connected to a handle by a
steel wire.
SPORTS In Exercises 55 and 56, use the following information.
During the hammer throw event, a hammer is swung around in a circle several
times until the thrower releases it. As the hammer travels in the path of the circle,
it accelerates toward the center. This acceleration is known as centripetal
acceleration. The speed s that the hammer is thrown can be modeled by the
formula s = Vl.2a, where a is the centripetal acceleration of the hammer prior
to being released.
55. Find the approximate
centripetal acceleration
(in meters per second per
second) when the ball is
thrown with a speed of
18 meters per second.
56. Find the approximate
centripetal acceleration
(in meters per second per
second) when the ball is
thrown with a speed of
24 meters per second.
57. LOGICAL REASONING Determine whether the statement is true or false.
Explain your reasoning.
36 is a solution of Vx = —6.
Chapter 12 Radicals and More Connections to Geometry
Standardized Test
Practice
Mixed Review
Maintaining Skills
Quiz 1
58. MULTIPLE CHOICE Which of the following is a solution of x = V30 — x?
(A) -6 Cl) 0 (© 5 (D) 30
59. MULTIPLE CHOICE Which of the following is a solution of x = Vx + 20?
CD -5 (3D -4 CB) 4 GD 5
QUADRATIC EQUATIONS Solve the equation. Write the solutions as
integers if possible. Otherwise, write them as radical expressions.
(Lesson 9.2)
60. x 2 = 36 61.x 2 = 11 62. lx 2 = 700
63. 25X 2 - 9 = 91 64. x 2 - 16 = -7 65. -16x 2 + 48 = 0
SPECIAL PRODUCT PATTERNS Find the product. (Lesson 10.3)
66. (x + 5) 2 67. (2x - 3) 2 68. (6 y - 4)(6y + 4)
69. (3x + 5_y)(3x — 5 y) 70. (x + ly) 2 71. (2 a — 9b) 2
PERFECT SQUARES Factor the expression. (Lesson 10.7)
72. x 2 + 18x + 81 73. x 2 — 12x + 36 74. 4X 2 + 28x + 49
RECIPROCALS Find the reciprocal of the mixed number. Write your
answer in lowest terms. (Skills Review p. 763)
75.
76. 4f
3
77 1 —
10
78.
79 '4
80. 8-7
0
CO
VO|^J
82.
Find the domain of the function. Then sketch its graph and find the
range. (Lesson12.1)
1. v = 10V^ 2 . y = \/x — 9 3 ~y = V2x — 1 4 - y = Vx — 2
Simplify the expression. (Lesson 12.2)
5 . 7VTo+ llVTo 6 . V 3 ( 3 V 2 + V 3 ) 7 . 4 V 7 + Vl 25 - V80
Solve the equation. Check for extraneous solutions. (Lesson 12.3)
8- Vx — 2 = 0 9- Vx — 8 = 0 10 . V3x + 2 + 2 = 3
11 . V3x — 2 + 3 = 7 12 . V77 — 4x = x 13 . x = V2x + 3
14. NOZZLE PRESSURE Using the flow rate equation r = 30 d 2 \fp given in
Exercises 37 and 38 on page 708, find the nozzle pressure for a hose that has a
flow rate of 250 gallons per minute and a diameter of 2.5 inches. (Lesson 12.3)
12.3 Solving Radical Equations
Rational Exponents
Goal
Evaluate expressions
involving rational
exponents.
Key Words
• cube root of a
• radical notation
• rational exponent
• rational exponent
notation
How large is the sphere used in women's shot put?
The metal sphere used in women’s
shot put is called a shot. In Exercise
46 you will find the size of this shot.
CUBE ROOT OF A NUMBER In Chapter 1 you learned how to cube a number.
Now we define a cube root.
If b 3 = a , then b is called a cube root of a.
For instance, 2 is a cube root of 8 because 2 3 = 8. In radical notation, a cube root
of a is written as Va. In general, for any integer n greater than 1,
if b n = a , then b is an nth root of a.
n /—
In radical notation, the nth root of a is written as Va.
Student HeCp
---\
► Reading Algebra
When the cube root of
a is written in rational
exponent notation, a 1/3 ,
it is read "a raised to
the one-third power."
L _ j
RATIONAL EXPONENT NOTATION Because ^/a • ^fa • tya = a, it is natural
to define \fa = a 113 . With this definition the product of powers property for
exponents holds for fractional exponents:
^r a . tTa = a l/3 . a l/3 . a l/3 = a (l/3 + 1/3 + 1/3) = a \ =a
More generally, ^/a = a lln for any a > 0 and integer n greater than 1. The value
of a lln is restricted to nonnegative numbers.
11 Find Cube and Square Roots
Find the cube root or square root,
a. 27 1/3 b. VlOOO c. 64 1/2
Solution
a. Because 3 3 = 27, you know that 27 1/3 = 3.
b. Because 10 3 = 1000, you know that ^1000 = 10.
c. Because 8 2 = 64 and 8 is a nonnegative number, you know that 64 1/2 = 8.
Chapter 12 Radicals and More Connections to Geometry
RATIONAL EXPONENTS A rational exponent does not have to be of the form —.
3 4
Other rational numbers, such as — and —, may also be used as exponents. For
integers m and n we have the rule ( a m ) n = a mn . This produces a basis for the
following definition of powers written with rational exponents.
RATIONAL EXPONENTS
2 Evaluate Expressions with Rational Exponents
Rewrite the expressions using rational exponent notation and radical notation,
a. 16 3/2 b. 8 4/3
Solution
a. Use rational exponent notation. 16 3/2 = (l6 1/2 ) 3 = 4 3 = 64
Use radical notation. 16 3/2 = (\/l6) 3 = 4 3 = 64
b. Use rational exponent notation. 8 4/3 = (8 1/3 ) 4 = 2 4 = 16
Use radical notation. 8 4/3 = (V/8 ) 4 = 2 4 = 16
Evaluate Expressions with Rational Exponents
Evaluate the expression without using a calculator.
1.^64 2 . 625 1/2 3 . 225 1/2 4 . 216 1/3
5 . 64 3/2 6 . (^27) 2 7 . (Vi) 5 8 . 1000 2/3
The multiplication properties of exponents presented in Lesson 8.1 can also be
applied to rational exponents.
Properties of Rational Exponents
Let a and b be nonnegative real numbers and let m and n be
rational numbers.
PROPERTY
EXAMPLE
d m • g n = Q m + n
3 1/2 . 33/2 = 3(1/2 + 3 / 2 ) = 3 2 = 9
(a m ) n = a mn
(43/2)2 = 4(3/2 • 2) = 4 3 = g 4
(ab) m = a m b m
CO
•
£
n 5
II
CD
n 5
•
4 ^
60
II
CO
•
I'D
II
O)
_>
12.4 Rational Exponents
Student HeCp
► More Examples
More exam Pl es
are available at
www.mcdougallittell.com
3 Use Properties of Rational Exponents
Evaluate the expression using the properties of rational exponents,
a, 5 1/3 • 5 2/3 b. (7 1/3 ) 6 c. (4.25) 1/2
Solution
a. Use the product of powers property.
51/3 . 5 2/3 = 5(1/3 + 2/3) = 53/3 _ 5I _ 5
b. Use the power of a power property.
( 71 / 3)6 _ 7(1/3 • 6) _ 72 _ 49
c. Use the power of a product property.
(4 • 25) 1/2 = 4 1/2 • 25 1/2 = 2 • 5 = 10
Use Properties of Rational Exponents
Evaluate the expression using the properties of rational exponents.
9 . (8 1/3 ) 2 10 . (4 • 16) 1/2 11. 4 1/2 • 4 3/2 12 . (3 1/2 ) 2
13 . (27 • 64) 1/3 1 4 . 2 5/2 • 2 1/2 15 . (6 2/3 )
3/2
16 . (64 • 81)
1/2
4 Use Properties of Rational Exponents
Simplify the variable expression (x • y 1/2 ) 2 Vx using the properties of
rational exponents.
Solution
Q Use the power of a product property.
© Write Vx in rational exponent notation.
© Use the product of powers property.
© Add the exponents.
(x • y 1/2 ) 2 Vx = (x 2
= x
2 .
•y
1 / 2 * 2 )v^
•x 1/2
= x 2 + 1/2 • y 1
= x 5,2 y
Use Properties of Rational Exponents
Simplify the expression.
17 . (x • y ,/2 ) 4 x
18 . (x 3/2 • _y) 2
19 . (y 3 ) 1/6
20 . (x 1/3 • x 5/3 ) 1/2
21 . (x 1/2 • y 1/3 ) 6
22 . ^(x 3
Chapter 12 Radicals and More Connections to Geometry
Exercises
Guided Practice
Vocabulary Check
1. Write “the cube root of 27” in both radical notation and rational exponent
notation.
Skill Check Evaluate the expression without using a calculator.
2. ^/\25 3. 49 1/2 4. ("v^) 5
6 . 121 1/2 7. 9 3/2 8 . "V / 343
5. 25 3/2
9. (V81) 3
Practice and Applications
RATIONAL EXPONENTS Rewrite the expression using rational
exponent notation.
10 . Vl4 11 . ^TT 12. (^5) 2 13. (Vl6) 5
RADICALS Rewrite the expression using radical notation.
14. 6 1/3 1 5. 7 1/2 1 6. 10 3/2 1 7. 8 7/3
EVALUATING EXPRESSIONS Evaluate the expression without using
a calculator.
18. V8
19. V 10,000
22 . 1 1/3
23. 256 1/2
26. 4 3/2
27. 125 2/3
20. 512 1/3 21.4 1/2
24. (Vl6) 4 25. (^27) 4
28. (VlOO) 3 29. (^64) 4
PROPERTIES OF RATIONAL EXPONENTS
30. 3 5/3 • 3 1/3
33. (6 1/3 ) 6
36. (2 3 • 3 3 ) 1/3
31.4 3/2 .4 1/2
34. (8 • 27) 1/3
37. (2 2/3 • 2 1/3 ) 6
Evaluate the expression.
32. ( 8 2/3 ) 1/2
35. (16 • 25) 1/2
38. (4 2 • 5 2 ) 1/2
! Student HeCp
^Homework Help
Example 1: Exs. 10-29
Example 2: Exs. 18-29
Example 3: Exs. 30-38
Example 4: Exs. 39-44
V, _ J
PROPERTIES OF RATIONAL EXPONENTS Simplify the variable
expression.
39. x 113 • x m 40. x • V + y 2 • rfx 3 41. ( y 1/6 ) 3 • Vx
42. ( 36 x 3 ) 1/2 43. {y • y 113 ) 312 44. (x 1/3 • y m ) 6 •
45. LOGICAL REASONING Complete the statement with always , sometimes,
or never.
If a and b are whole numbers, then Va 2 + b 2 is ? equal to a + b.
12.4 Rational Exponents
Standardized Test
Practice
Mixed Review
Maintaining Skills
■sfjiSia* J Volume of a Sphere
The formula for the volume of a sphere
4 q
is V = ~^7Tr, where r is the radius of the
sphere. Find the radius of a sphere that
has a volume of 33.5 cubic centimeters.
Solution
4 *
To find the radius of the sphere, first solve the equation V = —Trr* for r -
O Write the formula for the volume of a sphere.
@ Multiply each side by — and divide each side by tt.
0 Take the cube root of each side.
© Substitute 33.5 for V.
0 Evaluate the radicand.
0 Solve for r.
ANSWER ► The radius of the sphere is about 2 centimeters.
v = l^
r~ A v / 8I)
r~2
46. SHOT PUT The shot (a metal sphere) used in the women’s shot put has a
volume of about 524 cubic centimeters. Find the radius of the shot.
47. MULTIPLE CHOICE Evaluate the expression 100 3/2 .
(a) io CD ioo cd iooo CD 10,000
QUADRATIC EQUATIONS Solve the equation. Write the solutions as
integers if possible. Otherwise, write them as radical expressions.
(Lesson 9.2)
48. 16 + x 2 = 64 49. x 2 + 25 = 81 50. x 2 + 81 = 144
51. 4x2- 144 = 0 52.x 2 - 30 = -3 53. x 2 = ||
SOLVING EQUATIONS Solve the equation. (Lesson 10.4)
54. (x + 4) 2 = 0 55. (x + 4)(x - 8) = 0 56. x(x - 14) 2 = 0
FACTORS Determine whether the number is prime or composite. If it is
composite, give its prime factorization. (Skills Review p. 761)
57. 13 58. 28 59. 75 60. 99
61.18 62.33 63.69 64.80
Chapter 12 Radicals and More Connections to Geometry
DEVELOPING CONCEPTS
For use with
Lesson 12.5
Goal
Use algebra tiles to
complete the square.
Materials
• pencil
• algebra tiles
Question How can you use algebra tiles to complete the square?
Explore
Q You can use algebra tiles to
model the expression x 2 + 6x.
You will need one x 2 -tile
and sixx-tiles.
Student HeCp
► Logic Back
For help with algebra
tiles, see p. 567.
___ )
o
Arrange the x 2 -tile and the
x-tiles to form part of a square
Your arrangement will be
incomplete in one corner.
You want the length and width
of your "square" to be equal.
© To complete the square, you
need to add nine 1-tiles.
By adding nine 1-tiles,
you can see that
x 2 3 + 6x + 9 = (x + 3) 2 .
Think About It
1. Copy and complete the table by following the steps above.
Expression
Number of tiles to
complete the square
Number of tiles as
a perfect square
x 2 + 6x
9
3 2
x 2 + 4x
?
?
x 2 + 2x
?
?
2 . How is the number in the third column related to the coefficient of x?
3. Use the pattern you found in Exercise 2 to predict how many tiles you would
need to add to complete the square for the expression x 2 + 8x.
Developing Concepts
Completing the Square
Goal
Solve a quadratic
equation by completing
the square.
Key Words
• completing the square
• quadratic formula
• perfect square
trinomial
How far does a penguin leap?
Penguins leap out of the water
every few feet when swimming,
The distance a penguin leaps
can be modeled by a quadratic
equation, as you will see in
Example 4.
In Developing Concepts 12.5, page 715, you completed the square for
expressions of the form x 2 + bx when b = 2, 4, 6, and 8. In each case,
x 2 + bx + (j was modeled by a square with sides of length x +
( b\( b\
By using FOIL to expand I x + 2 )\ x + 2 /’ ^ 0U can s ^ ow ^ at P attern
holds for any real number b.
COMPLETING THE SQUARE
To complete the square of the expression x 2 + bx, add the square
of half the coefficient of x, that is, add (tH .
x 2 + bx +
2
1 Complete the Square
What term should you add to x 2 — 8x to create a perfect square trinomial?
/ _ 8 \ 2
The coefficient of x is —8, so you should add ( ) , or 16, to the expression.
: x 2 — 8x + 16
: (x — 4) 2
8x + ——
Complete the Square
Find the term that should be added to the expression to create a perfect
square trinomial.
1 . x 2 + 2x 2 . x 2 — 4x 3 - x 2 + 6x 4 - x 2 — lOx
Chapter 12 Radicals and More Connections to Geometry
Student HeCp
► Study Tip
When completing the
square to solve an
equation, remember that
you must always add the
term (^j to both sides ..
of the equation.
L j
2 Solve a Quadratic Equation
Solve x 2 + lOx = 24 by completing the square.
Solution
x 2 + lOx = 24
x 2 + lOx + 5 2 = 24 + 5 2
(x + 5) 2 = 49
x + 5 = ±7
x = —5 ± 7
Write original equation.
Add j , or 5 2 , to each side.
Write left side as perfect square.
Find square root of each side.
Subtract 5 from each side.
x= 2 or x — —12 Simplify.
ANSWER ► The solutions are 2 and —12. Check these in the original equation to
confirm that both are solutions.
Solve a Quadratic Equation
Solve the equation by completing the square.
5 . x 2 — 2x — 3 = 0 6 - x 2 — 12x + 4 = 0 7 . x 2 + 16x + 4 = 0
3 Develop the Quadratic Formula
The quadratic formula can be established by completing the square for the
general quadratic equation ax 2 + bx + c = 0, where a ^ 0.
O Write the original equation.
0 Subtract c from each side.
0 Divide each side by a.
_b 2 _
4 a 2
Add (— ] = to each side.
ax 2 + bx + c — 0
ax 2 + bx = —c
2 b c
X -i -X =-
a
2 , b
x H—x +
a
b 2
4 a 2
b 2
4 a 2
c_
a
© Write the left side of the equation
as a perfect square.
© Find the square root of each side.
b 2 - 4ac
4 a 2
b , / ~b 2 - 4 ~ac
e
Subtract— from each side.
2 a
b + Vfr 2 — 4 ac
2 a 2 a
0 Write the right side of the
equation as a single fraction.
This result is the quadratic formula.
x =
—b ± \/b 2 — 4ac
2 a
12.5 Completing the Square
H
J 4 Choose a Solution Method
PENGUINS The path followed by a penguin leaping out of the water is given
by h = — 0.05x 2 + 1.178x, where h is the vertical height (in feet) of the
penguin above the water and v is the horizontal distance (in feet) traveled over
the water. Find the horizontal distance traveled by this penguin when it reaches
a vertical height of 6 feet.
Solution
To find the horizontal distance when h = 6, solve the quadratic equation
6 = — 0.05x 2 + 1.178x. This equation cannot be factored easily and cannot be
solved easily by completing the square. The quadratic formula is a good choice.
—b ± V/? 2 - 4 ac
2 a
= -1.178 ± Vl.178 2 - 4(—0.05)(—6)
X 2(—0.05)
v ~ 7.4 or x ~ 16.1
Write quadratic formula.
Substitute values for a, b, and c.
Use a calculator.
ANSWER ► The penguin reaches a vertical height of 6 feet at horizontal
distances of about 7.4 feet and about 16.1 feet. Check these solutions
in the original equation.
Choose a method and solve the quadratic equation. Explain your choice.
8 . x 2 — 3 = 0 9 . 2x 2 = 8 10 - x 2 + 3x + 4 = 6
You have learned the following five methods for solving quadratic equations.
EEEEE2
Methods for Solving
Method
FINDING SQUARE ROOTS
(Lesson 9.2)
graphing (Lesson 9.5)
USING THE QUADRATIC
formula (Lesson 9.6)
FACTORING
(Lesson 10.5-10.8)
COMPLETING THE SQUARE
(Lesson 12.5)
Quadratic Equations
Comments
Efficient way to solve ax 2 + c = 0.
Can be used for any quadratic equation. Enables you to
approximate solutions.
Can be used for any quadratic equation.
Efficient way to solve a quadratic equation if the quadratic
expression can be factored easily.
Can be used for any quadratic equation, but is best suited for
quadratic equations where a = 1 and b is an even number.
Chapter 12 Radicals and More Connections to Geometry
\2M Exercises
Guided Practice
Vocabulary Check 1. Explain how to complete the square of the expression x 2 + bx.
2. LOGICAL REASONING Determine whether the statement is true or false.
Explain your reasoning.
To solve x 2 + 6x = 12 by completing the square, add 6 to both sides.
Skill Check Find the term that should be added to the expression to create a perfect
square trinomial.
3- x 2 + 20x 4. x 2 + 30x 5- x 2 — lOx
6- x 2 — 14x 7. x 2 — 22x 8- x 2 + 24x
9- Solve x 2 — 3x = 8 by completing the square. Solve the equation by using the
quadratic formula. Which method did you find easier?
Solve the quadratic equation by completing the square.
10- x 2 — 2x — 18 = 0 11. x 2 + lOx — 10 = 0
12- x 2 + 8x = —3 13. x 2 + 14x = —13
Choose a method and solve the quadratic equation. Explain your choice.
14. x 2 — x — 2 = 0 15. 3x 2 + 17x + 10 = 0 16. x 2 — 9 = 0
17. — 3x 2 + 5x + 5 = 0 18. x 2 + 2x — 14 = 0 19. 3x 2 - 2 = 0
Practice and Applications
PERFECT SQUARES Find the term that should be added to the
expression to create a perfect square trinomial.
20. x 2 - 12x
21.x 2 + 8x
22. x 2 + lOx
23. x 2 + 22x
24. x 2 + 14x
25. x 2 - 40x
26. x 2 + 4x
27. x 2 — 6x
28. x 2 + 16x
COMPLETING THE SQUARE Solve by completing the square.
29. x 2 - 8x + 12 = 0
32. x 2 + 4x = 12
35. x 2 + lOx = 39
38. x 2 — 6x — 11 = 0
41.x 2 - 4x - 1 = 0
30. x 2 2x — 3
33. x 2 + lOx = 12
36. x 2 + 16x = 17
39. x 2 — 2x = 5
42. x 2 + 20x + 3 = 0
31. x 2 + 6x — 16
34. x 2 + 8x = 15
37. x 2 - 24x = -
40. x 2 + 30x — 7
43. x 2 + 14x — 2
Student He dp
► Homework Help
Example 1: Exs. 20-28
Example 2: Exs. 29-55
Example 3: Exs. 29-56
Example 4: Exs. 57-76
\ ___ )
12.5 Completing the Square
Solve the quadratic equation.
SOLVING EQUATIONS
44. x 2 + 4x + 5 = 0
47. x 2 + 22x + 1 = 0
50. x 2 + 14x - 7 = 0
53. x 2 - 12x - 3 = 0
45. x 2 + lOx - 3 = 0
48. x 2 + 2x - 11 = 0
51.x 2 + 20x + 2 = 0
54. x 2 - 18x + 5 = 0
46. x 2 + 16x + 9 = 0
49. x 2 + 8x - 6 = 0
52. x 2 — 6x — 10 = 0
55. x 2 - 2x - 4 = 0
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 57-59 is available at
www.mcdougallittell.com
56. LOGICAL REASONING Explain why the quadratic formula gives real
solutions only if a A 0 and b 2 — 4ac > 0.
Geometry Linkfy In Exercises 57-59, make a sketch and write a quadratic
equation to model the situation. Then solve the equation.
57. In art class you are designing the floor plan of a house. The kitchen is
supposed to have 150 square feet of space. What should the dimensions of
the kitchen floor be if you want it to be square?
58. A rectangle is 2x feet long and x + 5 feet wide. The area is 600 square feet.
What are the dimensions of the rectangle?
59. The base of a triangle is x feet and the height is (4 + 2x) feet. The area of the
triangle is 60 square feet. What are the dimensions of the triangle?
CHOOSING A METHOD Choose a method and solve the quadratic
equation. Explain your choice.
60. x 2 — x — 12 = 0 61. x 2 — 9 = 0
63. x 2 + 5x — 14 = 0
66 . x 2 + 5x — 6 = 0
69. 2x 2 + lx + 3 = 0
72. 3x 2 - 48 = 0
64. x 2 — 2x = 2
67. x 2 — 6x + 7 = 0
70. 2x 2 - 200 = 0
73. x 2 + 3x + 4 = 1
62. x 2 4x — 8
65. 3x 2 + 5x — 12 = 0
68 . x 2 + 2 = 6
71.x 2 - 24x = 6
74. 3x 2 + lx + 2 = 0
75. DIVING The path of a diver
diving from a 10-foot high diving
board is
h = — 0.44x 2 + 2.61* + 10
where h is the height (in feet) of
the diver above water and x is the
horizontal distance from the end
of the board. How far from the
end of the board will the diver
enter the water?
76. VERTICAL MOTION Suppose you throw a ball upward from a height of
5 feet and with an initial velocity of 15 feet per second. The vertical motion
model h = —16^ 2 + I5t + 5 gives the height h (in feet) of the ball, where t is
the number of seconds that the ball is in the air. Find the time that it takes for
the ball to reach the ground (h = 0) after it has been thrown.
Chapter 12 Radicals and More Connections to Geometry
Standardized Test
Practice
Mixed Review
Maintaining Skills
77. MULTIPLE CHOICE Which of the following is a solution of the equation
2x 2 + 8x - 25 = 5?
(A) —Vl9 — 2 (DV17-2 (C)V2l-2 CDV17+1
O 1
78- MULTIPLE CHOICE What term should you add to x 2 — — x to create a
perfect square trinomial?
79. MULTIPLE CHOICE Solve x 2 + 8x - 2 = 0.
® -4 ± 3 V 2 CD -4 ± 2 V 2 CD 4 ± 3 V 2 CD 4 ± Vl6
SOLVING LINEAR SYSTEMS Solve the linear system. (Lessons 7.2, 7.3)
80- y = 4x 81. 3x + y = 12 82. 2x — y = 8
x + y = 10 9x — y = 36 2x + 2y = 2
QUADRATIC EQUATIONS Solve the equation. Write the solutions as
integers if possible. Otherwise, write them as radical expressions.
(Lesson 9.2)
83. 3x 2 - 147 = 0 84. x 2 - 5 = 20 85. x 2 + 2 = 83
86 . 9 + x 2 = 49 87. x 2 - 16 = 144 88. x 2 + 64 = 169
SOLVING GRAPHICALLY Use a graph to estimate the solutions of the
equation. Check your solutions algebraically. (Lesson 9.5)
89. x 2 + x + 2 = 0 90. — 3x 2 - x - 4 = 0 91. 2x 2 - 3x + 4 = 0
92. x 2 - x - 12 = 0 93. x 2 - 2x - 3 = 0 94. 2x 2 + lOx + 12 = 0
ZERO-PRODUCT PROPERTY Use the zero-product property to solve the
equation. (Lesson 10.4)
95. (x + 4)(x - 8) = 0 96. (x - 3)(x - 2) = 0 97. (x + 5)(x + 6) = 0
98. (x + 4) 2 = 0 99. (x - 3) 2 = 0 100. 6(x - 14) 2 = 0
FACTORING TRINOMIALS Factor the trinomial. (Lessons 10.5, 10.6)
101 . x 2 + x - 20 102 . x 2 - lOx + 24 103 . x 2 + 4x + 4
104 . 3X 2 - 15x+ 18 105 . 2x 2 — x — 3 106 . Ux 2 - 19x - 3
PERCENTS AND FRACTIONS Subtract. Write the answer as a fraction in
simplest form. (Skills Review p. 768)
108 . | - 80%
111. 26% - ^
114 . 100% - |
109 . 4 — 39%
112 . 75% - |
115 . 50% - |
12.5 Completing the Square
DEVELOPING CONCEPTS
'fhs yyihiis^ijYHzus 'Thavysm
For use with
Lesson 12.6
Goal
Work in groups to
investigate the
Pythagorean theorem
and its converse.
Materials
• graph paper
• scissors
• glue or tape
Question
tigs- '
If you are able to classify a triangle as acute, right,
or obtuse, what conclusions can you draw about the
lengths of its sides?
Explore
Q Cut graph paper into squares with the following side lengths:
3, 4, 5, 6, 7, 8, 10, 12, and 13.
Student HeCp
1 ^ -
►Vocabulary Tip
An obtuse triangle has
one angle measuring
between 90° and 180°.
An acute triangle has
three angles that each
measure between
0° and 90°.
^ _ >
© Create a triangle with side lengths a = 3,
b = 4, and c = 6, as shown. Label the
vertices A , B , and C, placing C opposite
the longest side.
© Using a protractor, classify the triangle
as acute, right, or obtuse.
© Repeat Steps 2 and 3 using the remaining
squares. Create one triangle with side
lengths a = 5, b = 12, c = 13 and one
triangle with side lengths a = 7, b = 8,
c = 10.
0 Compare the values of a 2 + b 2 with the values of c 2 for each of the three
triangles. Then copy and complete the table below.
Type of triangle
Side lengths
a 2 + b 2
<, >, or =
c 2
obtuse
3,4,6
25
?
36
?
5, 12, 13
?
?
?
?
7, 8, 10
?
?
?
Think About It
■ " i -
In Exercises 1 and 2, a, b, and c are the lengths of the sides of a triangle,
and c is the length of the longest side.
1. Repeat Steps 1-4 above with a number of different triangles. Be sure to
include acute triangles, right triangles, and obtuse triangles.
2 . Complete the following statements using <, >, or = as conjectures based
on your observations.
In an obtuse triangle, a 2 + b 2 ? c 2 .
In a right triangle, a 2 + b 2 ? c 2 . (Pythagorean theorem)
In an acute triangle, a 2 + b 2 ? c 2 .
Chapter 12 Radicals and More Connections to Geometry
Does the converse of the Pythagorean theorem hold true?
Explore
p 1 1 " 1 ■ ' «'
0 Select three of the graph paper squares and form a triangle. Label the vertices
A, B , and C, placing C opposite the longest side. Two triangles are shown.
© Compare the values of a 2 + b 2 with the values of c 2 for each triangle. Based
on your answers in Exercise 2 on page 722, classify the triangle as acute,
right, or obtuse. Then copy and complete the table below.
Side lengths
a 2 + b 2
II
o
A
V
c 2
Type of triangle
3,4,5
25
=
25
?
4, 6,7
52
>
49
?
Think About It
Let a, b, and c be the side lengths of a triangle with c the longest side.
1. Repeat Steps 1 and 2 above with a number of different triangles. Choose a
variety of lengths so a 2 + b 2 = c 2 is sometimes true, and sometimes not.
2_ Complete the following conjectures based on your observations.
If the sides of a triangle satisfy a 2 + b 2 = c 2 , then the triangle is
a ? triangle. (Converse of the Pythagorean theorem)
If the sides of a triangle satisfy a 2 + b 2 < c 2 , then the triangle is
a ? triangle.
If the sides of a triangle satisfy a 2 + b 2 > c 2 , then the triangle is
a ? triangle.
The Pythagorean Theorem
and Its Converse
Goal
use the Pythagorean What is the distance from home plate to second base?
theorem and its converse __ ■
Key Words
• theorem
• Pythagorean theorem
• hypotenuse
• legs of a right triangle
• converse
You will use the Pythagorean theorem
in Exercise 31 to find the distance from
home plate to second base of a standard
baseball diamond.
A theorem is a statement that can be proven to be true. The Pythagorean
theorem states a relationship among the sides of a right triangle. The hypotenuse
is the side opposite the right angle. The other two sides are the legs.
THE PYTHAGOREAN THEOREM
r-
If a triangle is a right triangle, then the leg
sum of the squares of the lengths of a
the legs a and b equals the square of
the length of the hypotenuse c.
a 2 + b 2 = c 2
L_
b leg
In Exercise 23 in Lesson 12.9, you will outline a proof of the Pythagorean theorem.
Student HeCp
► Study Tip
When you use the
Pythagorean theorem
to find the length of a
side of a right triangle,
you need only the
positive square root
because the length of
a side cannot be
negative.
^ _ /
1 Use the Pythagorean Theorem
a. Given a = 6 and b = 8, find c. Use the Pythagorean theorem: a 2 + b 2 = c 2 .
6 2 + 8 2 = c 2
100 = c 2
VlOO = Vc 2
10 = c 6 = 8
b. Given a = 5 and c = 6, find b. Use the Pythagorean theorem: a 2 + b 2 = c 2 .
5 2 + b 2 = 6 2
b 2 = 6 2 - 5 2
b 2 = 11
b = VII = 3.32
c = 6
Chapter 12.6 Radicals and More Connections to Geometry
Student HeCp
p Morel Examples
M°r e examples
IJfcL 2 are available at
www.mcdougallittell.com
2 Use the Pythagorean Theorem
A right triangle has one leg that is 3 inches longer than the other leg.
The hypotenuse is 15 inches. Find the unknown lengths.
Solution
Sketch a right triangle and label the sides. Let x be
the length of the shorter leg. Use the Pythagorean
theorem to solve for x.
x + 3
a 2 + b 2 = c 2
x 2 + (x + 3) 2 = 15 2
x 2 + x 2 + 6x + 9 = 225
2x 2 + 6x — 216 = 0
2(x - 9)(x + 12) = 0
x = 9 or x = — 12
Write Pythagorean theorem.
Substitute for a, b, and c.
Simplify.
Write in standard form.
Factor.
Zero-product property
ANSWER ^ Length is positive, so the solution x = —12 is extraneous.
The sides have lengths 9 inches and 9 + 3 = 12 inches.
EHmESB 3 Use the Pythagorean Theorem
A board game is a square 2 feet by 2 feet. What is the length of the diagonal
from one corner of the board game to the opposite corner?
Solution
The diagonal is the hypotenuse c of a right triangle. Each leg is 2 feet in length.
Write Pythagorean theorem.
Substitute 2 for a and 2 for b.
Simplify right side of the equation.
c = V8 ~ 2.8 Find square root of each side.
ANSWER ► The length from one corner of the board game to the opposite comer
is about 2.8 feet.
c 2 = a 2 + b 2
c 2 = 2 2 + 2 2
c 2 = 8
Find the hypotenuse of the right triangle with the given legs.
1 _ a — 12, b — 5 2 . a — 3, b — 4 3 . a = 12, b — 16
Solve for xto find the missing lengths of the right triangle.
12.6
The Pythagorean Theorem and Its Converse
Student HeCp
^
► Look Back
For help with if-then
statements, see p. 120.
V J
LOGICAL REASONING In mathematics an if-then statement is a statement of
the form “If p , then qf where p is the hypothesis and q is the conclusion. The
converse of the statement “If p , then < 7 ” is the related statement “If q , then pf
in which the hypothesis and conclusion are interchanged.
In many cases, a theorem is true, but its converse is false. For example, the
statement “If a = b, then a 2 = b 2 ” is true, while the converse “If a 2 = b 2 , then
a = b ” is false. In the case of the Pythagorean theorem, however, both the
theorem and its converse are true.
CONVERSE OF THE PYTHAGOREAN THEOREM
If a triangle has side lengths a, b, and c such that a 2 + b 2 = c 2 ,
then the triangle is a right triangle.
[ Student HeCp
► Study Tip
In a right triangle the
hypotenuse is always
the longest side.
V _ )
4 Determine Right Triangles
Determine whether the given lengths are sides of a right triangle: 15, 20, 25.
Solution Use the converse of the Pythagorean theorem. The lengths are sides
of a right triangle because
15 2 + 20 2 = 225 + 400 = 625 = 25 2 .
5 Use the Pythagorean Converse
You can take a rope and tie
12 equally spaced knots in it. You
can then use the rope to check that
a corner is a right angle. Why does
this method work?
Solution
You can use the rope to form a triangle with longest side of length 5 and other
sides of lengths 3 and 4. Check that
32 + 42 = 9 + 16 = 25 = 5 2 .
Therefore, by the converse of the Pythagorean theorem, the triangle is a
right triangle.
ANSWER ^ Because you can use the knots to form the sides of a right triangle,
one angle of the triangle must measure 90°. This is why you can
check with a rope that a corner is a right angle.
Use the Pythagorean Converse
Determine whether the given lengths are sides of a right triangle.
7. 5, 11, 12 8 . 5, 12, 13 9. 11.9, 12, 16.9
Chapter 12.6 Radicals and More Connections to Geometry
M3 Exercises
Guided Practice
Vocabulary Check
1. Complete: Sides of a right triangle that are not the hypotenuse are the ?
2 . State the hypothesis and the conclusion of the statement “If x is an even
number, then x 2 is an even number.”
Skill Check
Find the missing length of the right triangle if a and b are the lengths of
the legs and c is the length of the hypotenuse.
3 . a = 7, b = 24 4 . a — 5, c = 13
6 - a = 9, c = 41
7./?= 11, c = 61
5 . b= 15, c= 17
8 . a = 12, b = 35
Find each unknown length of the right triangle.
12, Explain how you can use the converse of the Pythagorean theorem to tell
whether three given lengths can be sides of a right triangle.
Practice and Applications
USING THE PYTHAGOREAN THEOREM Find the missing length of the
right triangle if a and b are the lengths of the legs and c is the length
of the hypotenuse.
14 . a = 10, b = 24 15 . b = 3, c = 7
13 . a = 3, c = 4
16.b = 9,c= 16
19 . a = 2, b = 8
22 . b= l,c = 3
17 . a = 5, c = 10
20 . a = 11,6 = 15
23 . a = 4, c = 1
18 . a = 14, c = 21
21.6 = 3,c= 10
24 . a = 8, c = 10
! Student HeCp
► Homework Help
Example 1: Exs. 13-24
Example 2: Exs. 25-30
Example 3: Exs. 31-35
Example 4: Exs. 36-41
Example 5: Exs. 42-44
MISSING LENGTH Find the unknown lengths of the right triangle.
12.6 The Pythagorean Theorem and Its Converse
31. BASEBALL The length
of each side of a baseball
diamond is 90 feet. What
is the diagonal distance c
from home plate to
second base?
2nd base a = 90 ft 1st base
32. DIAGONAL OF A FIELD A field hockey field is a rectangle 60 yards by
100 yards. What is the length of the diagonal from one corner of the field to
the opposite comer?
Student HeCp
► Homework Help
^ xtra
problem solving in
Exs. 33-34 is available at
www.mcdougallittell.com
DESIGNING A STAIRCASE
You are building the staircase
shown at the right.
33. Find the distance d between the
edges of each step.
34. The staircase will also have a
handrail that is as long as the
distance between the edge of the
first step and the edge of the top
step. How long is the handrail?
35. PLANTING A NEW TREE. You have just planted
a new tree. To support the tree, you attach four
guy wires from the trunk of the tree to stakes in
the ground. Each guy wire has a length of 7 feet.
Suppose you put the stakes in the ground 5 feet
from the base of the trunk. Approximately how
far up the trunk should you attach the guy wires?
DETERMINING RIGHT TRIANGLES Determine whether the given lengths
are sides of a right triangle. Explain your reasoning.
36. 2, 10, 11 37. 5, 12, 13 38. 12, 16, 20
39. 11, 60, 61 40. 7, 24, 26 41. 3, 9, 10
DETERMINING RIGHT TRIANGLES Determine whether the given lengths
are sides of a right triangle. Explain your reasoning.
45. CHALLENGE You have a rope with 24 equally spaced knots in it. Form a
triangle with the rope and give the length of each side. How can you use this
rope to check that a corner is a right angle?
Chapter 12.6 Radicals and More Connections to Geometry
Standardized Test
Practice
Mixed Review
Maintaining Skills
Quiz 2
46. MULTIPLE CHOICE Given the lengths of the three sides of a triangle,
determine which triangle is not a right triangle.
(A) a = 9, b = 40, c = 41 Cb) a = 3, b = 4, c = 5
Cg) a = 7, b = 24, c = 25 Co) a = 10,/? = 49, c = 50
PLOTTING POINTS Plot and label the ordered pairs in a coordinate plane.
(Lesson 4.1)
47. A( 2 , 5), 5(0, -1), C(3, 1) 48. A( 2 , -5), 5(2, 4), C(-3, 0)
49. A(— 1, — 2), 5(—4, 5), C(0, 2) 50. A(l, 4), 5(-2, -1),C(3, -1)
NUMBER OF X-INTERCEPTS Determine whether the graph of the
function intersects the x-axis in zero, one r or two points. (Lesson 9.7)
51. y = x 2 + 2x + 15 52. y = x 2 + 8x + 12 53. y = x 2 + x — 10
54. y = x 2 + 8x + 16 55. y = x 2 + 3v + 1 56. _y = x 2 — 8x — 11
ESTIMATING AREA Estimate the area of a rectangle whose sides are
given. First round each side length to the nearest whole number. Then
multiply to find the area. (Skills Review p. 775)
57. 5.1 by 7.2 58. 10.6 by 17.3 59. 5.1 by 9.9
60. 100.4 by 7.0 61. 17.3 by 2.8 62. 20.5 by 1.5
Evaluate the radical expression using the properties of rational
exponents. (Lesson 12.4)
1. 2 1/3 • 2 m 2. (36 • 49 ) 1/2 3. (3 1/2 ) 4
Solve the quadratic equation by completing the square. (Lesson 12.5)
4. x 2 — 6x + 7 = 0 5. x 2 + 4x — 1 = 0 6. x 2 + 2x = 2
Determine whether the given lengths are sides of a right triangle.
Explain your reasoning. (Lesson 12.6)
7.6,9, 11 8 . 12,35,37
9. 1 , 1,V2
10. DEPTH OF A SUBMARINE
The sonar of a Navy cruiser detects
a submarine that is 2500 feet away.
The point on the water directly above
the submarine is 1500 feet away from
the front of the cruiser. What is the
depth of the submarine? (Lesson 12.6)
12.6 The Pythagorean Theorem and Its Converse
The Distance Formula
Goal
Find the distance
between two points in a
coordinate plane.
Key Words
• distance formula
How far was the soccer ball kicked?
You can use the distance formula to
find the distance between two points
in a coordinate plane. In Example 3
you will find the distance that a
soccer ball was kicked.
To find a general formula for the distance
between two points A(x v yf) and B{x v y 2 ),
draw a right triangle as shown at the right.
Using the Pythagorean theorem, you can
write the equation
(x 2 - Xj) 2 + Cy 2 - Ji) 2 = d 1 .
Solving the equation for d leads to the
following distance formula.
THE DISTANCE FORMULA
The distance d between the points (x.,, y .,) and (x 2 , y 2 ) is
d = V(x 2 - x .,) 2 + (k 2 - Kl ) 2 .
i Find the Distance Between Two Points
Use the distance formula to find the distance between (1,4) and (—2, 3).
d = V(x 2 — x x ) 2 + (y 2 — y^) 2 Write distance formula.
= V(— 2 - l) 2 + (3 - 4) 2 Substitute.
= V(-3) 2 + (-1) 2 Simplify.
= V9 + 1
= VTo
~ 3.16
Evaluate powers.
Add.
Use a calculator.
Chapter 12 Radicals and More Connections to Geometry
Find the Distance Between Two Points
Find the distance between the points. Round your solution to the nearest
hundredth if necessary.
1.(2, 5), (0,4) 2. (-3, 2), (2,-2)
3. ( 8 , 0), (0, 6 )
4. (-4, 2), (-1,3)
Student HeCp
^
►Vocabulary Tip
Vertices is the plural of
vertex. A triangle has
three vertices.
v _ J
2 Check a Right Triangle
Determine whether the points
(3,2), (2, 0), and (-1,4) are
vertices of a right triangle.
Solution
Use the distance formula to find the lengths of the three sides.
d l = V(3 - 2) 2 + (2 - 0) 2 = Vl 2 + 2 2 = Vl + 4 = V5
d 2 = V[3 — (-1)] 2 + (2 - 4) 2 = V4 2 + (— 2) 2 = Vl6 + 4 = V20
d 3 = V[2 — (- 1)] 2 + (0 - 4) 2 = V3 2 + (—4) 2 = V9 + 16 = V25"
Next find the sum of the squares of the lengths of the two shorter sides.
^2 = (^" ) 2 (^20^ ) 2 Substitute for cf 1 and d 2 .
= 5 + 20 Simplify.
= 25 Add.
The sum of the squares of the lengths of the two shorter sides is 25, which is
equal to the square of the length of the longest side, (V25~ ) 2 .
ANSWER ► By the converse of the Pythagorean theorem, the given points are
vertices of a right triangle.
Check a Right Triangle
Determine whether the points are the vertices of a right triangle.
c
(3, 5)
J
/
Q
2
\
J
2
—-
--
(5,3)
1
(0,2)
r 1
5
5 x
12.7 The Distance Formula
DRAW A DIAGRAM To use the distance formula to find a distance in a real-life
problem, the first step is to draw a diagram with coordinate axes and assign
coordinates to the points. This process is called superimposing a coordinate
system on the diagram.
Student Hedp
► More Examples
More examples
^ are available at
www.mcdougallittell.com
3 Apply the Distance Formula
SOCCER A player kicks a soccer ball from a position that is 10 yards from a
sideline and 5 yards from a goal line. The ball lands 45 yards from the same
goal line and 40 yards from the same sideline. How far was the ball kicked?
Solution
Begin by superimposing a coordinate system on the soccer field as below.
Assuming the kicker is left of the goalie, the ball is kicked from the point
(10, 5) and lands at the point (40, 45). Use the distance formula.
d = V(x 2 - x x ) 2 + Cy 2 -
= V(40 - 10) 2 + (45 - 5) 2
= V30 2 + 40 2
= V900 + 1600
= V2500
= 50
Write the distance formula.
Substitute.
Simplify.
Evaluate powers.
Add.
Find the square root.
ANSWER ► The ball was kicked 50 yards.
Apply the Distance Formula
8. A player kicks a football from a position that is 15 yards from a sideline
and 25 yards from a goal line. The ball lands at a position that is 30 yards
from the same sideline and 65 yards from the same goal line. Find the
distance that the ball was kicked.
Chapter 12 Radicals and More Connections to Geometry
H Exercises
Guided Practice
Vocabulary Check 1 . The distance formula is related to which theorem?
Skill Check
Use the coordinate plane to estimate the distance between the two
points. Then use the distance formula to find the distance between the
points. Round your solution to the nearest hundredth.
2. (1, 5), (-3, 1) 3. (-3, -2), (4, 1) 4. (5, -2), (-1,1)
Determine whether the points are vertices of a right triangle.
5. (0, 0), (20, 0), (20, 21) 6- (4, 0), (4, -4), (10, -4)
7. (-2, 0), (-1, 0), (1, 7) 8. (2, 0), (-2, 2), (-3, -5)
9. SOCCER Suppose the soccer ball in Example 3 on page 732 lands in a
position that is 25 yards from the same goal line and 25 yards from the
same sideline. How far was the ball kicked?
Practice and Applications
FINDING DISTANCE Find the distance between the two points. Round
your solution to the nearest hundredth if necessary.
10. (2, 0), (8,-3)
13. (5, 8), (-2, 3)
16. (4, 5), (-1,3)
19. (7, 12), (-7, -4)
22. (-1,9), (0, 7)
11.(2, -8), (-3, 3)
14. (-3, 1), (2, 6)
17. (-6, 1), (3, 1)
20 . (2, 1), (8, 4)
23. (4, 11), (-5, 2)
12. (3, -2), (0, 3)
15. (—6, —2), (—3, —5)
18. (-2, -1), (3, -3)
21.(2, 1), (—4, 16)
24. (-10,-2), (1,7)
I Student HeCp
^Homework Help
Example 1: Exs. 10-24
Example 2: Exs. 25-30
Example 3: Exs. 31-37
^ _
RIGHT TRIANGLES Graph the points. Determine whether they are
vertices of a right triangle.
25. (4, 0), (2,1), (-1,-5)
27.(1, -5), (2, 3), (-3,4)
29. (-3, 2), (-3, 5), (0, 2)
26. (5, 4), (2, 1), (-3, 2)
28. (-1, 1), (—3, 3), (-7, -1)
30. (3, -1), (2, 4), (-3,0)
12.7 The Distance Formula
CARTOGRAPHERS prepare
maps using information from
surveys, aerial photographs,
and satellite data.
More about
^ h ' cartographers at
www.mcdougallittell.com
a
Geometry M In Exercises 31 and 32,
the diagram at the right.
31. Copy the diagram of triangle ABC on
graph paper. Find the length of each side
of the triangle.
32. Find the perimeter of triangle ABC to the
nearest hundredth.
In Exercises 33 and 34, use the following information.
You are planning a family vacation. Each
500
side of a square in the coordinate plane
that is superimposed on the map represents
400
Amusement
50 miles.
Park
300
(100,250)
33. How far is it from your home to the
•
amusement park?
200
34. You leave your home and go to
100
the amusement park. After visiting
Home
(0,0)
the amusement park, you go to the
°j
) 100 200
beach. You return home. How far
did you travel?
Beach
(450,450)
Campground
(350, 200)
Zoo
(450, 50)
MAPS In Exercises 35-37, use the map. Each side of a square in the
coordinate plane that is superimposed on the map represents 95 miles.
The points represent city locations.
35. Use the distance formula to estimate
the distance between Pierre, South
Dakota, and Santa Fe, New Mexico.
36. Use the distance formula to estimate
the distance between Wichita, Kansas,
and Santa Fe, New Mexico.
37. Use the distance formula to estimate
the distance between Pierre, South
Dakota, and Wichita, Kansas.
i
u
F
5 ier
re,:
SD
.(30, 420
'
VI
Wichita, KS
(i
70,
-1
5)
•
X
•
Santa Fe, NM
(
-31
05,
-IE
iO)
1
CHALLENGE In Exercises 38 and 39, use the distance formula to find
the perimeter of the geometric figure.
10
ky
(
2,6
)
(8, 8)
6
/
"A
/
\
2
/
\
\
(0,0)
(
10(14, 0)'*
Chapter 12 Radicals and More Connections to Geometry
Standardized Test
Practice
Mixed Review
Maintaining Skills
40. MULTIPLE CHOICE What is the distance between (— 6 , —2) and (2, 4)?
(A) 2V5 CD 2V7 CD 10 CD 28
41. MULTIPLE CHOICE The vertices of a right triangle are (0, 0), (0, 6 ), and
( 6 , 0). What is the length of the hypotenuse?
CD 6 CD 6V2 CD 36 CD 72
FACTORING Factor the expression. (Lesson 10.7)
42. m 2 - 25 43. 81x 2 - 144 44. 16/ 2 - 49
45. x 2 + I2x + 36 46. c 2 - 22c + 121 47. 9s 2 + 6s + 1
48. 4n 2 - 64 49. 72 - 50 <p 2 50. 60y 2 - 240
FACTORING COMPLETELY Factor the expression completely.
(Lesson 10.8)
51. 3y 3 + 15y 2 - ISy 52. 2^ — 98?
53. 2x 4 - 8x 2 54. c 3 + 2c 2 - 8c -16
SIMPLIFYING EXPRESSIONS Simplify the expression. (Lesson 11.3)
4x
15x
57.
—48.x 3
28
“■IT
— 12x 2
18x 3
— 3x 2 + 21 x
60.
35x — lx
56x 7
5s - ^
49x
DIVIDING POLYNOMIALS Find the quotient. (Lesson 11.3)
61. Divide (—Ax 1 — 2Ax) by —Ax. 62. Divide (7 p 5 + 18p 4 ) by p 4 .
63. Divide (9a 2 — 21a —36) by (a + 1 ). 64. Divide (An 2 —Ain + 45) by (An — 5).
ADDING RATIONAL EXPRESSIONS Simplify the expression.
(Lessons 11.5, 11.6)
65. - + X -^-
X X
66 .
+
4 a + 1 4 a + 1
2 12
67. f - + -
2x x
68 .
2x 5
x + 1 x + 3
70.
6x 2x + 4
x + 1 x + 1
FRACTIONS AND PERCENTS Write the fraction as a percent.
(Skills Review p. 769)
71.
2
5
7z ‘?
73 -y
74 4
75.
5
8
76 -^
11 -~k
78 4
12.7
The Distance Formula
The Midpoint Formula
Goal
Find the midpoint of a
line segment in a
coordinate plane.
Key Words
• midpoint
• midpoint formula
How are computer games designed?
You can use the midpoint formula to
find the midpoint of a line segment in
a coordinate plane. In Example 3 you
will locate the midpoint as part of
designing a computer game.
The midpoint of a line segment is the point on the segment that is equidistant
from its endpoints.
Student HeCp
► Study Tip
Midpoint can be
thought of as an
average.
i Find the Midpoint
Find the midpoint of the line segment connecting the points (—2, 3) and (4, 2).
Use a graph to explain the result.
THE MIDPOINT FORMULA
^ The midpoint between (x r yj and (x 2 , y 2 ) is |
/x 1 +x 2 K! + K 2 \
l 2 ' 2 /
Solution
Let (-2, 3) = (x v jj) and (4, 2) = (x 2 , y 2 ).
*i +x 2 y t + y 2 \ = ( -2 + 4 3 + 2 \_ /2 5\ / 5_\
2 ’ 2 / \ 2 ’ 2 / \ 2 ’ 2 / \ ’ 2 /
ANSWER ► The midpoint is (l,
From the graph, you can see that the point
1, appears to be halfway between
(—2, 3) and (4, 2). In Example 2 you will
use the distance formula to check a midpoint.
Find the Midpoint
Find the midpoint of the line segment connecting the given points.
1- (-2, 3), (4, 1) 2. (2, 5), (2, -1) 3. (0, 0), (4, 6) 4. (1, 2), (2, -2)
Chapter 12.8 Radicals and More Connections to Geometry
You can use the distance formula to check that the distances from the midpoint to
each given point are equal.
Student HeCp
p More Examples
More examples
are available at
www.mcdougallittell.com
2 Check a Midpoint
Use the distance formula to check the midpoint in Example 1.
Solution
The distance between ^1, and (—2, 3) is
= y <- 2 - ‘> 2 + ( 3 - if = ^<- 3 > 2 + (if
= J 9 + t =
V37
2 '
The distance between ^1, and (4, 2) is
d 2 = y <4 - 1)’-+( 2 - f ) 2 =+(- 1) 2
V37
2 •
ANSWER ► The distances from
1, £ ) to the ends of the segment are equal.
SOFTWARE ENGINEERS
design and develop computer
programs that are used to
perform desired tasks. These
programs are referred to as
computer software.
Apply the Midpoint Formula
COMPUTERS You are using software to design a computer game. You want to
place a buried treasure chest halfway between the points corresponding to a
palm tree and a boulder. Where should you place the treasure chest?
Solution
The palm tree is located at (200, 75). The
boulder is at (25, 175). Use the midpoint
formula to find the halfway point between
the two landmarks.
( x 1 + *2 Ti + y 2 \ = ( 25 + 200 175 + 75 \
V 2 9 2 y v 2 2 )
= (225 25o\
V 2 ’ 2 /
= (112.5, 125)
Ki
200
150
100
50
0
C
r
r it
-
_
■
■
<
25,175)
_
□
_
_
—
s
*
(200
2
i)
1
. .
.
_
"
) 50 100 150 200 x
ANSWER ^ You should place the treasure chest at (112.5, 125).
Apply the Midpoint Formula
5. In the computer video game in Example 3, you want to place another buried
treasure halfway between the boulder and the treasure chest. What are the
coordinates of the point?
12.8 The Midpoint Formula
MU Exercises
Guided Practice
Vocabulary Check 1 , What is meant by the midpoint of a line segment?
2 . Give two methods for checking the midpoint of a line segment.
Skill Check Find the midpoint of the line segment with the given endpoints.
3. (4, 4), (-1, 2) 4. (6, 2), (2, -3) 5. (-5, 3), (-3, -3)
6. (-4,4), (2, 0) 7. (0, 0), (0, 10) 8. (2, 1), (14, 6)
Find the midpoint of the line segment with the given endpoints. Then
show that the midpoint is the same distance from each given point.
9. (-2, 0), (6, 2) 10. (-2, 2) (2, -10) 11. (2, 6), (4, 2)
12. (-6,0), (-10,-2) 13. (-3, 6), (1,8) 14. (0, 0), (-8, 12)
Practice and Applications
FINDING THE MIDPOINT Find the midpoint of the line segment
connecting the given points.
15. (1, 2), (5, 4) 16. (0, 0), (0, 8) 17. (-1, 2), (7, 4)
18. (0, -3), (-4, 2) 19. (-3, 3), (2, -2) 20. (5, -5), (-5, 1)
21. (-1, 1), (-4, -4) 22. (-4, 0), (-1, -5) 23. (-4, -3), (-1, -5)
CHECKING A MIDPOINT Find the midpoint of the line segment
connecting the given points. Then show that the midpoint is the
same distance from each point.
24. (7, -3), (-1, -9) 25. (1, 2), (0, 0) 26. (3, 0), (-5, 4)
27. (5, 1), (1, -5) 28. (2, 7), (4, 3) 29. (-3, -2), (1, 7)
30. (-3, -3), (6, 7) 31. (-9, 17), (5, -7) 32. (-4, -2), (10, -6)
Student MeCp
^ --V
► Homework Help
Example 1: Exs. 15-23
Example 2: Exs. 24-32
Example 3: Exs. 33-37
Ge ometry Linkp In Exercises 33 and 34, use the diagram below.
33. Find the midpoint of each side of
the triangle.
34. Join the midpoints to form a new
triangle. Find the length of each
of its sides.
Chapter 12.8 Radicals and More Connections to Geometry
Student HeCp
► More Examples
Extra help with
-^py problem solving in
Ex. 35 is available at
www.mcdougallittell.com
35. History Link Pony Express stations were 10 to 15 miles apart. The
latitude-longitude coordinates of 2 former stations in Nevada are
(40.0° N, 115.5° W) and (39.9° N, 115.2° W). These stations were about
22 miles apart. Find the coordinates of the station halfway between them.
HIKING TRIP In Exercises 36 and 37, use the following information.
You and a friend go hiking. You hike 3 miles north and 2 miles west. Starting
from the same point, your friend hikes 4 miles east and 1 mile south.
36. At the end of the hike how far apart are you and your friend? HINT: Draw a
diagram on a grid.
37. If you and your friend want to meet for lunch, where could you meet so that
both of you hike the same distance? How far do you have to hike?
Standardized Test
Practice
38. MULTIPLE CHOICE What is the midpoint between (—2, —3) and (1, 7)?
®({.-2) ®(-{,2) ©({, 2 ) ®(-|,5)
ARRANGING LIKE TERMS Use linear combinations to solve the linear
system. Then check your solution. (Lesson 7.3)
39. 4x + 3y = 1 40. 3x + 5y = 6 41. 2x + 3y = 1
2x — 3 y = 1 —4x + 2y = 5 5x — 4y — 14
INTERPRETING ALGEBRAIC RESULTS Use the substitution method or
linear combinations to solve the linear system and tell how many
solutions the system has. (Lesson 7.5)
42. 2x + y = 3 43. 2x + 2y = 3 44. 2x + y = —4
4x + 2y = 8 4x + 2y = 6 y + 2x = 8
Maintaining Skills comparing fractions, decimals, and percents Complete the
statement using <, >, or =. (Skills Review pp. 768-771)
45.54% ? 0.54 46. | ? 6|% 47. @ 0.03 48.0.23 ?
12.8 The Midpoint Formula
Logical Reasoning:
Proof
Goal
Use logical reasoning
and proof to prove that a
statement is true or false.
Key Words
• postulate
• axiom
• theorem
• indirect proof
• counterexample
How can a lawyer prove that a client is not guilty?
Often lawyers use logical
reasoning to defend a client in
court. In Example 4 you will
use logical reasoning to prove
your client’s innocence.
LOGICAL REASONING Mathematics is believed to have begun with practical
“rules of thumb” that were developed to deal with real-life problems. Then, about
2500 years ago, Greek geometers (specialists in geometry) developed a different
approach to mathematics. Starting with a handful of properties that they believed
to be true, they insisted on logical reasoning as the basis for developing more
elaborate mathematical tools, or theorems.
AXIOMS The properties that mathematicians accept without proof are called
postulates or axioms. Many of the rules discussed in Chapter 2 fall in this
category. The following is a summary of the rules that underlie algebra.
Chapter 12 Radicals and More Connections to Geometry
DEFINITIONS In order to formulate the axioms and postulates of mathematics,
one needs a vocabulary of terms such as number , equal , addition , point , and /me.
Aside from their role in formulating axioms, these terms can also be used to
define other terms. For example, whole number and addition are used to define
integer and subtraction. Definitions do not need to be proved.
THEOREMS Recall that a theorem is a statement that can be proven to be true.
All proposed theorems have to be proved. For instance, you can use the basic
axioms to prove the theorem that for all real numbers b and c,c(—b) = —cb.
Once a theorem is proved, it can be used as a reason in proofs of other theorems.
Student HeCp
► Study Tip
When you are proving
a theorem, every step
must be justified by an
axiom, a definition,
given information, or
a previously proved
theorem.
I _ J
1 Prove a Theorem
Use the subtraction property, a — b =
theorem: c{a — b) — ca — cb.
a + (—/?), to prove the following
Solution c(a — b) = c[a + (—/?)]
Subtraction property
= ca + c(—b)
Distributive property
= ca + {—cb)
Theorem stated above
= ca — cb
Subtraction property
Prove a Theorem
1 . Use the associative and commutative properties to prove the following theorem.
If a, b, and c are real numbers, then (a + b) + c = (b + c) + a.
CONJECTURES A conjecture is a statement that is thought to be true but has
not yet been proved. Conjectures are often based on observations.
HH 2 Gold bach's Conjecture
Christian Goldbach (1690-1764) thought the following statement might be
true. It is now referred to as Goldbach’s Conjecture.
Every even integer, except 2, is equal to the sum of two prime numbers.
The following list shows that every even number between 4 and 26 is equal to
the sum of two prime numbers. Does this list prove Goldbach’s Conjecture?
4 = 2 + 2 6 = 3 + 3 8 = 3 + 5 10 = 3 + 7
12 = 5 + 7 14 = 3 + 11 16 = 3 + 13 18 = 5 + 13
20 = 3 + 17 22 = 3 + 19 24 = 5 + 19 26 = 3 + 23
Solution
This list of examples does not prove the conjecture. No number of examples
can prove that the rule is true for every even integer greater than 2. (At the
time this book was published, no one had been able to prove or disprove
Goldbach’s Conjecture.)
a
12.9 Logical Reasoning: Proof
Student HcCp
>
^ Look Back
For help with counter¬
examples, see p. 73.
L j
COUNTEREXAMPLES Sometimes a person makes a general statement they
suppose to be true. To show that a general statement is false, you need only
one counterexample.
3 Find a Counterexample
Show that the statement below is false by finding a counterexample.
For all numbers a and b, a + (—b) = (—a) + b.
Solution The statement claims that a + (—b) — (—a) + b for all values of
a and b. If we let a = 1 and b = 2, we find a + (~b) = 1 + (—2) = — 1, but
(—a) + b = (— 1) + 2 = 1. Since — 1 + 1, the counterexample a = 1 and
b = 2 shows that the general statement proposed above is false.
LAWYERS represent people
in criminal and civil trials by
presenting evidence
supporting their clients case.
They also give advice on
legal matters.
More about lawyers
4^ is available at
www.mcdougallittell.com
INDIRECT PROOF In this lesson you have used direct proofs to prove that
statements are true and counterexamples to prove that statements are false.
Another type of proof is indirect proof. To prove a statement indirectly, assume
that the statement is false. If this assumption leads to an impossibility, then you
have proved that the original statement is true. An indirect proof is also called
a proof by contradiction.
Use of Contradiction in Real Life
LAWYERS You are a lawyer defending a client accused of violating a law on
the north side of town at 10:00 A.M. on March 22. You argue that if guilty, your
client must have been there at that time. You have a video of your client being
interviewed by a TV reporter on the south side of town at the same time.
You argue that it would be impossible for your client to be in two different
places at the same time on March 22. Therefore your client cannot be guilty.
5 Use an Indirect Proof
Use an indirect proof to prove the following statement.
If & is a positive integer and a 2 is divisible by 2, then a is divisible by 2.
Solution Suppose the statement is false. Then there exists a positive integer
a such that a 2 is divisible by 2, but a is not divisible by 2. If so, a is odd and
can be written as a = 2n + 1.
a = 2n + 1 Definition of odd integer
a 2 = An 2 + 4/2+1 Apply FOIL to (2 n + 1)(2 n + 1).
a 2 = 2(2 n 2 + 2 ri)+ 1 Distributive property
a 2 is odd Definition of odd integer
The proof contradicts the assumption, thereby showing a is divisible by 2.
Chapter 12 Radicals and More Connections to Geometry
Student HeCp
p Morel Examples
M°r e examples
are available at
www.mcdougallittell.com
6 Use an Indirect Proof
Use an indirect proof to prove that V2 is an irrational number.
Solution
If you assume that \fl is not an irrational number, then \fl is rational and can
be written as the quotient of two integers a and b that have no common factors
other than 1 .
\fl — y- Assume \fl is a rational number.
b
2 = Square each side.
2 b 2 — a 2 Multiply each side by b 2 .
This implies that 2 is a factor of a 2 . Therefore 2 is also a factor of a. Thus a can
be written as 2 c.
lb 2 = (2c ) 2 Substitute 2c for a.
2 b 2 = 4 c 2 Simplify.
b 2 = 2c 2 Divide each side by 2.
This implies that 2 is a factor of b 2 and also a factor of b. So 2 is a factor of
both a and b. But this is impossible because a and b have no common factors
other than 1. Therefore it is impossible that \fl is a rational number. So you
can conclude that \fl must be an irrational number.
Use of Contradiction in Real Life
2 _ You are defending a client who is accused of violating a law near her home at
9:00 A.M. on June 5. Your client’s boss and coworkers testify that she arrived
at work at 9:15 A.M. on June 5. It takes your client 45 minutes to commute
from her house to work. Construct an argument to prove that your client is
not guilty.
iftl Exercises
Guided Practice
Vocabulary Check 1 . Explain the difference between an axiom and a theorem.
2 _ What is the first step in an indirect proof?
Skill Check In Exercises 3-8, state the basic axiom of algebra that is represented.
3- y(l) = y 4. 2x + 3 = 3 + 2x 5- 5(x + y) = 5x + 5y
6- (4 x)y = 4 (xy) 7. y + 0 = y 8. x + (— x) — 0
12.9 Logical Reasoning: Proof
Practice and Applications
9. STATING REASONS Copy and complete the proof of the statement:
For all real numbers a and b, (a + b) — b = a.
(a + b) — b = (a + b) + (~b) Definition of subtraction
= a + [b + (—b)\ Associative property of addition
= a + 0 _?_
Student HeCp
► Homework Help
Extra help with
problem solving in
Exs. 13-16 is available at
www.mcdougallittell.com
P Student HeCp
Homework Help
Example 1: Exs. 9-11
Example 2: Exs. 12,18
Example 3: Exs. 13-17
Example 4: Exs. 19-22
Example 5: Exs. 19-22
Example 6: Exs. 19-22
1 _
PROVING THEOREMS In Exercises 10 and 11, prove the theorem. Use
the basic axioms of algebra and the definition of subtraction given in
Example 1.
10, If a and b are real numbers, then a — b = —b + a.
11 - If a, b, and c are real numbers, then ( a — b)c — ac— be.
12- MAKING A CONJECTURE A student proposes the following conjecture:
The sum of the first n odd integers is n 2 .
She gives four examples: 1 = l 2 , 1 + 3 = 4 = 2 2 , l + 3 + 5 = 9 = 3 2 , and
1 + 3 + 5 + 7 = 16 = 4 2 . Do the examples prove her conjecture? Explain.
Do you think the conjecture is true?
FINDING A COUNTEREXAMPLE In Exercises 13-16, find a
counterexample to show that the statement is not true.
13- If a and b are real numbers, then (<a + b) 2 = a 2 + b 2 .
14- If a , b , and c are nonzero real numbers, then (a + b) + c = a + (b + c).
(Note: The counterexample shows that the associative property does not hold
for division.)
15- If a and b are integers, then a -r- b is an integer.
16- If a > 4, then \fa is not rational.
17. THE FOUR-COLOR PROBLEM
A famous theorem states that any map
can be colored with four different colors
so that no two countries that share a border
have the same color. No matter how the
map shown at the right is colored with
three different colors, at least two countries
having a common border will have the
same color. Does this map serve as a
counterexample to the following statement?
Explain.
Any map can be colored with
three different colors so that
no two countries that share a
border have the same color.
Brazil
Paraguay
Argentina
Chapter 12 Radicals and More Connections to Geometry
18 , Geo metry Explain how the diagrams below can be used to give a
geometrical argument to support the conjecture in Exercise 12 on page 744.
INDIRECT PROOF In Exercises 19-21, use an indirect proof to prove that
the conclusion is true.
19, Your bus leaves a track meet at 4:30 P.M. and does not travel faster than
60 miles per hour. The meet is 45 miles from home. Your bus will not get
you home in time for dinner at 5:00 P.M.
20- If a < b, then a + c < b + c.
21 - If ac > be and c > 0, then a > b.
22. PROOF USING THE MIDPOINT LetD
represent the midpoint between B and C,
as shown at the right. Prove that for any right
triangle, the midpoint of its hypotenuse is
equidistant from the three vertices of the
triangle. In order to prove this, you must first
find the distance between B and C. Using the
distance formula, you get BC = Vx 2 + y 2 ,
so BD and CD must be —Vx 2 + y 2 .
HINT: Use the distance formula to find the distance between A and D.
Standardized Test
Practice
23. CHALLENGE Explain how the following diagrams could be used to give a
geometrical proof of the Pythagorean theorem.
a
b
L
hfH
l"
i
h—i—*—i
24. MULTIPLE CHOICE What is the first step to prove the following theorem:
If & and b are real numbers and (x + a) = b, then x = b — a.
(A) x + (a — a) = b — a
CD x — b — a
Cc) (x + a) — a — b — a
Cp x + 0 — b — a
MULTIPLE CHOICE Which represents the distributive property?
CD (4 x)y = 4(xy)
<3D z(l) = z
(H) 4(jc + 1) = 4x + 4
CD y + 0 = y
12.9 Logical Reasoning: Proof
Mixed Review
Maintaining Skills
Quiz 3
PERCENTS Solve the percent problem. (Lesson 3.9)
26- How much is 15% of $15? 27. 100 is 1% of what number?
28. 6 is what percent of 3? 29. 5 is 25% of what number?
USING THE DISCRIMINANT Determine whether the equation has two
solutions, one solution , or no real solution. (Lesson 9.7)
30. x 2 — 2x + 4 = 0 31. 2x 2 + 4x — 2 = 0 32. 8 x 2 — 8x + 2 = 0
33. x 2 - 14x + 49 = 0 34. 3x 2 - 5x + 1 = 0 35. 6 x 2 - x + 5 = 0
SOLUTIONS Determine whether the ordered pair is a solution of the
inequality. (Lesson 9.8)
36. y > x 2 - 2x - 5, (1, 1) 37. y > 2x 2 - 8 x + 8 , (3, -2)
38. y < 2x 2 - 3x + 10, (-2, 20) 39. y > 4x 2 - 48x + 61, (1, 17)
OPERATIONS WITH FRACTIONS Evaluate the expression. Write the
answer as a fraction or as a mixed number in simplest form.
(Skills Review pp. 764-765)
41.
2 ^ J_ _ 5
7 ’ 14 4
10
4
5
45.
3
Use the distance formula to determine whether the points are the
vertices of a right triangle. (Lesson 12.7)
Find the distance between the two points. Round your solution to the
nearest hundredth if necessary. Then find the midpoint of the line
segment connecting the two given points. (Lessons 12.7, 12.8)
3. (1, 3), (7, -9) 4. (2, -5), ( 6 , -11) 5. (0, 0), ( 8 , -14)
6. (- 8 , - 8 ), (- 8 , 8 ) 7. (3, 4), (-3, 4) 8. (1, 7), (-4, -2)
Find a counterexample to show that the statement is not true.
(Lesson 12.9)
9. If a, b, and c are real numbers and a < b, then ac < be.
10. If a and b are real numbers, then —(a + b) = (— a ) — (—£>).
Chapter 12 Radicals and More Connections to Geometry
Chapter Summary
and Review
• square root function, p. 692
• Pythagorean theorem, p. 724
• midpoint formula, p. 736
\
• extraneous solution, p. 70S
• hypotenuse, p. 724
• postulate, p. 740
• cube root of 0, p. 710
• legs of a right triangle, p. 724
• axiom, p. 740
• rational exponent, p. 711
• converse, p. 726
• conjecture, p. 741
• completing the square, p. 716
• distance formula, p. 730
• indirect proof, p. 742
• theorem, p. 724
< _
• midpoint, p. 736
\ 2 .\
Functions Involving Square Roots
Examples on
pp. 692-694
To sketch the
graph of y = Vx — 1, note that
the rule is defined for all
nonnegative numbers. Make
a table of values, plot the points,
and connect them with a smooth
curve. The range is all numbers
greater than or equal to — 1 .
X
0
11
&
I
II
1
1
0
11
I
II
2
Vi
11
Si
1
n
4^
3
V
II
Si
l
n
<1
U 4
4
II
1
II
V
Find the domain of the function. Then sketch its graph and find the range.
1 . y = 11 Vx 2. y = 2Vx — 5 3. y = Vx + 3
1 2.2 Operations with Radical Expressions
Examples on
pp. 698-700
You can use radical operations and the distributive property to
simplify radical expressions.
4V20 - 3V5 = 4V4T5 - 3V5
= 4V2V2 • V5 - 3V5
= 8V5 - 3V5
= 5V5
Perfect square factor
Product property
Simplify.
Subtract like radicals.
Chapter Summary and Review
Chapter Summary and Review continued
Simplify the expression.
4. 6V2 - V2
7. V6(2V3 - 4V2)
5. V5 + V20 - V3
8 .
21
V3
12.3 Solving Radical Equations
Solve V3x- 2 = jc.
O Square both sides of the equation. (V3x — 2) 2 =
© Simplify the left side of the equation. 3x — 2 =
© Write in standard form. 0 =
0 Factor the quadratic equation. 0 =
© Solve for x. x = 2 oi
CHECK y Substitute 2 and 1 in the original equation.
V3(2) -222 V3(l) -2±1
2 = 2 /
1 = 1 /
ANSWER ^ The solutions are 2 and 1.
Solve the equation. Check for extraneous solutions.
1 0.2Vx — 4 = 0 11. V—4x - 4 = X
13. Vx - 1 = 5 14. 8 Vx - 16 = 0
12.4 Rational Exponents
1^22221^® Simplify the expression (x 2 • x 1/2 • y) 2 .
© Use the product of powers property. (x 2 • x 1/2 • y) 2
© Use the power of a product property.
© Simplify by multiplying exponents.
Evaluate the expression without using a calculator.
16.27 2/3 17. (^64) 2 18. 121 3/2
Simplify the expression.
20 . 5 1/3 • 5 5/3 21 . (4 • 121 ) 1/2
■ —
6 . (3 - Vl0)(3 + VlO)
Examples on
pp. 704-706
x 2
x 2 — 3x + 2
(x - 2)(x - 1)
x = 1
12 . Vx - 3 + 2 = 8
15. V5x + 36 = x
Examples on
pp. 710-712
= (X 5/2 • y ) 2
= x (5/2 • 2) . y 2
= x y
19. (V^) 4
22 . (l25 2/3 ) 1/2
Chapter Summary and Review continued^
12.5 Completing the Square
Examples on
pp. 716-718
Solve x 2 — 6x — 1 = 6 by completing the square.
x 2 — 6x = 7 Isolate x 2 -term and x-term.
x 2 — 6x + 9 = 7 + 9 Add = 9 to each side.
(jc - 3) 2 = 16
x 3 — ±4
Write left side as perfect square.
Find square root of each side.
Solve for x.
Solve the equation by completing the square.
23.x 2 — 4x — 1 = 7 24. x 2 + 20x + 19 = 0 25. x 2 — 16x + 8 = 0
Choose a method and solve the quadratic equation. Explain your choice.
26.4x 2 + 8x + 8 = 0 27. x 2 - x - 3 = 0 28. 3x 2 - x + 2 = 0
12.6
Pythagorean Theorem and its Converse
Examples on
pp. 724-726
Given a = 6 and c = 12, find b.
O Write the Pythagorean theorem. a 2 + b 2 = c 2
© Substitute 6 for a and 12 for c. 6 2 + b 2 = 12 2
© Subtract 6 2 from each side and simplify. b 2 = 108
O Find square root of each side. b = 6V3
Find the missing length of the right triangle.
Determine whether the given lengths are sides of a right triangle. Explain
your reasoning.
34.
10
Chapter Summary and Review
Chapter Summary and Review continued
M.1A2.Z The Distance and Midpoint Formulas
Examples on
pp. 730-732, 736-737
Find the distance d and the midpoint m between (—6, —2) and (4, 3).
d = V(x 2 - x x ) 2 + (y 2 - )+
= V[4 - (—6)] 2 + [3 - (-2)] 2
= VlO 2 + 5 2
= Vl25
= 5V5
(x l +x 2 y 1 + y 2 ^
m =
2 9 2
-6 + 4 -2 + 3
2
1
= 1-^2
Find the distance between the two points. Round to the nearest hundredth.
35 . (8, 5) and (11, -4) 36 . (-3, 6) and (1, 7) 37 . (-2, -2) and (2, 8)
38 . Use the distance formula to decide whether the points (—4, 1), (0, —2), and
(—4, —2) are the vertices of a right triangle.
Find the midpoint of the line segment connecting the given points. Use a
graph to check the result.
39 . (-1, -3) and (5, 1) 40 . (0, 4) and (-2, 4) 41 . (9, -5) and (-10, -8)
\2A logical Reasoning: Proof
Prove that for all numbers a and b, (a + b) — b = a
(a + b) — b = (a + b) + (— b)
= a+[b + (~b)\
— a + 0
= a
Definition of subtraction
Associative property of addition
Inverse property of addition
Identity property of addition
42 . Which basic axiom of algebra is represented by
Examples on
pp. 740-743
T
43 . Prove that ( c)(—b ) = —cb for all real numbers c and b.
Chapter 12 Radicals and More Connections to Geometry
Find the domain of the function. Then sketch its graph and find the range
of the function.
1. y = 12Vx 2. y = V2x + 7 3. y = \ / 3x — 3 4. y = Vx — 5
Simplify the expression.
5. 3V2 — V2 6. (4 + V7)(4 - V7) 7. ^ 8. J
9-^f 10 . 2 ^vTI 11 .( 8 - V5X8 + V5) 12.V3(Vl2 + 4)
Solve the equation. Check for extraneous solutions.
13. Vy + 6 = 10 14. V2 m + 3-6 = 4 15. n = V9« - 18 16. p = V-3p + 18
Simplify the variable expression using the rules for rational exponents.
17. x m • x 3/2 18. V25X 3 19. (x 1/3 ) 2 • Vy 20. (x 2 • x 1/3 ) 3/2
Solve the equation by completing the square.
21. x 2 — 6x = —5 22. x 2 — 2x = 2 23. x 2 + 16x - 1 = 0
Find the missing length of the right triangle if a and b are the lengths of
the legs and c is the length of the hypotenuse.
24. a = 7, b = 24 25. a = 5, c = 13 26. = 15, c = 17
27. a = 30, b = 40 28. a = 6, c = 10 29. b = 12, c = 15
Determine whether the given lengths are sides of a right triangle. Explain your reasoning.
30.
8
In Exercises 33-35, use the diagram shown at the right.
33. Use the distance formula to find the length of each side of
the parallelogram.
34. Use your answers from Exercise 33 to find the perimeter of
the parallelogram.
35. Find the coordinates of the midpoint of each side of
the parallelogram.
36. Prove that if a , b, and c are real numbers and a + c = b + c, then a = b.
Chapter Test
Chapter Standardized Test
Test Tip Learn as much as you can about a test ahead of time, such as
the types of questions and the topics that the test will cover.
6 . Which of the following is the simplest
What is the value of y — 2
whenx = 8?
form of (vy 1/3 x 2/3 ) 3 ?
^^ 3V7
^^ 7
(5) x 6 y
CD A 1/9
® 16
® 8
CD * 5 y
CD -r 5 .y l0/3
7- What term should you add to x 2 —
2 . What is the range of the function
y = Vx + 7?
(A) All positive real numbers
Cg) All real numbers
CD All real numbers greater than
or equal to 7
CD All real numbers less than 7
3- Which of the following is the value of the
expression 5V7 + V448 + Vl75 - V63?
CD 15V7 CD 16V7
CD 18V7 CD 20 V 7
4. Which of the following is the simplest
f °mx> f 3_V6 ?
,_ v 6 + Vl2
CD
C® 3
6 + 2V6
^ 15
CD
6 + 2V6
6 + Vl2
15
5. Which of the following is a solution of the
equation x = V880 — 18x?
(A) -22 CD 0
CD 22 CD 40
create a perfect square trinomial?
(A) -36 CD -9
CD 9 CD 81
8 . What is the length of the missing side of
the triangle?
CD 10 CD 11 9
CD 12 CD 13 \j_
9 . What is the distance between points
P and (2?
6
0
P{~ 5, 6 '
2
21
1
-10 -6
2 1
r 2 6 X
CD V33
CD V73
CD V65
CD Vrn
10- Use the graph in Exercise 9. Find the
midpoint of the line segment connecting
the points P and Q.
(A)
CD
^3
2 ’
-3
,2
CD
CD
-7
2 ’
-7
,4
Chapter 12 Radicals and More Connections to Geometry
11. Choose the missing reason in the following proof that for all real numbers
a and b, —(a + b) = (— a ) + ( —b).
STATEMENTS
1 - a and b are real numbers
2. —{a + b) — (—1 )(a + b )
3. = (-l)a + (-l)fe
4. = (~a) + (—fo)
(A) Definition of subtraction
Cb) Associative property of addition
Cc) Inverse property of addition
Cd) Distributive property
CE) None of these
REASONS
1. Given
2_ Multiplication property of — 1
3. ?
4_ Multiplicative property of — 1
12, Which graph best represents the function y = 3Vx — 2?
13, Which of the following triangles is a right triangle?
Chapter Standardized Test
Cumulative Practice
Write the sentence as an equation or an inequality. Then use mental
math to solve the equation or the inequality. (1.4—1.5)
1- The quotient of m and 7 is greater than or equal to 16.
2 . The sum of 4 and the second power of b is equal to 104.
3. The distance t you travel by train is 3 times the distance d you live from the
train station. You drive 3 miles to get from your house to the train station.
Evaluate the expression for the given value of the variable. (2.2—2.6, 2.8)
4. 3 + v + (—4) whenx = 5 5. 2x + 12 — 5 whenx — 9 6. 3.5 — (—x) whenx = 1.5
7. — (—3) 2 (x) when x = 1 8. 6x(x + 2) when x = 2 9. (8x + 1)(—3) when x = l
10. ^ |(jc)(jc)(— x)\ whenx — 4 11. x + - whenx = 8 12. (—5)^ —^xj whenx = 6
Solve the equation. Round your solution to the nearest hundredth.
(3.1—3.4, 3.6)
13. -|(x - 5) = 12 14. lx - (3x - 2) = 38
16. 8(x + 3) — 2x = 4(x — 8) 17. 11 + 6.23x = 7 + 5.5lx
In Exercises 19 and 20, use the graph. (4.7, 5.3, 5.6)
19. Write an equation of a line passing through the point (2, —2)
and parallel to the line shown.
20. Write an equation of a line passing through the point (—4, 2)
and perpendicular to the line shown. Graph the equation in
the same coordinate plane to check your answer.
Determine whether the relation is a function. If it is a function, give the
domain and the range. (4.8)
22.
Write in standard form the equation of the line described below. (5.1-5.2)
4 1
25. Slope = y-intercept = —3 26. (— 1,2), m = —
Input
Output
-1
-1
1
-1
3
1
5
3
15. —x + 7 = —lx — 5
18. -3(2.9 - 4.lx) = 9.2x + 6
Chapter 12 Radicals and More Connections to Geometry
Solve the inequality. Then graph the solution. (6.3—6.5)
27. -3 < -4x + 9 < 14 28. I 3x + 16 I + 2 < 10 29. 3x - 4 > 5 or 5x + 1 < 11
Solve the linear system. (7.2-7.3)
30. 4y = 8x + 16 31. — 2x H- 3y = 15 32. y = 5x — 2
2y = lljc - 7 lOx - lly = 9 3x + ly = 5
Simplify. Then evaluate the expression when a — 1 and b = 2. (8.1-8.2, 8.4)
33.
b*
36. 4b 3 • (2 + b ) 2
34. 3<2 4 • a 3
4<2 3 Z? 3
37.
35. {~a 3 ){2b 2 ) 3
(5 ab 2 )~ 2
ab
38
- 3 ^
Determine whether the equation has two solutions, one solution , or no
real solution. Then solve the equation. (9.2, 9.6-9.7,10.5)
39. 6x 2 + 8 = 34 40. 4X 2 - 9x + 5 = 0 41. 3x 2 + 6x + 3 = 0
Completely factor the expression. (10.5-10.7)
42.x 2 + 6x + 8 43. x 2 - 24x - 112
45. 4x 2 + 12x + 9 46. x 2 + lOx + 25
Solve the equation. (10.4-10.8)
48. (3x + l)(2x + 7) = 0 49. 6x 2 - x - 7 = 8 50. x 2 - 4x + 4 = 0
51. 4x 2 + 16x + 16 = 0 52. x 3 + 5x 2 - 4x - 20 = 0 53. x 4 + 9x 3 + 18x 2 = 0
44. 3x 2 + 17x — 6
47. x 2 - 14x + 49
Simplify the expression. (11.3-11.7)
54.
4x
12x 2
55.
2x + 6
x 2 - 9
56.
3x
x 2 — 2x — 24
x — 6
6x 2 + 9x
57.
x 2 — 6x + 8
x 2 — 2x
- (3jc - 12)
58.
4
x + 2
+
15x
3x + 6
59.
3x
x + 4
x
X - 1
Simplify the expression. (12.2)
60.4V7 + 3V7 61.9V2-12V8 62. V6(5V3 + 6) 63.
Solve the equation by completing the square. (12.5)
64.x 2 + 24x = -3 65. x 2 - 12x = 19
67.x 2 - 6x - 13 = 0 68 . x 2 + 16x - 1 = 0
66 . x 2 + 20x = —7
69. x 2 + 22x + 5 = 0
Find the distance between the two points. Round your solution to the
nearest hundredth if necessary. Then find the midpoint of the line
segment connecting the two points. (12.7-12.8)
70. (3, 0), (—5, 4)
74. (-1,2), (6, 9)
71.(2, 7), (4, 3)
75. (0, 4), (10, 11)
72. (5, 1), (1, -5)
76. (-5, -7), (5, 7)
73. (6, 2), (-2, -3)
77.(1,-1), (3, 10)
Cumulative Practice
Chapters
Materials
• graph paper
• metric ruler
• graphing calculator
(optional)
Before the pediment on top of
the Parthenon in Athens was
destroyed, the front of the
building fit almost exactly into a
golden rectangle.
10-12 Project
Investigate
Iden Ratio
OBJECTIVE Explore what the golden ratio is and how it is used.
Over the centuries, the golden rectangle has fascinated artists, architects, and
mathematicians. For example, the golden rectangle was used in the original
design of the Parthenon in Athens, Greece. A golden rectangle has the special
shape such that when a square is cut from one end, the ratio of length to width
of the remaining rectangle is equal to the ratio of length to width of the original
rectangle.
b =
golden rectangle
Investigating The Golden Ratio
smaller golden
rectangle
From the picture, the large rectangle has a ratio of length to width a + while
the small b-by-a rectangle that remains after cutting off the b-by-b square has a
b
ratio of length to width — . For a golden rectangle, the ratio of length to width of
the large rectangle is equal to the ratio of the small rectangle. In other words,
a + b
b
a *
b
Let r = — represent the ratio of length to width of a golden rectangle. This
ratio r is called the golden ratio. To derive the exact value of r, rewrite the
equality above.
2
r - r
■+1=±
n .. a + b a , -
Rewrite — t— as -r + 1.
a
b b
+ 1 = r
Substitute r for
a
. + r = r 2
Multiply each side by r.
■-1=0
Write equation in standard form.
- V5 1 - V5
2 or r = 2
Use quadratic equation to solve for r.
Since r > 0, the golden ratio r is given by r = * + or about 1.618034.
Chapter 12 Radicals and More Connections to Geometry
Constructing Golden Rectangles
O On graph paper,
draw a 1-by-l
square.
© On one side of
the square add
another l-by-l
square.
© Build a 2-by-2
square on the
longest side of the
l-by-2 rectangle.
© Build a 3-by-3 square
on the longest side of
the 3-by-2 rectangle.
None of the rectangles in Steps 1-4 are golden rectangles. It is not possible
to construct a golden rectangle with integer side lengths. However, it is
possible to construct rectangles with integer side lengths whose ratios of
length to width are very close to the golden ratio.
1 _ Continue the pattern from Steps 1-4 to draw the next four rectangles.
2 _ Copy and complete the table. If necessary, round to four decimal places.
length b
3
5
8
13
21
34
width a
2
3
5
8
13
21
b
a
1.5
1.6667
?
?
?
?
3. How do the ratios in your table compare to the golden ratio?
Presenting The Results
Write a report or make a poster to present your results. Include a sketch of a
golden rectangle and include your answers to Exercises 1-3. Then describe
what you learned about the golden ratio and the golden rectangle.
Extending The Project
• The average chicken egg fits inside a golden rectangle. Measure the
lengths and widths of six eggs and find the approximate ratio of length to
width for each. Then find the average of these ratios.
• Find some rectangular objects that you think may have a length to width
ratio close to the golden ratio. Measure them to see if they approximate
golden rectangles. You might try a picture frame, a $1 bill, or a TV screen.
Project
Skills Review Handbook _ _ pages 759-782
I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II
• Decimals
759-760
• Factors and Multiples
• Fractions
761-762
763-766
• Writing Fractions and Decimals
• Comparing and Ordering Numbers
• Perimeter, Area, and Volume
• Estimation
767-769
770-771
772-773
774-776
• Data Displays
• Measures of Central Tendency
• Problem Solving
Extra Practice for Chapters 1-12
777-779
780
781-782
pages 783-794
End-of-Course Test
pages 795-796
Tables
pages 797-802
• Symbols
• Formulas
797
798
• Properties
• Squares and Square Roots
• Measures
799
800-801
802
Glossary
English -to-Spanish Glossary
Index
...
Selected Answers
Student Resources
Skills
Decimals
To add and subtract decimals, you can use a vertical format. When you do this,
line up the decimal places. Use zeros as placeholders as needed to help keep the
decimal places aligned correctly. The steps are similar to those used for adding
and subtracting whole numbers.
■*:f:!Sldf Add 3.7 + 0.77 + 9.
SOLUTION Write the addition problem in vertical form. Line up the decimal
points. Use zeros as placeholders.
3.70
0.77
+ 9.00
13.47
ANSWER t 3.7 + 0.77 + 9 = 13.47
Subtract 21.32 - 18.78.
SOLUTION Write the subtraction problem in vertical form. Line up the
decimal points.
21.32
- 18.78
2.54
ANSWER} 21.32 - 18.78 = 2.54
Decimal multiplication is similar to multiplication with whole numbers. When
multiplying decimals, you need to know where to put the decimal point in the
product. The number of decimal places in the product is equal to the sum of the
number of decimal places in the factors.
Multiply 6.84 x 5.3.
SOLUTION Write the multiplication problem in vertical form. When
multiplying decimals, you do not need to line up the decimal points.
6.84
two decimal places
X 5.3
one decimal place
2052
34200
36.252
three decimal places
ANSWER ► 6
.84 X 5.3 = 36.252
Vs
Skills Review Handbook
SKILLS REVIEW
SKILLS REVIEW
You can divide decimals using long division. The steps for dividing decimals
using long division are the same as the steps for dividing whole numbers using
long division. When you use long division to divide decimals, line up the decimal
place in the quotient with the decimal place in the the dividend. If there is a
remainder, write zeros in the dividend as needed and continue to divide.
■SfJiSIsH J Divide 0.085 -4 0.2.
SOLUTION Write the problem in long division form.
0.2)0.085
Move the decimal points in the divisor and dividend the same number of places
until the divisor is a whole number. Then divide.
r- Line up decimal place in quotient with
0.425 decimal place in dividend.
0.2)0.085 2)0.850-- Write a zero in dividend so you can
0.8 continue to divide.
Move decimal points 5
one place to the right. 4
To
10
0
ANSWER ► 0.085 4 - 0.2 = 0.425
V, __
Practice
Find the sum.
1.7.92 + 6.5
2. 12.36 + 9
3. 28.012 + 94.3
4. 19.9 + 93.8 + 5.992
5. 9.02 + 8 + 8.7
6. 2.25 + 7.789 + 4.32
Find the difference.
7. 3.42 - 2.4
8. 0.88 - 0.39
9. 2.91 - 0.452
10. 15 - 6.32 - 1.44
11. 10.24 - 3.1 - 0.07
12. 94.48 - 16.7 - 42.902
Find the product.
13. 6.25 X 6.5
14. 0.26 X 9.58
15. 0.15 X 24
16.64 X 3.51
17. 183.62 X 2.834
18. 510.375 X 80.2
Find the quotient.
19. 133.6 4- 8
20. 57.3 4 - 0.003
21. 231.84 4 - 12.6
22. 100.38 4 - 21
23. 84.4 4 - 0.02
24. 2712.15 - 35
25. You bought a shirt for $24,
a pair of pants for $25.99,
and a pair of shoes for
$12.45. How much did you spend all together? If you
give the cashier $70,
how much change will you receive?
H
Factors and Multiples
The natural numbers are all the numbers in the sequence 1, 2, 3, 4, 5,... .
When two or more natural numbers are multiplied, each of the numbers is a
factor of the product. For example, 3 and 7 are factors of 21, because 3*7 = 21.
A prime number is a natural number that has exactly two factors, itself and 1.
To write the prime factorization of a number, write the number as a product of
prime numbers.
Write the prime factorization of 315.
SOLUTION Use a tree diagram to factor
the number until all factors are prime
numbers. To determine the factors, test
the prime numbers in order.
3
ANSWER ► The prime factorization of 315
is 3 • 3 • 5 • 7, or 3 2 • 5 • 7.
315
/ \
3 105
/ / \
3 3 35
/ / / \
3 5 7
A common factor of two natural numbers is a number that is a factor of both
numbers. For example, 7 is a common factor of 35 and 56, because 35 = 5 • 7
and 56 = 8 • 7. The greatest common factor (GCF) of two natural numbers is
the largest number that is a factor of both.
Find the greatest common factor of 180 and 84.
SOLUTION First write the prime factorization of each number. Multiply the
common prime factors to find the greatest common factor.
180 = 2 • 2 • 3 • 3 • 5 84 = 2 • 2 • 3 • 7
ANSWER^ The greatest common factor of 180 and 84 is 2 • 2 • 3 = 12.
A common multiple of two natural numbers is a number that is a multiple of
both numbers. For example, 42 is a common multiple of 6 and 14, because
42 = 6 • 7 and 42 = 14 • 3. The least common multiple (LCM) of two natural
numbers is the smallest number that is a multiple of both.
Find the least common multiple of 24 and 30.
SOLUTION First write the prime factorization of each number. The least
common multiple is the product of the common prime factors and all the
prime factors that are not common.
24 = 2 • 2 • 2 • 3 30 = 2- 3- 5
ANSWER ► The least common multiple of 24 and 30 is 2 • 3 • 2 • 2 • 5 = 120.
Skills Review Handbook
SKILLS REVIEW
SKILLS REVIEW
The least common denominator (LCD) of two fractions is the least common
multiple of their denominators.
5 1
Find the least common denominator of the fractions „ and w-
o b
Solution
Begin by finding the least common multiple of the denominators 8 and 6.
Multiples of 8: 8, 16,32, 40, 48, 56, 64, 12,...
Multiples of 6: 6, 12, 18,(24), 30, 36, 42, 48, 54,. . .
The least common multiple of 8 and 6 is 24.
ANSWER The least common denominator of ^ and \ is 24.
o O
Practice -
List all the factors of the number.
4. 35
8 . 49
1.18 2.10 3.77
5. 27 6. 100 7. 42
Write the prime factorization of the number if it is not a prime number. If a
number is prime, write prime.
9.27 1 0.24 11.32 1 2.61
13.55 1 4.68 1 5.148 1 6.225
List all the common factors of the pair of numbers.
17.15,22 18.36,54 19.5,20
21.9,36 22.24,25 23.20,55
Find the greatest common factor of the pair of numbers.
25.25, 30 26. 32, 40 27. 17, 24
29.14,28 30.65,39 31.102,51
Find the least common multiple of the pair of numbers.
33.5,7 34.7,12 35.16,26
37.9,15 38.12,35 39.6,14
Find the least common denominator of the pair of fractions.
„„ 1 11 4 7 1 3
41 - 3 , 12 42 ‘ 9 ’ 12 43 ' 6 ’ 10
45 — — 40 — — 47 — —
4’70 10 ’ 24 3 ’ 17
20. 14, 21
24. 12, 30
28. 35, 150
32. 128, 104
36. 5, 10
40. 20, 25
44.
1 _ 9 _
8 ’ 14
4 27
48. — —
15 ’ 40
Student Resources
Fractions
A fraction is in simplest form if its numerator and denominator have a greatest
common factor of 1. To simplify a fraction, divide the numerator and
denominator by their greatest common factor.
Simplify the fraction ||.
SOLUTION The greatest common factor of 28 and 63 is 7. Divide both the
numerator and denominator by 7.
28 = 28 | 7 = 4
63 63 -7 9
14
Rewrite the improper fraction ^-asa mixed
number.
SOLUTION Begin by dividing 14 by 3. The remainder will be the numerator
of the mixed number’s fraction.
14
3
4R2
3JI4 -.4§
Rewrite the mixed number 5y as an improper fraction.
SOLUTION To find the numerator of the improper fraction, multiply the whole
number by the denominator and add the numerator of the fraction. The
denominator of the improper fraction will be the same as the denominator of
the mixed number.
- 3 = 5*7 + 3 = 38
/ 7 7
Two numbers are reciprocals of each other if their product is 1. Every number
except 0 has a reciprocal.
2 3 2 3
— X — = 1, so — and — are reciprocals.
To find the reciprocal of a number, write the number as a fraction. Then
interchange the numerator and the denominator.
Find the reciprocal of 3^.
Solution 3
l
11
4
Write 3-r as a fraction.
4
13 4
— — Interchange numerator and denominator.
ANSWER ► The reciprocal of 3^ is y^-.
CHECK /3± X ± = SI X ± = 11211 = 1
13
13 4X13
Skills Review Handbook
SKILLS REVIEW
SKILLS REVIEW
To add or subtract two fractions with the same denominator, add or subtract
the numerators.
a. Add | +
3 4 3 + 4
a -5 + 5 = ~
b. Subtract^ -4
7 i 2
= 5- ° r *5
2 _2_ _ 7-2
10 10 10
_ _5_
10
2 • 5
Add numerators.
Simplify.
Subtract numerators.
Simplify.
Factor.
Simplify.
3 1
Find the least common denominator of the fractions and Then
b Z
rewrite the fractions with the least common denominator.
Solution
Begin by finding the least common multiple of the denominators 5 and 2.
Multiples of 5: 5,©, 15, 20, 25, 30, 35, 40,. . .
Multiples of 2: 2, 4, 6, 8,©, 12, 14, 16, 18,. . .
The least common multiple of 5 and 2 is 10. Now rewrite each fraction with a
common denominator of 10.
3 * 2
5 • 2
1 • 5
2 • 5
_ 6 _
10
J_
10
Multiply by
Multiply by
To add or subtract two fractions with different denominators, write equivalent
fractions with a common denominator.
Add | + f.
5 6
1 , 5 = 18 25
5 6 30 30
18 + 25
30
43 , 13
— or 1 —
30’ 30
Use the LCD, 30.
Add numerators.
Simplify.
Student Resources
To add or subtract mixed numbers, you can first rewrite them as fractions.
Subtract
3 4
-2 _ 9 I = II _ 9
3 4 3 4
= 44 _ 27
12 12
= 44 - 27
12
17 , 5
= l2- 0rl 'i2
Rewrite mixed numbers as fractions.
Use the LCD, 12.
Subtract numerators.
Simplify.
To multiply two fractions, multiply the numerators and multiply the
denominators.
Multiply
5 = 3X5
6 4X6
= _15
24
? * 8
5
8
Multiply numerators and multiply denominators.
Simplify.
Factor numerator and denominator.
Simplify fraction to simplest form.
To divide by a fraction, multiply by its reciprocal.
a. Divide 4 5
3.5 3^6
a -i-6 = 4 X 5
= 3X6
4X5
= J_8
20
b. 2- r
4— = —
6 2
_9_
10
. 25
‘ 6
_ 1 _ 6 _
“ 2 X 25
5X6
2 X 25
30
50
6‘
b. Divide 4-
2 6
The reciprocal of ~ is
Multiply numerators and denominators.
Simplify.
Write mixed numbers as fractions.
The reciprocal of ^ is
Multiply numerators and denominators.
Simplify.
Skills Review Handbook
SKILLS REVIEW
SKILLS REVIEW
Practice -
Find the reciprocal of the number.
1. 7
2 -t*
3 J-
12
4 -f
5 —
20
6 . 100
7 —
13
«■?
9 ‘ 1 I
10 . 2|
11 —
12 ^
IZ - 17
13 - 6 f
14. Id}
15 —
lb. 7
16. 4|
Add or subtract. Write the answer as a fraction
or a mixed number in
simplest form.
17 — + —
,/- 6 6
18 5 _ 3
18 ' 8 8
4 1
9 9
„ 5
20 l2
+ A
12
21 ‘2 + 8
3 1
22 ' 5 10
23 4 + l
24 ^
24
7
12
2 S. 5 i- 2 f
3 1
26 ‘ ^7 + 2
27 - 4 f - 2 f
3 3
28 '7 + 4
1 7
29 7— + —
7 2 10
3 °. 5f - 2 }
31 - 4 I - 4
32. 9§
+ 3-
3
Multiply or divide. Write the answer as a fraction or a mixed number in
simplest form.
3s - \ x \
2 4
34 -3 X 5
35 -f X T?
3 7
36- y
37 -i x i
2 3
38. 1 3 X -
39. 3 X 2|
40. 5~
x4
41 -i + !
42 — 4 - —
12 2
4 2
43 - f - 1
44 —
16
«- 4 hi
46. 2| h- l|
47. 3§ 4- 4
48. 7 i
+ 2 ?
Add, subtract, multiply, or divide. Write the answer as a fraction or a
mixed number in simplest form.
15 1
49 ' 16 8
50. f X l|
51 — — —
13 13
24
52 —
D 25
4
53 4-l
54 — — —
10 5
55.| x !
“■j + i
57.4 X f
58. 9f + 3 j
„ 4 l
59. — • —
5 2
60. 6y
— 2—
Z 5
6 , 4 + l
62. X i
33 4 x !
64 f-
4
5
Student Resources
Writing Fractions and Decimals
A fraction can be written as a decimal by dividing the numerator by the
denominator. If the division stops with an exact quotient, then the decimal form
of the number is a terminating decimal. If the resulting quotient includes a
decimal digit or group of digits that repeats over and over, then the decimal form
of the number is a repeating decimal.
Write the fraction as a decimal.
Solution
Divide the numerator by the denominator.
0.45
a. 20Wo6
9
ANSWER — = 0.45, a terminating decimal.
0.636363. . .
b. 11)7.000000. . .
7
ANSWER — = 0.636363. .., a repeating decimal. Write a repeating
7 —
decimal with a bar over the digits that repeat: — = 0.63.
Solution
Write the decimal as a fraction.
a. 0.12 b. 0.18
a. To write a terminating decimal as a fraction, use the name for the last
place to the right of the decimal as the denominator. The first place to the
right is tenths, the second place is hundredths, and so on.
0.12 = Write as hundredths.
= Simplify.
ANSWER ► 0.12 = ^
b. x = 0.181818. . .
lOOx = 18.181818. . .
- x= 0.181818...
99x = 18
_ 18
X 99
_ _ 2 _
11
Let x represent the repeating decimal.
Multiply x by 10 n where n is the number of digits that
repeat. (Here, n = 2.)
Subtract x from lOOxto eliminate repeating decimal.
Divide each side by 99.
Simplify.
ANSWER ^ 0.18
_ 2 _
11
Skills Review Handbook
SKILLS REVIEW
SKILLS REVIEW
Practice
Write the fraction as a decimal.
- 1
2 -To
3 A
25
4
50
1 ‘ 4
e-f
7 ^
'■ 11
8 ^
8 ' 37
Write the decimal as a fraction. Simplify if possible.
9. 0.5
10 . 0.16
11 . 0.289
12. 0.1234
13.0.7
14. 0.15
15. 0.613
16. 0.5840
Fractions, Decimals, and Percents
To write a percent as a decimal, move the decimal point two places to the left and
remove the percent symbol.
Write the percent as a decimal.
a. 85% = 85% = 0.85
Ks
c. 427% = 427% = 4.27
b. 3% = 03% = 0.03
Ky
d. 12.5% = 12.5% = 0.125
To write a percent as a fraction in simplest form, first write the percent as a
fraction with a denominator of 100. Then simplify if possible.
Write the percent as a fraction or a mixed number.
a. 71% =
71
100
b. 10% =
10
c. 4%
4
100
25
d. 350%
100
350 _ 7 _ 0 1
100 2 3 2
To write a decimal as a percent, move the decimal point two places to the right
and add a percent symbol.
Write the decimal as a percent.
a. 0.93 = 0.93 = 93%
c. 0.025 = 0.025 = 2.5%
b. 1.47 = 1.47 = 147%
d. 0.005 = 0.005 = 0.5%
Student Resources
To write a fraction as a percent, first determine whether the denominator of the
fraction is a factor of 100. If it is, rewrite the fraction with a denominator of 100.
If not, divide the numerator by the denominator.
Write the fraction as a percent.
a. ^ 25 is a factor of 100, so write ly = ^ t ^ = 68%.
b.
8 is not a factor of 100, so divide: 1-^8 = 0.125 = 12.5%.
\J
6 is not a factor of 100, so divide: 1 -h 6 = 0.1666. . . = 0.167
= 1 k5%.
You should memorize the relationships in this chart.
Equivalent percents, decimals, and fractions
1% = 0.01 =
33 j% = 0.3 = |
66|% = 0.6 = 1
10 % = o.i = -jT
40% = 0.4 = |
75% = 0.75 = |
20% = 0.2 = ~
50% = 0.5 = |
80% = 0.8 = y
25% = 0.25 = r
60% = 0.6 = |
100% = 1
Practice
Write the percent as a decimal and as a fraction or a mixed number in
simplest form.
1 . 63%
2. 7%
3. 24%
4. 35%
5. 17%
6 . 125%
7. 45%
8 . 250%
9. 33.3%
10. 96%
11.62.5%
12 . 725%
13. 5.2%
14. 0.8%
15. 0.12%
Write the decimal as a percent and as a fraction or a mixed number in
simplest form.
16. 0.39
17. 0.08
18. 0.12
19. 1.5
20. 0.72
21.0.05
22 . 2.08
23. 4.8
24. 0.02
25. 3.75
26. 0.85
27. 0.52
28. 0.9
29. 0.005
30. 2.01
Write the fraction or mixed number as a decimal and as a percent.
si 4
oo 13
32 ' 20
„ 11
33 ‘ 25
35 -l
36.2!
37.5}
38 ^
38 ' 20
39.}
40 - 3 2 ?
Skills Review Handbook
SKILLS REVIEW
SKILLS REVIEW
Comparing and Ordering Numbers
When you compare two numbers a and b , a is either less than , equal to ,
or greater than b. To compare two whole numbers or decimals,
compare the digits of the two numbers from left to right. Find the
first place in which the digits are different.
a is less than b.
a<b
a is equal to b.
a = b
a is greater than b.
a> b
Compare the two numbers. Write the answer using <, >, or =.
a. 27.52 and 27.39 b. -4.5 and -4.25
Solution
a. 27.52 and 27.39
ANSWER ► 5 > 3, so 27.52 > 27.39.
You can picture this on a number line. The numbers on a number line
increase from left to right.
V
27.39 27.52
i i i i i i ii i i i i i : I I I I I
27.30 27.40 27.50 27.60
27.52 is greater than 27.39.
27.52 is to the right of 27.39.
b. Begin by graphing —4.5 and —4.25 on a number line.
-4.5 -4.25 -4.5 is less than -4.25.
i—i—i—i—i—i—i—i—:—i—i—►
-6 -5 -4 -4.5 is to the left of -4.25.
ANSWER ► From the number line, —4.5 is to the left of —4.25,
so -4.5 < -4.25.
To compare fractions that have the same denominator, compare the numerators. If
the fractions have different denominators, first rewrite the fractions to produce
equivalent fractions with a common denominator.
Write the numbers
3 7
4' 8'
and ^ in order from least to greatest.
Solution
The LCD of the fractions is 24.
3 _ 3 • 6 _ 18 7 _ 7 • 3 _ 21 5 = 5 • 2 = 10
4 4 * 6 24 8 8 * 3 24 12 12 • 2 24
10 18 21 5 3 7
Compare the numerators: 10 < 18 < 21, so — < — < —, or J2 < 4 < ~8 m
5 3 7
ANSWER In order from least to greatest, the fractions are — , — , and —
IZ 4 o
Student Resources
3 2
Compare 4^ and 4^. Write the answer using <, >, or =.
SOLUTION The whole number parts of the mixed numbers are the same, so
compare the fraction parts.
The LCD of | and | is 12.
3 * 3
4 • 3
9_
12
2 = 2 * 4
3 3-4
_ 8 _
12
9 8 3 2
Compare the numerators: 9 > 8, so yy > —. or — > —.
ANSWER ^ Since y- > j, it follows that 4y > 4~.
’3
3 2
4^ is greater than 4^.
3 2
4^ is to the right of 4^.
Practice
Compare the two numbers. Write the answer using <, >, or =.
1. 12,428 and 15,116
4. -16.82 and -14.09
7. 1005.2 and 1050.7
W ‘j3 md j3
13. -y and -yy
4 2
16. y and y
2 . 905 and 961
5. 0.40506 and 0.00456
8 . 932,778 and 934,112
11. 17 j and nf
4 o
14. 4 and -|
17. 42yand4lJ
J O
19.32,227 and 32,226.5
5 ,2
20 . y and y
3. -140,999 and -142,109
6 . 23.03 and 23.3
9. -0.058 and -0.102
1 2 4 and !
15. -y and y
18. 508.881 and 508.793
21 . -17|and -17^
6 7
Write the numbers in order from least to greatest.
22 .1207, 1702, 1220, 1772
24. -23.12, -23.5, -24.0, -23.08, -24.01
15 3 5
26.4.07, 4.5, 4.01,4.22
29 ^ 1 5 I 15
5’ 4’ 3’ 8’ 16
27.
23. -45,617, -45,242, -40,099, -40,071
25. 9.027, 9.10, 9.003, 9.3, 9.27
3 3 3 3 3
3’ 6’ 8’ 4
28.
5’ V 4’ 10’ 7
30. 14 9 , 15 3 , 14 6 , 15 4 31. § , y, 1 3 , n
32. You need a piece of trim that is 6— yards long for a craft project. You have
3
a piece of trim that is 6— yards long. Is the trim you have long enough?
Skills Review Handbook
SKILLS REVIEW
SKILLS REVIEW
Perimeter, Area, and Volume
The perimeter P of a figure is the distance around it.
P = 4 + 2 + 4 + 5
= 15
P — £ + w + £ + w
= 21+2 w
Find the perimeter of a rectangle with length 14 centimeters
and width 6 centimeters.
Solution p = 21 + 2w = (2 x 14) + (2 x 6) = 28 + 12 = 40
ANSWER ► The perimeter is 40 centimeters.
A regular polygon is a polygon in which all the angles have the same measure
and all the sides have the same length. The perimeter of a regular polygon can be
found by multiplying the length of a side by the number of sides.
regular (equilateral) triangle
regular pentagon
s/\s
The area A of a figure is the number of square units enclosed by the figure.
rectangle
w
triangle
\>
□A
A = length X width A = side X side
= £ X w = s X s
= £w
A = — X baseX height
= s 2
= -XbXh
= ^bh
Student Resources
Volume is a measure of how much space is occupied by a solid figure. Volume
is measured in cubic units.
One such unit is the cubic centimeter (cm 3 ). It is the amount of space occupied by
a cube whose length, width, and height are each 1 centimeter.
7
/
7
A
3
/]/\t
V= 6X2X3 v = i XwXh
= 36 cubic units
Find the volume of a rectangular prism with length 8 feet,
width 5 feet, and height 9 feet.
Solution V = ixwx/z = 8x5x9 = 360
ANSWER ► The volume is 360 cubic feet (ft 3 ).
Practice
Find the perimeter.
1 - 10
0.5 in.
21ft
0.75 in.\
V0.75 in.
28 ft
3.5 m
0.5 in.
3.5 m
5- a square with sides of length 18 ft
Find the area.
7- a square with sides of length 29 yd
9. a square with sides of length 3.5 in.
11 - a triangle with base 8 in. and height 5 in.
Find the volume.
13. a cube with sides of length 25 ft
14. a cube with sides of length 4.2 cm
15. a rectangular prism with length 15 yd, width 7 yd, and height 4 yd
16. a rectangular prism with length 7.3 cm, width 5 cm, and height 3.2 cm
17. a rectangular prism with length 5.3 in., width 4 in., and height 10 in.
6 . a rectangle with length 6 m and width 7 m
8. a rectangle with length 7 km and width 4 km
10. a rectangle with length 24 ft and width 6 ft
12. triangle with base 7.2 cm and height 5.3 cm
Skills Review Handbook
SKILLS REVIEW
Estimation
You can use estimation to provide a quick answer when an exact answer is not
needed. You also can use estimation to check if your answer is reasonable. Three
methods of estimation are rounding, front-end estimation, and using compatible
numbers.
To round, decide to which place you are rounding.
• If the digit to the right of that place is less than 5, round down.
• If the digit to the right of that place is greater than or equal to 5, round up.
Estimate the difference of 688 and 52 by rounding to the
nearest ten.
Solution
688
- 52
690
- 50
640
Round 688 to the nearest ten.
Round 52 to the nearest ten.
Subtract.
ANSWER ► The difference of 688 and 52 is about 640.
Estimate the quotient of 110.23 and 10.85 by rounding to
the nearest whole number.
Solution
10 Round 110.23 and 10.85 to the nearest
110.23 -r- 10.85-► llJTIO whole numbers and divide.
ANSWER ^ The quotient of 110.23 and 10.85 is about 10.
To use front-end estimation, add the front digits. Then estimate the sum of the
remaining digits, and add that sum to the front-end sum.
Use front-end estimation to estimate the cost of 3 shirts
marked $14.96, $11.78, and $8.25.
Solution
Add the front digits. Estimate whats left.
$14.96 $0.96 about $1
$ n - 78 f' 78 about $1
+ $8.25 $0.25 _
$33 $2
ANSWER ► The cost of the shirts is about $33 + $2 = $35.
Student Resources
There are two methods to estimate products and quotients. You can use rounding
or compatible numbers. Compatible numbers are numbers that are easy to
compute mentally.
Use compatible numbers to estimate the product of
116.11 and 41.09.
Solution
116.11 —► 115 Use compatible numbers 115 and 40
X 41.09 —* X 40 since they are easy to multiply.
4600
ANSWER ► The product of 116.11 and 41.09 is about 4600.
You can estimate the area of a figure by placing it on a grid. Count the number
of squares that are completely covered by the figure. Then count the number of
squares that are partially covered. You can assume that on average a partially
covered square is about half covered. So you can estimate the total area of the
figure by adding the number of squares that are totally covered to one-half the
number of squares that are partially covered.
Estimate the area of the figure shown to the nearest
square unit.
Solution
First count the number of squares
that are completely covered.
Then count the number of squares
that are partially covered.
There are 9 squares that are There are 18 squares that are
completely covered. partially covered.
So an estimate for the area of the figure can be calculated as follows:
Area = 9 + j(lS) = 9 + 9
18
ANSWER ► The area of the figure is approximately 18 square units.
Skills Review Handbook
775
SKILLS REVIEW
SKILLS REVIEW
Practice
Round to the nearest ten or hundred to estimate the sum or difference.
1. 36 + 11
4. 16 + 23 + 74
7. 58 - 39
10.65 - 42 - 12
2. 249 + 782
5. 108 + 92 + 345
8 . 1375 - 911
11. 1059 - 238 - 111
Use front-end estimation to estimate the sum.
13.15.98 + 6.46 14. 62.36 + 44.68
16.533.2 + 37.2 17. 912.14 + 428.13
19.24.22 + 4.53 + 12.31 20. 16.1 + 34.2 + 25.2
22.113.73 + 97.1 + 65.18 23. 88.9 + 86.19 + 92.14
Use rounding to estimate the product or quotient.
25.52 X 48 26.27 X 414
28.42 X 6.1 29.10.34 X 2.69
31.642 — 219 32.121 -57
34.77 - 3.84 35. 58.9 - 14
40.536.2 X 22.1
43.68.66 - 2.96
46.948.68 - 47.96
41. 498.75 X 13.55
44. 995.88 - 102.34
47. 1487.81 - 28.65
Estimate the area of the figure to the nearest square unit.
50.
1
_
I
3. 1585 + 791
6 . 1023 + 5062 + 3873
9. 2014 - 389
12. 8375 - 3847 - 1224
15. 156.22 + 324.72
18. 588.61 + 120.37
21. 59.31 + 71.21 + 78.47
24. 0.4 + 120.46 + 584.53
27. 602 X 53
30. 108.8 X 435
33. 838 - 22
36. 40.32 - 1.25
Use compatible numbers to estimate the product or quotient.
37.74.94 X 11.6 38. 397.25 X 41.37
39. 3997.63 X 18.87
42. 2465.83 X 68.52
45. 523.12 - 51.87
48. 148.64 - 14.71
51.
r
V
Student Resources
Data Displays
A bar graph can be used to display data that fall into distinct categories. The
bars in a bar graph are the same width. The height or length of each bar is
determined by the data it represents and by the scale you choose.
In 1998, baseball player Mark McGwire hit a record 70 home runs,
The table shows the locations to which the home runs were hit.
Draw a bar graph to display the data. ►Source: Stats Inc.
O Choose a scale. Since the data range from 0 to 31, make the scale
increase from 0 to 35 by fives.
© Draw and label
the axes. Mark
intervals on
the vertical
axis according
to the scale
you chose.
© Draw a bar for
each category.
© Give the bar
graph a title.
Home Run Field Location
Number of
home runs
-i. ro o
o o o o
left left- center right- right
center center
Field location
Field location
Number
of runs
left
31
left-center
21
center
15
right-center
3
right
0
A histogram is a bar graph that shows how many data items occur within
given intervals. The number of data items in an interval is the frequency.
The table shows the distances of McGwire's home runs.
Draw a histogram to display them.
SOLUTION Use the same method you used for drawing the bar graph
above. However, do not leave spaces between the bars.
O Since the frequencies range from 4 to 27, make the scale increase from
0 to 30 by fives.
© Draw and label
the axes. Mark
intervals on the
vertical axis.
e Draw a bar for
each category.
Do not leave
spaces between
the bars.
© Give the
histogram a title.
Distance (ft)
Distance
(ft)
Frequency
300-350
4
351-400
24
401^50
27
451-500
11
501-550
4
Skills Review Handbook
H
SKILLS REVIEW
A line graph can be used to show how data change over time.
A science class recorded the highest temperature each day from
December 1 to December 14. The temperatures are given in the table.
Draw a line graph to display the data.
Date
1
2
3
4
5
6
7
Temperature (°F)
40
48
49
61
24
35
34
Date
8
9
10
11
12
13
14
Temperature (°F)
42
41
40
22
20
28
30
O Choose a scale.
0 Draw and label the axes. Mark
evenly spaced intervals on
both axes.
© Graph each data item as a point.
Connect the points.
O Give the line graph a title.
v.
Daily High Temperature Dec. 1 - Dec. 14
Date
A circle graph can be used to show how parts relate to a whole and to each other.
The table shows the number of sports-related
injuries treated in the hospital emergency room in
one year. Draw a circle graph to display the data.
O Find the total number of injuries.
56 + 34 + 22 + 10 + 28 = 150
To find the degree measure for each sector of the
circle, write a fraction comparing the number of
injuries to the total. Then multiply the fraction
by 360°. For example:
Football: ^ • 360° = 81.6°
0 Draw a circle. Use a protractor to
draw the angle for each sector.
© Label each sector.
O Give the circle graph a title.
Sports-related Injuries
Basketball 56 /•
Football 34
(
—\ Skating/
w
•"'T Hockey 22
\ Track and
Other 28
field 10
Related sport
Number of injuries
basketball
56
football
34
skating/hockey
22
track and field
10
other
28
Student Resources
Practice
In 1998, baseball player Sammy Sosa hit 66 home runs. The tables show the
field locations and distances of his home runs. ►Source: Stats Inc.
1. The location data range from 10 to
22. The scale must start at 0. Choose
a reasonable scale for a bar graph.
2 . Draw a bar graph to display the
field locations of Sosa’s home runs.
3. The distance data range from 1 to
16. The scale must start at 0. Choose
a reasonable scale for a histogram.
4. Draw a histogram to display the
distances of Sosa’s home runs.
Field location
Number of runs
left
12
left-center
22
center
10
right-center
11
right
11
For Exercises 1 and 2
Distance (ft)
Frequency
326-350
5
351-375
12
376-400
14
401-425
16
426-450
14
451-475
1
476-500
4
For Exercises 3 and 4
5. There are 150 runs at a ski resort: 51 expert runs, 60 intermediate runs,
and 39 beginner runs. Draw a circle graph to display the data.
6 . A patient’s temperature (in degrees Fahrenheit) was taken every 3 hours from
9 A.M. until noon of the following day. The temperature readings were 102°F,
102°F, 101.5°F, 101.1°F, 100°F, 101°F, 101.5°F, 100°F, 99.8°F, and 99°F.
Draw a line graph to display the data.
Choose an appropriate graph to display the data. Draw the graph.
7.
Value of One Share of Company stock
Year
1994
1995
1996
1997
1998
1999
Value ($)
15
18
16
12
10
15
8 .
Passenger Car Stopping Distance (dry road)
Speed (mi/h)
35
45
55
65
Distance (ft)
160
225
310
410
9.
Fat in One Tablespoon of Canola Oil
Type of fat
Number of grams
saturated
22
polyunsaturated
10
monounsaturated
11
► Source: U.S. Department of Agriculture
10 .
Enrollment in Capital City Schools by Age
Age
4-6
7-9
10-12
13-15
16-18
Enrollment
912
2556
4812
2232
1502
Skills Review Handbook
H
SKILLS REVIEW
SKILLS REVIEW
Measures of Central Tendency
A measure of central tendency for a set of numerical data is a single number
that represents a “typical” value for the set. Three important measures of central
tendency are the mean , the median , and the mode.
• The mean, or average, of a data set is the sum of the values in the set divided
by the number of values in the set.
• The median of a data set with an odd number of values is the middle value
when the values are written in numerical order. The median of a data set with
an even number of values is the mean of the two middle values when the values
are written in numerical order.
• The mode of a data set is the value or values in the set that occur most often. If
no value occurs more often than any of the others, there is no mode.
Find the mean, median, and mode of the following data set.
10,12, 7,11,20, 7, 8,19, 9,5
Solution
To find the mean, divide the sum of the data values by the number of
data values.
Mean = 10+12 + 7 + 11+20 + 7 + 8+19 + 9 + 5 = M = 10 g
Since there are an even number of values, find the median by writing the data
values in numerical order and finding the mean of the two middle values.
5, 7, 7, 8, 9,10, 11, 12, 19, 20 Median = 9 + 10 = y- = 9.5
The mode is the number that occurs most often in the data set.
Mode = 7
Practice -
Find the mean, median, and models) of the data set.
1.0, 0, 0, 0, 0, 1,2, 2, 4,4
2 . 3, 1, 1, 8, 2, 1, 3, 5, 3
3. 10, 15, 20, 25, 30, 35, 40, 45, 50
4. 14, 10, 45, 38, 60, 14, 23, 35, 68, 50
5. 376, 376, 386, 393, 487, 598, 737, 745, 853
6 . 101, 76, 52, 50, 26, 7, 13, 1000
Student Resources
Problem Solving
One of your primary goals in mathematics should be to become a good problem
solver. It will help to approach every problem with an organized plan.
STEP O understand the problem. Read the problem carefully. Organize the
information you are given and decide what you need to find. Determine
whether some of the information given is unnecessary, or whether
enough information is given. Supply missing facts, if possible.
step © make a plan to solve the problem. Choose a strategy. (Get ideas
from the list given on page 782.) Choose the correct operations. Decide
whether a graphing calculator or a computer is necessary.
step © carry out the plan to solve the problem. Use the strategy and
any technology you have chosen. Estimate before you calculate, if
possible. Do any calculations that are needed. Answer the question that
the problem asks.
step O check to see if your answer is reasonable. Reread the problem and
see if your answer agrees with the given information.
0
e
How many segments can be drawn between 7 points,
no three of which lie on the same line?
You are given a number of points, along with the information that no three
points lie on the same line. You need to determine how many segments
can be drawn between the points.
Some strategies to consider are: draw a diagram, solve a simpler problem
and look for a pattern.
Consider the problem for fewer points.
2 points
1 segment
3 points
3 segments
4 points
6 segments
5 points
10 segments
Look for a pattern. Then continue the pattern to find the number of
segments for 7 points.
Number of points
2
3
4
5
6
7
Number of segments
1
3
6
10
15
21
ANSWER ► Given 7 points, no three of which lie on the same line,
21 segments can be drawn between the points.
O You can check your solution by making a sketch.
Skills Review Handbook
SKILLS REVIEW
SKILLS REVIEW
In Step 2 of the problem solving plan, you may want to consider the
following strategies.
Problem Solving Strategies
• Guess, check, and revise. When
• Draw a diagram or a graph. When
• Make a table or an organized list. When
• Use an equation or a formula. When
• Use a proportion. When
• Look for a pattern. When
• Break the problem into simpler parts. When
• Solve a simpler problem. When
• Work backward. When
you do not seem to have enough information.
words describe a picture.
you have data or choices to organize.
you know a relationship between quantities.
you know that two ratios are equal.
you can examine several cases.
you have a multi-step problem.
easier numbers help you understand a problem.
you are looking for a fact leading to a known result.
Practice -
1_ Tasha bought salads at $2.75 each and cartons of milk at $.80 each. The total
cost was $16.15. How many of each did Tasha buy?
2 _ A rectangular garden is 45 feet long and has perimeter 150 feet. Rows of
plants are planted 3 feet apart. Find the area of the garden.
3. If five turkey club sandwiches cost $18.75, how much would seven
sandwiches cost?
4. How many diagonals can be drawn from one vertex of a 12-sided polygon?
5. Nguyen wants to arrive at school no later than 7:25 A.M. for his first class.
It takes him 25 minutes to shower and dress, 15 minutes to eat breakfast,
and at least 20 minutes to get to school. What time should he plan to get
out of bed?
6 . There are 32 players in a single-elimination chess tournament. That is, a
player who loses once is eliminated. Assuming that no ties are allowed, how
many games must be played to determine a champion?
7. Andrea, Betty, Joyce, Karen, and Paula are starters on their school basketball
team. How many different groups of three can be chosen for a newspaper
photo?
8. Carl has $135 in the bank and plans to save $5 per week. Jean has $90 in the
bank and plans to save $10 per week. How many weeks will it be before
Jean has at least as much in the bank as Carl?
9. The Peznolas are planning to use square tiles to tile a kitchen floor that is
18 feet long and 15 feet wide. Each tile covers one square foot. A carton of
tiles costs $18. How much will it cost to cover the entire kitchen floor?
■ “
Extra Practice
Chapter1
Evaluate the expression for the given value of the variable. (Lesson 1.1)
1 - 15a when a = 7 2. 7 + x when x =15 3- — when c = 32
Evaluate the expression for the given value(s) of the variable(s). (Lesson 1.2)
4. 3 y 2 when y = 5 5. (4x) 3 when x = 2 6. 6x 4 when x = 4
7. a 4 — 5 when a — 3 8- (x + 2) 3 when x = 4 9 . (c — d) 2 when c = 10 and d — 3
Evaluate the expression. (Lesson 1.3)
10.33 -12- 4 11. 10 2 - 4 + 6 12. 10 2 - (4 + 6) 13. 2 + 21 - 3 - 6
14. 3 + 7 • 35 + 5 15. 15 - (6 — 1) — 2 16. [(5 • 8) + 8] - 16 17. g 9 ‘ 72 ^
5 + 8—6
Use mental math to solve the equation. (Lesson 7.4 )
18.x + 7 = 13 19. n — 4 = 8 20. 3y = 21 21.^ = 6
Check to see if the given value of the variable is or is not a solution of the
inequality. (Lesson 1.4)
22. y + 10 < 22; y = 12 23. 6 n > 25; n = 5 24. 3t < 12; t = 4
25.4 + x> 11; x = 6 26. 48 - g < 4; g = 16 27. a - 5 > 3; a = 9
Write the sentence as an equation or an inequality. Let x represent the
number. (Lesson 1.5)
28. The product of a number and 4 is less than or equal to 36.
29. 16 is the difference of 20 and a number.
30. SPORTS Your friend’s score in a game is 48. This is twice your score. Write
a verbal model that relates your friend’s score to your score. What is your
score? (Lesson 1.6)
31. WIRELESS INDUSTRY The table shows the estimated number of cellular
telephone subscribers (in millions) in the United States. Make a bar graph
and a line graph of the data. (Lesson 1.7)
Year
1993
1994
1995
1996
1997
1998
1999
Subscribers (millions)
13
19
28
38
49
61
76
► Source: Cellular Telecommunications Industry Association
Make an input-output table for the function. Use 0,1, 2, 3, 4, and 5 as
values for x. (Lesson 1.8)
32. y = 8 — 2x 33. y = lx + 1 34. y = 3(x - 4)
Extra Practice
EXTRA PRACTICE
EXTRA PRACTICE
Chapter 2
Graph the numbers on a number line. Then write two inequalities that
compare the numbers. (Lesson 2.1)
1.-7, 8 2. 3,-5 3. -4,-7 4. 0,-3
Evaluate the expression. (Lesson 2.2)
5. |-3 | 6. — 14 | 7. 18.5 |
Find the sum. (Lesson 2.3)
9.-3 + 8 10.18 + 27 11. 5 +(-7)
13. -4 + 13 + (-6) 14. 15 + (-12) + (-4) 15. -2 + (-9) + 8
12. -4 + (-11)
16. 17 +(—5) + 15
Find the difference. (Lesson 2.4)
17. -8-5 18.-3-(-7) 19.4.1-6.3
20 . -
2
5
3
5
21.6 - 13
22. 5 - (-2)
23.-10 - (-3.5) 24.-2 - 14
Evaluate the expression. (Lesson 2.4)
25.-6 -(-3)- 4 26.-15 - 4 - 12 27. 2 - 5 -(-18)
Find the product. (Lesson 2.5)
28. -6( - 7) 29. -5(90) 30. 4(—1.5)
32. (-4) 3 33. -(3) 4 34. -(-2) 5
WHALES In Exercises 36 and 37, suppose a whale is diving at a rate of
about 6 feet per second. (Lesson 2.5)
36. Write an algebraic model for the displacement d (in feet) of the whale after
t seconds.
37. What is the whale’s change in position after diving for 15 seconds?
Use the distributive property to rewrite the expression without
parentheses. (Lesson 2.6)
38. 6 (y + 5) 39. 4 (a — 6) 40. (3 + w)2
42. —3(r - 5) 43. -(2 + 0 44. (x + 4)(-6)
Simplify the expression by combining like terms if possible. If not possible,
write already simplified. (Lesson 2.7)
46. 3x + lx 47. 8 r - r 2 48. 6 + 2y - 3
49. w + 2w + 4w — 4 50. 7 + 5r — 6 + 4r 51 . m 2 + 3 m — 2m 2 — m
41. (4jc + 3)2
45. (y - 3)1.5
35. 3(—8)(—2)
Find the quotient. (Lesson 2.8)
52.18 4- (-2) 53. -48 4- 12
54. 16
55.
-22
3
Student Resources
Chapter 3
Solve the equation. (Lesson 3.1)
00
II
\o
1
2. 5 + n — —10
3. 3 = r — 14
4. ~4 — 5 + q
5. 8 = x- (-1)
6. t ~ 4 = -7
7, m + 6 = 9
8.-2 = r - (-5)
Use division to solve the equation. (Lesson 3.2)
9. lx = 35
10. —15m = 150
11. 6a = 3
12. -144 = —12?
Use multiplication to solve the equation. (Lesson 3.2)
„ x _ . _ y _ 2 8 _ t . t _ 3
13 ‘5 _ 4 14 ‘ 10 ~~ 5 15 ‘ 6 _ 14 16 ' -8 ~~ 8
Solve the equation. (Lesson 3.3)
17. 6x + 8 = 32
18. 2x - 1 = 11
19. 4m + 8m — 2
20. 2x — 3(x + 4) = -1
21.|(m - 1) = -5
22. + 3) = 4
Solve the equation. (Lesson 3.4)
23. —6 + 5x = 8x — 9 24. 8r + 1 = 23 - 3 r 25. 2w + 3 = 3w + 1
26. 3a + 12 = 4a — 2a + 1 27. 5x + 6 = 2x + x + 2 28. 6d — 2d = 10 d + 6
Solve the equation. (Lesson 3.5)
29.4(a + 3) = 3(a + 5) 30. 8(r - 2) + 6 = 2(r + 1)
31.6(x - 1) = 5(2x + 3) - 15 32. ^(4 q + 12) = 2 + 3(6 - q)
Solve the equation. Round the result to the nearest hundredth. Check the
rounded solution. (Lesson 3.6)
33. —26x - 59 = 135 34. 18.25J - 4.15 = 2.75 d 35. 2.3 - 4.8w = 8.2w + 5.6
In Exercises 36 and 37, use the distance formula d = rt. (Lesson 3.7)
36. Solve the formula for rate r.
37. You ride your bike for 3 hours and travel 36 miles. Use the formula you
wrote in Exercise 36 to find your average speed.
Find the unit rate. (Lesson 3.8)
38. 33 ounces in 6 cans of juice
40. Hike 10.5 miles in 3 hours
Solve the percent problem. (Lesson 3.9)
42. How much money is 40% of $800?
44. 24 is what percent of 60?
39. Earn $50.75 for working 7 hours
41. 16 grams of protein in 8 granola bars
43. 15% of 320 meters is what length?
45. What number is 30% of 150?
Extra Practice
EXTRA PRACTICE
Chapter 4
Plot and label the ordered pairs in a coordinate plane. (Lesson 4.1)
1- A(2, 4), B(—2, 0), C(5, -2) 2 . A(4, 4), 5(0, -2), C(-3, -3) 3. A(4, -4), 5(2.5, 5), C(-3, 2)
4. A(0, —1), 5(1, -3), C(3, 1) 5. A(—4, -2),5(-2,4), C(4, 0) 6.A(-3, -4), 5(1, -1), C(-1, 1)
Use a table of values to graph the equation. (Lesson 4.2)
7. y = 5x + 1 8 . y = — 2x + 4 9. 4x + y = — 8
ll.j — 2x — — 5 12 . y = 3x — 1 13. y = —2x + 1
10 - 2 y-x = -1
14. 5y — lOx = 20
Graph the equation. (Lesson 4.3)
15. y= -2 16. x = 3
Find the x-intercept of the line. (Lesson 4.4)
19. 5x + 37 = — 5 20. 2x — y = 6
Find the /-intercept of the line. (Lesson 4.4)
23. 37 = 2x — 5 24. 37 = 2x + 14
i? .x=4
21 . 637 + 2x = 12
25. 37 = 6 — 3x
18. y = 5
22 . 8x + 237 = -16
26. lOx - 15)7 = 30
Find the slope of the line that passes through the points. (Lesson 4.5)
27. ( 6 , 1) and (-4, 1) 28. (2, 2) and (-1, 4) 29. (-4, 2) and (-3, -5)
30. (4, 5) and (2, 2) 31 . (3, 6 ) and (3,-1) 32. (0, 6 ) and (3, 0)
In Exercises 33-40, the variables xand / vary directly. Use the given values
to write an equation that relates xand /. (Lesson 4.6)
33. x = 6 , y = 18 34. x = 4, 37 = 1 35. x = 8 , y = —7 36. x = — 1, 37 = —20
37.x = —2, 37 = —2 38. x = 8, y = —4 39. x = 2, y = —6
Write the equation in slope-intercept form. Then graph the equation.
(Lesson 4.7)
40. x = 5, 37 = 2
41.x — 37 = 1
44. 2x — 4)7 + 6 = 0
42. —3x + 2y — 6
45. 2x + 2)7 + 2 = 4)7
43. x + 37 + 4 — 0
46. 5x — 3)7 + 2 = 14 — 4x
Determine whether the relation is a function. If it is a function, give the
domain and the range. (Lesson 4.8)
47. Input Output
48. Input Output
49.
Input
Output
4
-2
0
0
4
2
2
4
Student Resources
Chapter5
Write in slope-intercept form the equation of the line described below.
(Lesson 5.1)
1- m = 2, b = 1 2. m = —3, b — —2 3. m — b — —3 4. m = —4, fo = 0
Write in point-slope form the equation of the line that passes through the
given point and has the given slope. (Lesson 5.2)
5. (—1, 0), m = 3 6- (5, 2), m — —2 7. (3, 6), m = 0
9. (—3, — 1), m = 4 10- (1, 5), m = 8 11- (2, — 1), m = ^
Write in slope-intercept form the equation of the line that passes through
the given points. (Lesson 5.3)
13. (3, -2) and (5, 4) 14. (5, 1) and (0, -6) 15. (-2, -1) and (4, -4)
16. (-1, 7) and (5, 7) 17. (-3, 5) and (-6, 8) 18. (5, 2) and (1, 4)
Write in standard form an equation of the line that passes through the
given point and has the given slope. (Lesson 5.4)
19. (5, —2), m = 3 20. (—2, 5), m = 5 21. (—4, 3), m = —— 22. (5, 7), m = ^
23. (0, 8), m = —1 24. (-1, -7), m = 4 25. (3, 6), m = -2 26. (4, 5), m = -5
In Exercises 27-29, use the following information. You buy $10.00 worth of
apples and oranges. The apples cost $.80 a pound and the oranges cost $1.00
a pound. (Lesson 5.5)
27. Write an equation in standard form that represents the different amounts
(in pounds) of apples A and oranges R that you could buy.
28. Copy the table. Then use the linear equation to complete the table.
Pounds of apples, A
0
1
2
3
4
5
Pounds of oranges, R
?
?
?
?
?
?
8. (—2, 1), m — — 5
12 . (-4, 3), m — —
29. Plot the points from the table and sketch the line.
Determine whether the lines are perpendicular. (Lesson 5.6)
30. y = x — 2, y = ~x + 4
32. y — ^x — 1, y = —2x + 2
31. y = ^x — 5, y = ~x + 5
33. 3 y = —2x + 12, y — —^x — 12
Write in slope-intercept form the equation of the line passing through the
given point and perpendicular to the given line. (Lesson 5.6)
35. (-1, 4), y = —x — 1
34. (1, 2), y = x + 2
36. (3, —2), y = 1
Extra Practice
H
EXTRA PRACTICE
EXTRA PRACTICE
Chapter 6
Solve the inequality. Then graph the solution. (Lesson 6.1)
1. x + 1 < 2 2. r + 5 > —4 3. 3>y-4 4. 8 + t <-2
Solve the inequality. Then graph the solution. (Lesson 6.2)
5. 9x > 36 6. 5w<—15 7. |<2 8.-|n > 4
Solve the inequality. (Lesson 6.3)
9. 2x + 5 > 3 10. — 3x - 7 < 2 11. 4(x + 5) > 10
12. 3x + 8 > — 2x + 3 13. 4(x - 2) < 3x + 1 14. -(x + 5) < -4x - 11
Write an inequality that represents the statement. Then graph the
inequality. (Lesson 6.4)
15. x is greater than —5 and less than 2.
16. x is greater than or equal to 4 and less than 6.
17. x is less than or equal to 5 and greater than —3.
18. x is less than 6 and greater than or equal to — 1.
Solve the inequality. Then graph the solution. (Lesson 6.4)
19. 3<x + 4<8 20. —36 < 6x < 12 21. -2 < 2x - 4 < 10 22. 0 < 5x - 6 < 9
Solve the inequality. Then graph the solution. (Lesson 6.5)
23. x — 3 < — 2 or x + 2 > 6 24. x + l>4or2x + 3<5
25. 2x + 1 > 9 or 3x — 5 < 4 26. —4x + 1 > 17 or 5x — 4 > 6
Solve the equation and check your solutions. If the equation has no
solution, write no solution. (Lesson 6.6)
27. |x | = 14
28.
x | = —10
29.
x | = 12
30.15x | = 15
31.
10 + x | —4
32.
<N
II
00
1
33.15x - 3 | =2
34.
2x + 3 | =9
35.
x — 4| +4 = 7
Solve the inequality. Then graph and check the solution. (Lesson 6.7)
36. | x | > 2
37.
00
VI
K
38.
x - 5 | < 10
39.16x | <30
40.
4 + x| >8
41.
4x + 5 | >3
42.110 - 4x| <2
43.
6x — 5 | + 1 < 8
44. |
3x + 4 | - 6 > 14
Graph the inequality in a coordinate plane. (Lesson 6.8)
45. j > -2
46. x — y < 0 47. x +
IV
U3
48. 4y + x < 4
49.x - 3y<0
50. 3_y — 2x < 6 51. 5x -
-3y>9
52. 2y — x > 10
Student Resources
Chapter 7
Estimate the solution of the linear system graphically. Then check the
solution algebraically. (Lesson 7.1)
1 - y = 5 2. x = 0 3- x + y = 10
x = —2 y = 3x + 7 x — y = —2
Use the substitution method to solve the linear system. (Lesson 7.2)
5. x = 5 y
2x + 3 y = —13
9 - —s — t — — 5
3s + 4t — 16
6 . y — — 2 x
x + y = 7
10 . 5x- 8 y = -17
3x — y — 5
7. x + y = 9
x - y = 3
11. 2 / 7 ? + n = 7
4m + 3/7 = — 1
Use linear combinations to solve the linear system. Then check your
solution. (Lesson 7.3)
13.x + y — 6 14. 3x +
x — y = 2 2 x —
17. — x + y = —15 18. 2x +
x + 4y = 5 3y +
3y = 6 15. 4x - 5 y= 10
3y = 4 2x + 5 y= -10
3y = 15 19. y = 2x — 3
5x = 12 3x — 5 y = 1
4. —2x + 4y = 12
5x — 2y = 10
8 . 2a + 3b = 3
a — 6b = —6
12 . 5a + Z? = 4
la -\- 5b — 11
16. 2x + 8 }/ = 9
x-y = 0
20 . — 4x - 15 = 5y
2y — 11 — 5x
Choose a solution method to solve the linear system. Explain your choice,
and then solve the system. (Lesson 7.4)
21 .x - 2 y = -10
3x + y = 5
22 . 5x + 3}/ = 15
4x — 3 y= 12
23. y = —2x — 6
y = -4
24. x + y = 8
x-y = 4
25. 2x — 3y = 6
x + y = 3
26. 2x + y — -8
6 x + y — —2
27. 5x - y = 10
2x + y = 4
28. — 4x + 3j = 1
— 8 x + 4y = -4
29. STUDENT THEATER You sell 20 tickets for admission to your school play
and collect a total of $104. Admission prices are $6 for adults and $4 for stu¬
dents. How many of each type of ticket did you sell? (Lesson 7.4)
method to tell how many solutions the system has.
Use the graphing
(Lesson 7.5)
30. x + y = 4
2x + 3y = 9
34. y = —3x
6 y — x = 38
31. x + y = 6
3x + 3y = 3
35. 2x — 3y = 3
6 x — 9y = 9
32. x + 2y = 5
3x — 15 = —6 y
36. 3x + 6 = 7y
x + 2y = 11
33. 12x — y = 5
— 8 x + y = — 5
37. 3x — 8 y = 4
6 x — 42 = 16 y
Graph the system of linear inequalities. (Lesson 7.6)
38. y > 0
39. y >
x < 0
y<
42. x < 5
43 .y>
x > 1
y *
<N
1
Al
y^
y< 7
X + 1
40. x > 1
x + 3
y + x<5
x — 4
44. y > x — 3
—x — 1
y < x + 2
0
x < 3
41. y + 2 < -x
2y — 4 > 3x
45. 3x - 1 < 5
—x + y < 10
—5x + 2 < 12
Extra Practice
EXTRA PRACTICE
EXTRA PRACTICE
Chapter Z
Simplify the expression. (Lesson 8.1)
1. 7 2 • 7 3
2. (2 3 ) 4
3. (12x) 3
5. (m 3 ) 2
6. (4r) 2 • r
7. (7x 2 ) 2 • 2x 3
Rewrite as an expression with positive exponents. (Lesson 8.2)
9. x~ 4
10. 2x“ 2
11.x 3 y 2
13.^j
3y -3
14. _j
r 1
15. (4y~ 2 ) 2
4. (-3 cd) 4
8 . (3x) 3 (—5y) 2
12. A
16.
(5x)
-2
Graph the exponential function. (Lesson 8.3)
17. j = 5 X 18. j = —3 X
Simplify the quotient. (Lesson 8.4)
2 11 _ s 1
19. y
21
25.
2 8
(~ 4) 2
(-4) 5
22 . x 5 • -4
„3
26.
23.
27.
-l
20. y
24.
28.
-2
Simplify the expression. Use only positive exponents. (Lesson 8.4)
29.
32.
2 x 4 y 2 3 x 2 y
- • -
xy 4x
3x 2 ;y 2 j 2
2x
2
XV
30.
33.
16rV # rs
-2rs 2 * -8
4a x h?_
a 4 b~ 2
-2
31.
34.
3x 2 z‘
2 xz
&
(« 5 ) 4
■ 2^4 \ 3
Write the number in decimal form. (Lesson 8.5)
35.4.813 X 10“ 6 36. 3.11 X 10 4
39.5.0645 X 10 1 40. 1.2468 X 10“ 3
37. 8.4162 X 10 -2
41. 2.34 X 10“ 8
Write the number in scientific notation. (Lesson 8.5)
43.5280 44.0.0378 45.11.38
47.827.66 48.0.208054 49. 16.354
51.3.95 52.78.4 53.0.008
INTEREST You deposit $1100 in an account that pays 5% interest
compounded yearly. Find the balance at the end of the given time period.
(Lesson 8.6)
38. 9.43 X 10°
42. 6.09013 X 10 10
46. 33,000,000
50. 0.000891
54. 67,000
55,1 year
56, 10 years
57, 15 years
58. 25 years
59. DEPRECIATION A piece of equipment originally costs $120,000. Its value
decreases at a rate of 10% per year. Write an exponential decay model to rep¬
resent the decreasing value of the piece of equipment. (Lesson 8.7)
Student Resources
Chapter 4
Evaluate the expression. Give the exact value if possible. Otherwise,
approximate to the nearest hundredth. (Lesson 9.1)
1. V3 2. V625 3. -VlOO 4. ±V676
5. Vl5 6. — Vl25 7. V220 8. ±V90
Solve the equation or write no solution. Write the solutions as integers if
possible. Otherwise, write them as radical expressions. (Lesson 9.2)
9. x 2 = 25 10. 4x 2 - 8 = 0 11.x 2 =-16 12. x 2 + 1 = 1
13.3x 2 - 48 = 0 14. 6x 2 + 6 = 4 15. 2x 2 - 6 = 0 16. x 2 - 4 = -3
17. FALLING OBJECT A ball is dropped from a bridge 80 feet above a river.
How long will it take for the ball to hit the surface of the water? Round your
solution to the nearest tenth. (Lesson 9.2)
Simplify the expression. (Lesson 9.3)
18.V60
19. V88
23.#
V5
20 . V250 21.VU2
24. 2^| 25. |V27
Sketch the graph of the function. Label the coordinates of the vertex.
(Lesson 9.4)
26. y — 3x 2 27. y — x 2 — 4 28. y — —x 2 — 2x 29. y = x 2 — 6x + 8
30. y = 4x 2 + 4x — 5 31. y = x 2 — 2x + 3 32. y = — x 2 + 3x + 2 33. y = — 3x 2 + 12x — 1
Solve the equation algebraically. Check your solution by graphing.
(Lesson 9.5)
34.x 2 — 6x = —5x 35. x 2 + 5x = —6 36. x 2 — 3x = 4
37.x 2 + 3x = 10 38. x 2 — 9 = 0 39. —2x 2 + 4x + 6 = 0
Write the quadratic equation in standard form. Then solve using the
quadratic formula. (Lesson 9.6)
40.x 2 + x = 12 41. x 2 12 — 4x 42. 3x 2 + llx = 4
44. x 2 — 3x — 4 = —6 45. — x 2 — 5x = 6 46. x 2 — 8 = lx
43. —x 2 + 5x = 4
47. 10 - 2x 2 = -x
Determine whether the equation has two solutions , one solution , or no
real solution. (Lesson 9.7)
48. 3x 2 + 14x - 5 = 0 49. 4x 2 + 12x + 9 = 0 50. x 2 + lOx + 9 = 0
52.5x 2 + 125 = 0 53. x 2 - 2x + 35 = 0 54. 2x 2 - x - 3 = 0
51. 2x 2 + 8x + 8 = 0
55. —3x 2 + 5x — 6 = 0
Sketch the graph of the inequality. (Lesson 9.8)
56. y > -x 2 + 4 57. y > 4x 2 58. y < 5x 2 + lOx 59. y < -x 2 + 4x + 5
Extra Practice
EXTRA PRACTICE
EXTRA PRACTICE
Chapter 10
Use a vertical format or a horizontal format to add or subtract. (Lesson 10.1)
1. (7x 2 - 4) + (x 2 + 5) 2. (3x 2 - 2) - (2x - 6x 2 ) 3. (8x 2 - 3x + 7) + (6x 2 - 4x + 1)
4. (— t? + 3 z) + (~z 2 — 4z — 6) 5. (5x 2 + lx — 4) — (4x 2 — 2x) 6. (3a + 2a 4 — 5) — (a 3 + 2a 4 4- 5a)
Find the product. (Lesson 10.2)
7. x(4x 2 — 8x + 7) 8. — 3x(x 2 + 5x — 5)
11 . (d- 1 )(d + 5) 12 . (3z + 4)(5z - 8)
Find the product. (Lesson 10.3)
15 . (x + 9) 2 16 . (~c - d) 2
19. (4x + 5) 2 20. (5p — 6q) 2
Solve the equation. (Lesson 10.4)
23. (x + 3)(x + 6) = 0 24. (x — ll) 2 = 0
27. (6 n - 9)(n - 7) = 0 28. 3(x + 2) 2 = 0
9. 56 2 (3£ 3 - 26 2 +1) 10. (t + 9)(2f + 1)
13. (x + 3)(x 2 — 2x + 6) 14. (3 + 2s — s 2 )(s — 1)
17. (a — 2 )(a + 2) 18. (—7 + m)(—7 — m)
21 . (2a + 3b)(2a — 3b) 22 . (lOx — 5j)(10x + 5)0
25. (z — 1 )(z + 5) = 0 26. w(w — 4) = 0
29. (2d ~ 2)(4 d - 8) = 0 30. x(3x + 1) = 0
Find the x-intercepts and the vertex of the graph of the function. Then
sketch the graph of the function. (Lesson 10.4)
31. y = (x — 8)(x — 6)
32. y = (x + 4)(x - 4)
33. y = (x — 5)(x — 7)
34. y =
(x + l)(x + 6)
35. y = (~x + 5)(x — 9)
36. v = (—x + l)(x + 5)
37. _y = (x — 3)(x + 1)
38. y =
( x - 3)(x + 7)
Solve the equation by factoring. (Lesson 10.5)
39.x 2 + 6x + 9 = 0
40. x 2 + 2x — 35 = 0
41. x 2 — 12x = —36
42. -x 2
1
II
04
43.x 2 - 15x = -54
44. —x 2 +14x = 48
45. x 2 2x — 24
46. x 2 -
5x + 4 = 0
Solve the equation by factoring. (Lesson 10.6)
47. 2x 2 + x - 6 = 0
48. 2x 2 + lx — —3
49. 9x 2 + 24x = —16
50. 20x 2 + 23x + 6 = 0
51. 4x 2 - 5x = 6
52. 3x 2 — 5 = — 14x
53. 3x 2 — 17x = 56
54. 12x 2
+ 46x — 36 = 0
Factor the expression. (Lesson 10.7)
55.x 2 - 1
56. 9b 2 — 81
57. 121 - x 2
58. 12 -
- 27x 2
59. t 2 + It + 1
60. x 2 + 20x + 100
61. 64y 2 + 48.y + 9
62. 20x 2
- lOOx + 125
Factor the expression completely. (Lesson 10.8)
63.x 4 - 9x 2
64. m 3 + 11m 2 + 28 m
65. x 4 + 4x 3 — 45x 2
66. x 3 + 2x 2 — 4x — 8
67. — 3j 3 - 15v 2 - 12 y
68. x 3 - x 2 + 4x - 4
69. 7x 6 - 2 lx 4
70. 8f 3
- 3 1 2 + 16 1 — 6
71. GEOMETRY The width of a box is 2 feet less than the length. The height is
8 feet greater than the length. The box has a cubic volume of 96 cubic feet.
What are the dimensions of the box? (Lesson 10.8)
Student Resources
Chapter 11
Solve the equation. Check your solutions. (Lesson 11.1)
1.
m
12
11
10
~ x _ 8
2 x
Q 3_ _ x + 2
5 “ 6
5 + t
c 2 — 16 c — 4
8
t - 3
c + 4
6 .
x + 15
-9
16
x — 10
The variables xand / vary directly. Use the given values to write an
equation that relates xand y. (Lesson 11.2)
7. x = 4, y = 12
8 . x = 5, y = 10
9. x = 16, y = 4
10. x = 21, y = 7
The variables xand y vary inversely. Use the given values to write an
equation that relates xand y. (Lesson 11.2)
11.x = 3, y = 5
12. x = l,y = 1
13. x = 4, y
14. x = 5.5, y = 6
Simplify the expression. If not possible, write already in simplest form.
(Lesson 11.3)
15.
12x 4
19.
42x
8x 2
16.
5x 2 - 15x 3
lOx
17.
x + 6
12x 3
20 .
x + 2
21 .
x 2 + lx + 6
4-y
18.
8x + 15
x - 3
y 2 - 16
22 .
x 2 - 9x + 18
x 2 - 4x - 12
Write the product in simplest form. (Lesson 11.4)
23.
3x 15
18 *
24.
r + 5z + 6
z + z
z + 3
25.
10x 2
x 2 - 25
• (x - 5)
Write the quotient in simplest form. (Lesson 11.4)
1 . 6x 5x . x
26.
27.
4 * 16 x 2 _ 6x + 9
Simplify the expression. (Lesson 11.5)
30.
x — 3
28.
x 2 + 5x — 36
x 2 - 81
+ (x 2 - 16)
__ 3 , 2
29 ' 5x + 5x
3x 4x — 1
x + 2 x + 2
31.
3x + 2
x — 1
X — 1
32.
6x
2x — 1 2x — 1
Simplify the expression. (Lesson 11.6)
33.4 - -
x z X
34. —
3x
x + 2
9x 2
35.^4 +
36. 1 + l + j-
3x 2 4x
37. ^ + 3 + 5
x 2 - 25
38.
x + 8 x — 3
4* — 1 3x
3x + 2
x
Solve the equation. Check your solutions. (Lesson 11.7)
39.
25
40.
1
x - 3
4 2
42.- + f
x 3
6
x
43.
x + 9
11
41.
4 + x
x — 5 x — 5
= 7
44.
3x
5
x — 1
+ 1 =
X 2 + 3x - 4
Extra Practice
EXTRA PRACTICE
EXTRA PRACTICE
Chapter 12
Find the domain of the function. Then sketch its graph and find the range.
(Lesson 12.1)
1 - y = ?>Vx 2 .y = V5x 3, y — Vx — 5
4. y = Vx + 1
5. j = Vx — 2
6 .y = Vx + 3
Simplify the expression. (Lesson 12.2)
9. 3V5 + 2V5 10. 8V7 - 15V7
13. V3(7 - V6)
14. (4 + VlO) 2
7.y = V3x + 2
11 . 2 V 8 + 3V32
4
15.
8 . y = V4x - 3
12. V20 - V45 + V80
3
V24
16.
5 - V2
Solve the equation. Check for extraneous solutions. (Lesson 12.3)
17. Vx— 11=0 18. V2x - 1 + 4 = 7 19. Vx + 10 = 2
22. 4Vx + 5 = 21
20 . 12 = V3x + 1 + 7
21 . x = V4x — 3
25. (8 1/4 ) 8
Evaluate the expression. (Lesson 12.4)
23. 4 2/3 • 4 4/3 24. (27 1/2 ) 2/3
Simplify the variable expression. (Lesson 12.4)
27. x 1/4 • x 1/2 28. (x 2 )" 4 29. (x • y m f •
2\m
26. (2 2 • 3 2 )
30. (x • x 1/3 ) 3/4
Solve the quadratic equation by completing the square. (Lesson 12.5)
31.x 2 4- lOx = 56 32. x 2 + 2x = 3 33. x 2 + 6x + 8 = 0
34.x 2 - 12* = 13 35. x 2 - 6x = 16 36. x 2 - lOx - 39 = 0
Find the missing length of the right triangle if a and b are the lengths of the
legs and c is the length of the hypotenuse. (Lesson 12.6)
37. a = 1, b = 1 38. a = 1, c = 2 39. b = 6, c = 10
40. a = 1, b = 10 41. b = 15, c = 25 42. a = 30, c = 50
Find the distance between the two points. Round your solution to the
nearest hundredth if necessary. (Lesson 12.7)
43. (7, -6), (-1, -6) 44. (5, 2), (5, -4) 45. (12, -7), (-4, 2) 46. (-4, -5), (-8, 9)
47.(5, 8), (0, -3) 48.(10,-1), (4, 11) 49. (-3,-8), (-1,-4) 50. (12, 1 1), (9, 15)
Find the midpoint of the line segment connecting the given points. Then
show that the midpoint is the same distance from each point. (Lesson 12.8)
51. (0,4), (4, 5) 52. (-3, 3), (6,-1) 53. (1, 0), (4,-4) 54. (0, 0), (3,-2)
55. (-2, 0), (2, 8) 56. (3, 7), (-5, -9) 57. (6, 2), (4, 10) 58. (4, -6), (-8, 3)
59. INDIRECT PROOF Use an indirect proof to prove that the following conclu¬
sion is true. If xy = 0, then either x = 0 or y = 0. (Lesson 12.9)
Student Resources
ijjiJ-S-f- £3011=1 EMI
Variables, Expressions, and Properties
Evaluate the variable expression when a — l.b — l.c— -4, and d— 1.
*\. a 2 — 3b + be 2. |c + d\ 3- — d — (—c)
5.
b — c
6- + 2d)
7. c 3 d
Simplify the expression. Name each property that you used.
9. —ab + ba 10. 0 + V2 11. 5(x + 4)
13. 7 -3 • 7 5 • 7 3
5
14. (2v 2 ) 4
15.
Linear Equations and Inequalities
Solve the equation or inequality.
17. 4s — 6 = 18 18.0.2 b - 1.3 >6.7
20 . 4m — 2(5 — m) = 14
21 . 9 + \k = 14
4. -6(f )(</)
8 . 5(2“ 4 )
12. — 1 • « + 0 • n
v-3
16.
19. j-p - 1 < 11
22 . 7(a + 5) — —(2a + 1)
23. 0.15x + 5.01 = 1.44
24. -7 > 5 — 2y
25. 0 < 1 -c<=r
26. 4 1 < —12 or — t < —4
27. 2 - x = 1
28. \2n + 5 | > 3
Linear Systems
Solve the system of linear equations. Then check your solution.
29. 4x — y = 6 30. 5p + 3q = 4
x + 3y = 8 Ip + 2q = 21
Graph the system of linear inequalities.
32. 2x + 3y> -6 33. x + 4y > 0
_y > 3x — 13 y>0
31. 6a — 9b = 18
/? = —a + 2
34. 3% - y > 1
.y >x
End-of-Course Test
END-OF-COURSE TEST
END-OF-COURSE TEST
Solve the quadratic equation. Write the exact solution.
35. a 2 + 5 = 37 36. x 2 + 2x = 35 37. 2v 2 — 6v — 9 = 0
38. Sketch the graph of y < x 2 + 3x.
Polynomials and Factoring
Add, subtract, or multiply.
39. ( t 2 + 3t — 2) — (t + 6) 40. (x + 2) + (x 2 — 6x — 1)
41. (x + 2)(x 2 — 6x — 1) 42. (9c — 5)(9 c + 5)
Factor the expression completely.
43./ + y - 30 44. z 3 - 3z 2 + 2z 45. 8 + 21 n 3
Rational Expressions and Equations
Write the expression in simplest form.
46.
x 2 - 6x + 9
4x - 12
47.
x 2 - 7x + 6
2x - 12
49.
3x
x 2 - 3x
z + 1
6z 2
Solve the equation.
52.
d
d + 4
d-5
d + 1
J5
4^
4x
3x - 3
48.
6yl 2 ^ 4yl 3
4^+8 k 2 - 4
51.
x — 2
+
x — 2
x — 1
54.
n
n — 1
+
2
ft + 1
= 2
Radical Expressions and Equations
Simplify the expression.
55. Vl8 • V2
58.
8
2 + V3
56. V98 • V8
59. 4 5/2 • 4 1/2
Solve the equation. Check for extraneous solutions.
61. Vx + 4 = 0 62. V4x - 3 = 3
57. 2V6(5 - V6)
60. (100 2 ) 1/4
63. Vx + 2 = x
Student Resources
Table of Symbols
Symbol
Page
Symbol
Page
♦, (a)(b)
multiplied by or times (X)
3
is approximately equal to
163
a n
nth power of a
9
a
b
ratio of a to b, or a:b
177
...
continues on
9
a
h
rate of a per b, where a and b are
177
o
measured in different units
( )
parentheses
10
%
percent
183
[ ]
brackets
10
( X,y )
ordered pair
203
—
equal sign, is equal to
24
m
slope
229
?
Is this statement true?
24
k
constant of variation
236
is not equal to
24
b
y-intercept
243
<
is less than
26
fix)
the value of/at v
254
<
is less than or equal to
26
71
pi, an irrational number
445
>
is greater than
26
approximately equal to 3.14
>
is greater than or equal to
26
a~ n
\’ a ± 0
a n
449
O
degree(s)
67
c x 10 w
scientific notation, where
1 < c < 10 and n is an integer
469
—a
the opposite of a
71
the positive square root of a
499
\a\
absolute value of a
71
when a > 0
plus or minus
499
i
a
reciprocal of a, a =f= 0
113
Tables
TABLES
TABLES
Table of Formulas
Geometric Formulas
Perimeter of a polygon
P = a + b + ...+z where a, b, . . ., z = side lengths
Area of a triangle
A = j-bh where b = base and h = height
Area of a square
A = s 1 where s = side length
Area of a rectangle
A = tw where t = length and w = width
Area of a trapezoid
A = ^ ■h(b l + b 2 ) where h = height and b v b 2 = bases
Volume of a cube
V = .s’ 3 where s = edge length
Volume of a rectangular prism
V = Iwh where i = length, w = width, and h = height
Circumference of a circle
C = nd where 71-3.14 and d = diameter
C = 27t r where 71-3.14 and r = radius
Area of a circle
A = nr 2 where 71-3.14 and r = radius
Surface area of a sphere
S = 47rr 2 where 71-3.14 and r = radius
Volume of a sphere
V = ~^7Zr where ll - 3.14 and r = radius
Other Formulas
Average speed
r = y where r = average rate or speed, d = distance, and t = time
Algebraic Formulas
Slope formula
y 2 " y x
m = where m = slope and (x v yA and (jc 2 , y 2 ) are two points
x 2 x x
Quadratic formula
The solutions of ax 2 + bx + c — 0 are x — 0
9 2 a
when a ^ 0 and b z — 4 ac > 0.
Pythagorean theorem
a 2 + b 2 = c 2 where a, b = length of the legs and c = length
of the hypotenuse of a right triangle
Distance formula
The distance between (x p y x ) and (x 2 , y 2 ) is V(v 2 — x x ) 2 + (y 2 — y x ) 2 .
Midpoint formula
(x x + v 2 y x + y 2 \
The midpoint between (x v y x ) and (x v y 2 ) is 1 2 , 2 1.
Student Resources
Table of Properties
Basic Properties
Addition
Multiplication
Closure
a + b is a unique real number.
ab is a unique real number.
Commutative
a + b = b + a
II
Associative
(a + b) + c = a + (b + c)
( ab)c = a(bc)
Identity
a + 0 = a,0 + a = a
a( 1) = a , 1(a) = a
Property of zero
a + (— a) = 0
o
II
o
Property of negative one
(—1 )a = —a or a(— 1) = —a
Distributive
a(b + c) = ab + ac
or (b + = ba + ca
Properties of Equality
Addition
If a = b, then a + c = b + c.
Subtraction
If a = b, then a — c = b — c.
Multiplication
If a = b, then ca = cb.
Division
If a = b and c A 0, then — = —.
c c
Properties of Exponents
Properties of Radicals
Product of Powers
a m . a n = a m + n
Product Property
Vab = Va • Vb
Power of a Power
Power of a Product
(a m ) n = a m ‘ n
(.a • b) m = a m • b m
a m
Quotient Property
II
^il»i
%
O
Quotient of Powers
— - = a m ~ n ,a =£ 0
a n
Properties of Proportions
Power of a Quotient
Negative Exponent
Zero Exponent
(a\ m a m u , n
\bJ b m
a~ n = — , a A 0
a n
a° = 1, a =f= 0
Reciprocal
Cross-multiplying
t r a c . i b d
If t — then -- = -.
b d a c
If T = then ad = be.
b d
Special Products and Their Factors
Sum and Difference Pattern
Square of a Binomial Pattern
(<a + b)(a — b) = a 2 — b 2
(«a + b) 2 = a 2 + lab + b 2
(<a — b) 2 = a 2 — lab + b 2
Properties of Rational Expressions
Multiplication
Division
Addition
Subtraction
a c_
b # d
ac
bd
a d
a , b_ _ a + b a_ , c_ _ ad + be
c ’ b d bd
c c
a_ _ b_
c c
a — b a
c ’ b
c_
d
ad — be
bd
H
Tables
TABLES
TABLES
Find a decimal approximation of V54.
Solution
Find 54 in the numbers’ column. Read across that line to the square roots’ column.
This number is a three-decimal place approximation of V54, so V54 ~ 7.348.
Estimate V3000.
Solution
Find the two numbers in the squares’ colu mn th at 3000 is between. Read across
these two li nes to the numbers’ column; V3000 is between 54 and 55, but closer
to 55. So, V3000 » 55. A more accurate approximation can be found using a
calculator: 54.772256.
No.
Square
Sq. Root
51
2601
7.141
52
2704
7.211
53
2809
7.280
54
2916
7.348
55
3025
7.416
Student Resources
Table of Squares and Square Roots
No.
Square
Sq. Root
101
10,201
10.050
102
10,404
10.100
103
10,609
10.149
104
10,816
10.198
105
11,025
10.247
106
11,236
10.296
107
11,449
10.344
108
11,664
10.392
109
11,881
10.440
110
12,100
10.488
111
12,321
10.536
112
12,544
10.583
113
12,769
10.630
114
12,996
10.677
115
13,225
10.724
116
13,456
10.770
117
13,689
10.817
118
13,924
10.863
119
14,161
10.909
120
14,400
10.954
121
14,641
11.000
122
14,884
11.045
123
15,129
11.091
124
15,376
11.136
125
15,625
11.180
126
15,876
11.225
127
16,129
11.269
128
16,384
11.314
129
16,641
11.358
130
16,900
11.402
131
17,161
11.446
132
17,424
11.489
133
17,689
11.533
134
17,956
11.576
135
18,225
11.619
136
18,496
11.662
137
18,769
11.705
138
19,044
11.747
139
19,321
11.790
140
19,600
11.832
141
19,881
11.874
142
20,164
11.916
143
20,449
11.958
144
20,736
12.000
145
21,025
12.042
146
21,316
12.083
147
21,609
12.124
148
21,904
12.166
149
22,201
12.207
150
22,500
12.247
No.
Square
Sq. Root
1
1
1.000
2
4
1.414
3
9
1.732
4
16
2.000
5
25
2.236
6
36
2.449
7
49
2.646
8
64
2.828
9
81
3.000
10
100
3.162
11
121
3.317
12
144
3.464
13
169
3.606
14
196
3.742
15
225
3.873
16
256
4.000
17
289
4.123
18
324
4.243
19
361
4.359
20
400
4.472
21
441
4.583
22
484
4.690
23
529
4.796
24
576
4.899
25
625
5.000
26
676
5.099
27
729
5.196
28
784
5.292
29
841
5.385
30
900
5.477
31
961
5.568
32
1024
5.657
33
1089
5.745
34
1156
5.831
35
1225
5.916
36
1296
6.000
37
1369
6.083
38
1444
6.164
39
1521
6.245
40
1600
6.325
41
1681
6.403
42
1764
6.481
43
1849
6.557
44
1936
6.633
45
2025
6.708
46
2116
6.782
47
2209
6.856
48
2304
6.928
49
2401
7.000
50
2500
7.071
No.
Square
Sq. Root
51
2601
7.141
52
2704
7.211
53
2809
7.280
54
2916
7.348
55
3025
7.416
56
3136
7.483
57
3249
7.550
58
3364
7.616
59
3481
7.681
60
3600
7.746
61
3721
7.810
62
3844
7.874
63
3969
7.937
64
4096
8.000
65
4225
8.062
66
4356
8.124
67
4489
8.185
68
4624
8.246
69
4761
8.307
70
4900
8.367
71
5041
8.426
72
5184
8.485
73
5329
8.544
74
5476
8.602
75
5625
8.660
76
5776
8.718
77
5929
8.775
78
6084
8.832
79
6241
8.888
80
6400
8.944
81
6561
9.000
82
6724
9.055
83
6889
9.110
84
7056
9.165
85
7225
9.220
86
7396
9.274
87
7569
9.327
88
7744
9.381
89
7921
9.434
90
8100
9.487
91
8281
9.539
92
8464
9.592
93
8649
9.644
94
8836
9.695
95
9025
9.747
96
9216
9.798
97
9409
9.849
98
9604
9.899
99
9801
9.950
100
10,000
10.000
Tables
TABLES
TABLES
Table of Measures
Time
60 seconds (sec) = 1 minute (min) 365 days
60 minutes = 1 hour (h) 52 weeks (approx.)
24 hours = 1 day ^ months_
7 days = 1 week 10 years
4 weeks (approx.) = 1 month 100 years
= 1 year
= 1 decade
= 1 century
Metric
United States Customary
Length
Volume
10 millimeters (mm) = 1 centimeter (cm)
100 cm~~I 1 ( .
i aaa 1 meter (m)
1000 mmj
1000 m = 1 kilometer (km)
100 square millimeters = 1 square centimeter
(mm 2 ) (cm 2 )
10,000 cm 2 = 1 square meter (m 2 )
10,000 m 2 = 1 hectare (ha)
1000 cubic millimeters = 1 cubic centimeter
(mm 3 ) (cm 3 )
1,000,000 cm 3 = 1 cubic meter (m 3 )
1000 milliliters (mL) = 1 liter (L)
1000 L = 1 kiloliter (kL)
1000 milligrams (mg) = 1 gram (g)
1000 g = 1 kilogram (kg)
1000 kg = 1 metric ton (t)
Degrees Celsius (°C) .
0°C = freezing point of water
37°C = normal body temperature
100°C = boiling point of water
Temperature
Length
12 inches (in.)
36 in.
3ft
5280 ft
1760 yd
= 1 foot (ft)
= 1 yard (yd)
= 1 mile (mi)
144 square inches (in. 2 ) = 1 square foot (ft 2 )
9 ft 2 = 1 square yard (yd 2 )
43,560 ft 2 "
4840 yd^_
= 1 acre (A)
Volume
1728 cubic inches (in. 3 ) = 1 cubic foot (ft 3 )
27 ft 3 = 1 cubic yard (yd 3 )
Liquid Capacity
1 fluid ounces (fl oz)
2c
2 pt
4 qt
1 cup (c)
1 pint (pt)
1 quart (qt)
1 gallon (gal)
Weight
16 ounces (oz) = 1 pound (lb)
20001b = 1 ton (t)
Temperature
Degrees Fahrenheit (°F)
32°F = freezing point of water
98.6°F = normal body temperature
212°F = boiling point of water
Student Resources
Appendix I
Data Displays and
Describing Data
Goal Make stem-and-leaf plots and box-and-whisker plots; Describe data
using mean, median, mode, and range.
A stem-and-leaf plot is an arrangement of digits that is used to display and order
numerical data.
i Make a Stem-and-Leaf Plot
The following data show the ages of the 27 residents of a community in
Alaska. Make a stem-and-leaf plot to display the data.
45
1
52
42
10
40
50
40
7
46
19
35
3
11
31
6
41
12
43
37
8
41
48
42
55
30
58
Solution
Use the digits in the tens’ place for the stems and the digits in the ones’ place
for the leaves. You can order the leaves to make an ordered stem-and-leaf plot.
Unordered stem-and-leaf plot
0
1
2
3
4
5
Key: 415 = 45
1 7 3 6 8
0 9 12
5 17 0
5200613182
2 0 5 8
Ordered stem-and-leaf plot
0
1
2
3
4
5
1 3 6 7 8
0 12 9
0 15 7
001 1223568
0 2 5 8
Key: 415 = 45
CHOOSING PLACE VALUES Your choice of place values for the stems and the
leaves will depend on the data. For data between 0 and 100, the leaves are the
digits in the ones’ place. Include a key that explains how to interpret the digits.
Make a Stem-and-Leaf Plot
1. Use the stem-and-leaf plot at the
right to order the data set.
6 3 8 1 7 5 3
8 2 5 0 7 7 0
8 3 14 7
Key: 3 | 8 = 38
2 . Make an ordered stem-and-leaf plot for the following data:
22, 34, 11, 55, 13, 22, 30, 21, 39,48, 38, 46
Appendix 1
MEASURES OF CENTRAL TENDENCY The mean, the median, and the mode
are three commonly used measures of central tendency. A measure of central
tendency describes a typical number in a data set.
m
Student ffeCp
► Skills Review
For help with mean,
median, and mode, see
p. 780.
E2ISQ19 2 Find the Mean, Median, and Mode
Find the measure of central tendency of the ages of the residents of the
community in Alaska given in Example 1 on page 803.
a. mean b. median c. mode
Solution
a. To find the mean, add the 27 ages and divide by 27.
mean =
1 + 3 + • • • + 55+ 58
27
ANSWER ^ The mean age is about 32.
Student HeGp
^
►Study Tip
Recall that when a
data set has an even
number of values, the
median is the mean of
the two middle values
when the values are
written in numerical
order.
L „ __ J'
b- To find the median, write the ages in order and find the middle number.
To order the ages, use the ordered stem-and-leaf plot in Example 1.
1
3
6
7
8
10
11
12
19
30
31
35
37
40
40
41
41
42
42
43
45
46
48
50
52
55
58
ANSWER ^ From this list, you can see that the median age is 40. Half of the
ages fall below 40 and half of the ages are 40 or older.
c. To find the mode, use the ordered list in part (b).
ANSWER ► There are three modes, 40, 41, and 42.
RANGE You can describe how spread out data are by finding the range. The
range of a data set is the difference between the greatest value and the least value.
EHmEHI 3 rind the Range
Find the range of the ages of the residents of the community in Alaska given in
Example 1 on page 803.
Solution
To find the range, subtract the least age from the greatest age:
58 — 1 = 57
ANSWER ► The range of the ages is 57.
Find the Mean , Median, Mode, and Range
Appendix 1
Find the mean, the median, the mode, and the range of the data.
3 . 2, 2, 2, 2, 4, 4, 5 4 . 5, 10, 15, 1, 2, 3, 7, 8
BOX-AND-WHISKER PLOT A box-and-whisker plot is a data display that
divides a set of data into four parts. The median or second quartile separates the
set into two halves: the numbers that are below the median and the numbers that
are above the median. The first quartile is the median of the lower half. The
third quartile is the median of the upper half.
4 Find Quartiles
Use this set of data: 11, 19, 5, 34, 9, 25, 28, 16, 17, 11, 12, 7.
a. Find the first, second, and third quartiles of the data.
b. Draw a box-and-whisker plot of the data.
Solution
a. Begin by writing the numbers in increasing order. You must find the second
quartile before you find the first and third quartiles.
Second quartile: 12 + 16 = 14
5,7, 9,| 11, 11, 12,| 16, 17, 19, |25, 28, 34
= 10 Third quartile: —
19 + 25
b. Draw a number line that includes the least number and the greatest number
in the data set.
Plot the least number, the first quartile, the second quartile, the third
quartile, and the greatest number. Draw a line from the least number to the
greatest number below your number line. Plot the same points on that line.
The “box” extends from the first to the third quartile. Draw a vertical line
in the box at the second quartile. The “whiskers” connect the box to the
least and greatest numbers.
5 10 14
Least First Second
number quartile quartile
22
Third
quartile
34
Greatest
number
v
Find the first, second, and third quartiles of the data. Then draw a box-
and-whisker plot.
5 . 9, 20, 30, 10, 18, 11,22, 10, 20
6 . 12, 30, 19, 15, 18, 22
Appendix 7
Exercises
Make a stem-and-leaf plot for the data. Use the result to list the data in
increasing order.
1.60, 74, 75,63, 78, 70, 50,
74, 52, 74, 65, 78, 54
3. 4,31,22, 37, 39,24, 2, 28,
1, 26, 28, 30, 28, 3, 20, 20, 5
5. 87, 61, 54, 77, 79, 86, 30, 76, 52,
44, 48, 76, 87, 68, 82, 61, 84,
33, 39, 68, 37, 80, 62,81,76
2.24, 29, 17,50, 39,51, 19, 22,
40, 45, 20, 18, 23, 30
4 . 15, 39, 13,31,46, 9,38, 17, 32,
10, 12, 45, 30, 1, 32, 23, 32, 41
6 . 48, 10, 48, 25, 40, 42, 44, 23,
21, 13, 50, 17, 18, 19,21,57,
35, 33, 25, 50, 13, 12, 46
Find the mean, the median, the mode, and the range of the data.
7 . 4, 2, 10, 6, 10, 7, 10
9 . 8, 5, 6, 5, 6, 6
11 . 5 , 3, 10, 13, 8, 18, 5, 17, 2,
7, 9, 10, 4, 1
13 . 3.56, 4.40, 6.25, 1.20, 8.52, 1.20
8 . 1,2, 1,2, 1,3, 3,4,3
10 . 4, 4, 4, 4, 4, 4
12 . 12, 5, 6, 15, 12, 9, 13, 1,4, 6,
8, 14, 12
14 . 161, 146, 158, 150, 156, 150
15. Tell which set of data is shown by the box-and-whisker plot.
0 5 10 15 ] 20
I
i o o o
A. 0, 10, 15, 18, 25 B. 5, 10, 15, 18, 30
25 30
I
-.
C. 5, 10, 15, 18,25
Find the first, second, and third quartiles of the data.
16 . 12, 5, 3, 8, 10, 7, 6, 5
18 . 1, 12, 6, 5,4, 7, 5, 10, 3,4
17 . 20, 73,31,53, 22, 64, 47
19 . 2.3, 5.6, 3.4, 4.5, 3.8, 1.2, 9.7
Draw a box-and-whisker plot of the data.
20 . 6, 7, 10, 6, 2, 8, 7, 7, 8
21 . 10,5,9, 50, 10, 3,4, 15,20,6
22 . 12, 13, 7, 6, 25, 25, 5, 10, 15, 10, 16, 14, 29
23 . 8, 8, 10, 10, 1, 12, 8, 6, 5, 1, 9, 10
Create a collection of 16 numbers that could be represented by the box-
and-whisker plot.
24 .
10 13 19 27
25.
-•
•—
— i
i - i
>-
•
38
106
124
150 162
193
Appendix 1
26 . Use a stem-and-leaf plot (months as stems, days as leaves) to write the
birthdays in order from earliest in the year to latest (1 = January,
2 = February, and so on). Include a key with your stem-and-leaf plot.
10-11
4-14
7-31
12-28
4-17
2-22
8-21
1-24
9-12
1-3
4-30
10-17
6-5
1-25
5-10
12-9
4-1
8-26
12-15
3-17
4-30
2-3
11-11
6-13
11-4
6-24
6-3
4-8
2-20
11-28
27 . Create two different sets of data, each having 10 ages. Create one set so that
the mean age is 16 and the median age is 18. Create the other set so that the
median age is 16 and the mean age is 18.
In Exercises 28-32, use the following information.
The table shows the number of shutouts
that ten baseball pitchers had in their
careers. A shutout is a complete game
pitched without allowing a run.
28 . Find the mean and the median for the
set of data.
29 . Write the numbers in decreasing order.
30 . Does the set of data have a mode?
If so, what is it?
31. What number could you add to the
set of data to make it have more than
one mode? Explain why you chose
the number.
32 . Draw a box-whisker plot of the data.
Pitcher Shutouts
Warren Spahn
63
Christy Mathewson
80
Eddie Plank
69
Nolan Ryan
61
Bert Blyleven
60
Don Sutton
58
Grover Alexander
90
Walter Johnson
110
Cy Young
76
Tom Seaver
61
In Exercises 33-36, use the box-and-whisker plot that shows the lengths
(in hours) of commuters' trips to work.
| I I I I | I M I | I I I I | I I I I | I I I I | I I I I | »
0
I
4 =
0.5
1.0
1.5
2.0
2.5
3.0
i
I
33 . How long is the median trip to work?
34 . What is the range of the lengths of commuters’ trips to work?
35 . Compare the number of people whose trip is 0-0.5 hour long to the number
of people whose trip is 1-3 hours long. Explain your reasoning.
36 . Do more people travel 1-3 hours than travel 0.5-1 hour? Explain.
Appendix 7
Appendix 2
Probability and Odds
Goal Find the probability of an event and the odds of an event.
The probability of an event is a measure of the likelihood that the event will
occur. It is a number between 0 and 1, inclusive.
---
P= 0 P = 0.25 P= 0.5 P = 0.75 P= 1
Impossible Unlikely Occurs half the time Quite likely Certain
The possible results of an experiment are called outcomes. An event is a
collection of outcomes. The outcomes for an event you wish to have happen are
called favorable outcomes.
Theoretical probability is based on knowing the number of equally likely
outcomes of an experiment. When all of the outcomes are equally likely, you can
use the following rule to calculate the theoretical probability of an event.
Theoretical probability P
Number of favorable outcomes
Total number of outcomes
■Elista* i Find the Probability of an Event
a. You toss two coins. What is the probability P that both are heads?
b. An algebra class has 17 boys and 16 girls. One student is chosen at random
from the class. What is the probability P that the student is a girl?
Solution
a. There are four possible outcomes that
are equally likely.
_ Number of favorable outcomes _ 1
Total number of outcomes 4
b. Because the student is chosen at random, it is equally likely that any of the
33 students will be chosen. Let “number of girls” be the favorable
outcome. Let “number of students” be the total number of outcomes.
Number of favorable outcomes _ Number of girls
Total number of outcomes
Number of students
f - 0.485
Find the Probability of an Event
1. A bag contains 5 bagels: plain, plain, whole wheat, raisin, and poppy seed. If
a bagel is chosen at random, what is the probability of choosing a plain bagel?
Appendix 2
EXPERIMENTAL PROBABILITY Another type of probability is experimental
probability. This type of probability is based on repetitions of an actual
experiment and is calculated by the following rule.
Experimental probability P =
Number of favorable outcomes observed
Total number of trials
Activity Investigating Experimental Probability
Partner Activity Toss three coins 20 times and record the number of
heads for each of the 20 tosses.
© Use your results to find the experimental probability of getting three
heads when three coins are tossed.
© Combine your results with those of all the other groups in your class.
Then use the combined results to find the experimental probability of
getting three heads when three coins are tossed.
© Find the theoretical probability of getting three heads when three coins
are tossed. How does it compare with your results from Step 2?
SURVEYS A survey is a type of experiment. Probabilities based on the outcomes
of a survey can be used to make predictions.
2 Use a Survey to Find a Probability
Use the circle graph at the right
showing the responses of
500 teens to a survey asking
“Where would you like to live?”
If you were to ask a randomly
chosen teen this question, what
is the experimental probability
that the teen would say “large
city?”
Where Would You Like to Live?
Large city 155
Small town 90
Suburbs 100
Rural/ranch 70
Wilderness area 85
Solution
Let “choosing large city” be the favorable outcome. Let “number surveyed” be
the total number of trials.
Experimental probability P =
Number choosing large city
Number surveyed
= ^55
500
= 0.31
Appendix 2
Tug
Ul
Use a Survey to Find a Probability
2 - The circle graph at the right
shows the responses of
200 adults to a survey asking
“What is your favorite color?”
If you were to ask a randomly
chosen adult this question,
what is the experimental
probability that the adult
would say “red?”
What is Your Favorite Color?
nthni* R m
uiner o •
Red 55-•
•- Blue 80
,
• oreen ou
FINDING ODDS When all outcomes are equally likely, the odds that an event
will occur are given by the formula below.
_ Number of favorable outcomes
Number of unfavorable outcomes
Student HeCp
►Study Tip
Odds are always read
as the ratio of one
quantity to another. For
example, | is read as
"four to three," not as
"four thirds."
3 Find the Odds of an Event
You randomly choose an integer from 0 through 9.
a. What are the odds that the integer is 4 or more?
b. What are the odds that the integer is 1 or less?
Solution
a. There are 6 favorable outcomes: 4, 5, 6, 7, 8, and 9. There are
4 unfavorable outcomes: 0, 1,2, and 3.
_ Number of favorable outcomes _ 6 _ 3
Number of unfavorable outcomes 4 2
ANSWER ► The odds that the integer is 4 or more are 3 to 2.
b. There are 2 favorable outcomes: 0 and 1. There are 8 unfavorable
outcomes: 2, 3, 4, 5, 6, 7, 8, and 9.
Odds =
Number of favorable outcomes
Number of unfavorable outcomes
2
8
1
4
ANSWER ► The odds that the integer is 1 or less are 1 to 4.
Find the Odds of an Event
3 , You roll a number cube. What are the odds of rolling a 3 or higher?
4 . You write each of the 26 letters of the alphabet on a separate piece of paper
and place them all in a paper bag. What are the odds of randomly choosing a
vowel (A, E, I, O, U) from the bag?
T
Appendix 2
Exercises
1 - Using the results of a student lunch survey, you determine that the
probability that a randomly chosen student likes green beans is 0.38. Is this
probability theoretical or experimental?
2 . The probability that an event will occur is 0.4. Is it more likely that the event
will occur, or is it more likely that the event will not occur?
3 . The odds that an event will occur are 3 to 4. Is it more likely that the event
will occur, or is it more likely that the event will not occur?
Tell whether the event is best described as impossible , unlikely; likely to
occur half the time , quite likely , or certain. Explain your reasoning.
4 . The probability of rain is 80%, or 0.8.
5 . The odds in favor of winning a race are
6. The odds of being chosen for a committee are 1 to 1.
Suppose it is equally likely that a teacher will chose any day from
Monday, Tuesday, Wednesday, Thursday, and Friday to have the next test.
7 . What is the probability that the next test will be on a Friday?
8. What are the odds that the next test will be on a day starting with the
letter T?
choosing a red marble from the given
Find the probability of randomly
bag of red and white marbles.
9 . Number of red marbles: 16
Total number of marbles: 64
11. Number of white marbles: 7
Total number of marbles: 20
10, Number of red marbles: 8
Total number of marbles: 40
12, Number of white marbles: 24
Total number of marbles: 32
Find the odds of randomly choosing the indicated letter from a bag that
contains the letters in the name of the given state.
13 . S; MISSISSIPPI 14 . N; PENNSYLVANIA
15. A; NEBRASKA 16. G; VIRGINIA
You toss two coins.
17. What is the theoretical probability that only one is tails?
18. Use the theoretical probability to find the odds that only one is tails.
19. You toss a six-sided number cube 20 times. For twelve of the tosses the
number tossed was 3 or more.
a. What is the experimental probability that a number tossed is 3 or more?
b. What are the odds that a number tossed is 3 or more?
Appendix 2
20 . A sea turtle buries 90 eggs in the sand. From the 50 eggs that hatch,
37 turtles do not make it to the ocean. What is the probability that an egg
chosen at random hatched and the baby turtle made it to the ocean?
In Exercises 21-23, use the graph.
21. What is the probability that a pet¬
owning household chosen at random
owns a dog?
22 . What is the probability that a pet¬
owning household chosen at random
does not own a fish?
23 . There are approximately 98.8 million
households in the United States. If a
household is chosen at random, what
are the odds that the household owns
a pet?
In Exercises 24 and 25, use the table, which shows the percent of citizens
from various age groups who changed homes within the United States
from 1995 to 1996.
24 . What is the
probability
that a citizen from
the 15-19 age group
changed homes?
25 . What are the odds
that a citizen from
the 25-29 age group
moved to a home in
a different state?
Percent of U.S. Citizens of Given Ages who Moved
Age
group
Total
Same
county
Different county,
same state
Different
state
15 to 19
15
10
3
2
20 to 24
33
21
6
5
25 to 29
32
20
7
5
30 to 44
16
10
3
3
In Exercises 26-28, use the table, which shows the number of
earthquakes of magnitude 4.0 or greater in the western United States
since 1900. The magnitude of an earthquake indicates its severity.
26 . What is the probability that the
magnitude of an earthquake is from
6.0 to 6.9?
27 . What is the probability that the
magnitude of an earthquake is not
from 4.0 to 4.9?
28 . What are the odds that the magnitude
of an earthquake is from 7.0 to 7.9?
Magnitude
Number of
earthquakes
8 and higher
i
7.0-7.9
18
6.0-6.9
129
5.0-5.9
611
4.0-4.9
3171
29 . Your cousin spills spaghetti sauce on her shirt and asks to borrow a clean
shirt from you for the rest of the day. You decide to let her choose from a
selection of 4 sweatshirts, 1 hockey shirt, 8 T-shirts, and 3 tank tops. If it is
equally likely that your cousin will choose any shirt, what are the odds that
she will choose a sweatshirt?
Appendix 3
Fitting a Line to Data
Goal Find a linear equation that approximates a set of data points.
In this lesson you will learn how to write
a linear model to represent a collection
of data points.
Usually there is no single line that
passes through all of the data points, so
you try to find the line that best fits the
data, as shown at the right. This is called
the best-fitting line.
There is a mathematical definition of the best-fitting line that is called least
squares approximation. Many calculators have a built-in program for finding the
equation of the best-fitting line. In this lesson, however, you will be asked to use
a graphical approach for drawing a line that is probably close to the best-fitting
line.
Activity
ll
Approximating a Bast-Fitting Line
With your group, use the
following steps to approximate
a best-fitting line.
o Carefully plot the following
points on graph paper.
(0, 3.3), (0, 3.9), (1,4.2), (1,4.5),
(1,4.8), (2, 4.7), (2, 5.1), (3, 4.9),
(3, 5.6), (4, 6.1), (5, 6.4), (5,7.1),
(6, 6.8), (7, 7.5), (8, 7.8)
© Use a ruler to sketch the line that you think best approximates the data
points. Describe your strategy.
© Locate two points on the line. Approximate the x-coordinate and the
/-coordinate for each point. (These do not have to be two of the
original data points.)
© Use the method from Lesson 5.3 to find an equation of the line that
passes through the two points.
Appendix 3
i Approximate a Best-Fitting Line
The data in the table show the forearm lengths and
foot lengths (without shoes) of 18 students in an
algebra class. After graphing these data points,
draw a line that corresponds closely to the data.
Write an equation of your line.
Solution
Let x represent the forearm length and let
y represent the foot length. To begin, plot
the points given by the ordered pairs. Then
sketch the line that appears to best fit the points.
Forearm and Foot Length
y
26
'E 24
o
f> 22
I 20
18
0
Forearm length (cm)
Forearm
length
Foot
length
22 cm
24 cm
20 cm
19 cm
24 cm
24 cm
21 cm
23 cm
25 cm
23 cm
18 cm
18 cm
20 cm
21 cm
23 cm
23 cm
24 cm
25 cm
20 cm
22 cm
19 cm
19 cm
25 cm
25 cm
23 cm
22 cm
22 cm
23 cm
18 cm
19 cm
24 cm
23 cm
21 cm
24 cm
22 cm
22 cm
Next, find two points that lie on the line. You might choose the points (19, 20)
and (26, 26). Find the slope of the line through these two points.
m
m
m
y izi i
X 2 — X x
26 - 20
26 - 19
6
7
Write slope formula.
Substitute.
Simplify.
~ 0.86
Decimal approximation.
To find the y-intercept of the line, substitute the values m = 0.86, x = 19, and
y = 20 in the slope-intercept form.
y = mx + b Write slope-intercept form.
20 = (0.86)(19) + b Substitute 0.86 for m, 19 for x, and 20 for y.
20 = 16.34 + b Simplify.
3.66 = b Solve for b.
ANSWER ^ An approximate equation of the best-fitting line is y = 0.86x + 3.66.
In general, if a student has a long forearm, then that student also has
a long foot.
V.
Appendix 3
Approximate a Best-Fitting Line
Draw a scatter plot of the data. Draw a line that corresponds closely to
the data and write an equation of the line.
-3
8
-2
6
-1
5
0
3
1
2
2
0
1.1
5.1
1.7
5.5
2.2
5.9
2.6
6.3
3.3
7.5
3.5
7.6
3 . Use the equation of the line from Example 1 to predict the foot length of a
student with a forearm length of 27 centimeters.
Exercises
Copy the graph and draw a line that corresponds closely to the data.
Write an equation of your line.
k y
i
1
►
-]
L
}
L
3 X
— V
•
y*
^ •
3
-]
[
]
L x
— l
»•
D
•
<
>
> •
-]
[
L
3 x
— 1
•
Tell whether you could use a best-fitting line to model the data. Explain
your reasoning.
Appendix 3
Appendix 3
Draw a scatter plot of the data. Draw a line that corresponds closely to
the data and write an equation of the line.
1.0
3.8
1.5
4.2
1.7
5.3
2.0
5.8
2.0
5.5
1.5
6.7
3.0
7.1
3.4
8.1
4.0
8.5
4.1
8.9
4.8
9.6
5.2
9.8
3.0
9.9
3.5
9.7
3.7
8.6
4.0
8.1
4.0
8.4
4.5
7.4
5.0
6.8
5.4
5.8
6.0
5.6
6.1
5.2
6.8
4.3
7.2
3.5
In Exercises 14 and 15,
use the following
information.
The median base salary for
players in the National
Football League from 1983
to 1997 is shown in the
scatter plot at the right. In
the scatter plot, y represents
the salary and x represents
the number of years since
1980.
Median Base Salary in Football
14. Find an equation of the line that you think closely fits the data.
15. Use the equation from Exercise 14 to approximate the median base salary
in the year 2010.
In Exercises 16-19, use the following
information.
As people grow older, the size of their
pupils tends to get smaller. The average
diameter (in millimeters) of one person’s
pupils is given in the table.
16. Draw a scatter plot of the day
diameters and another of the night
diameters. Let x represent the
person’s age and let y represent
pupil diameters.
17. Find an equation of the line that closely
fits the day and the night sets of data for pupil diameters.
18. Do the two lines have the same slope? Explain your answer in the context of
the real-life data.
19. Use your equations to approximate the pupil diameters of the person during
the day and at night at age 25.
Sample Pupil Diameters
Age (years)
Day
Night
20
4.7
8.0
30
4.3
7.0
40
3.9
6.0
50
3.5
5.0
60
3.1
4.1
70
2.7
3.2
80
2.3
2.5
Glossary
Q>
absolute value (p. 71) The distance between zero and
the point representing a real number on the number
line. The symbol | a | represents the absolute value of
a number a.
absolute value equation (p. 355) An equation of the
form | ax + b \ — c.
absolute value inequality (p. 361) An inequality that
has one of these forms: | ax + b \ < c, \ ax + b \ < c,
| ax + b | > c, or | ax + b \ > c.
addition property of equality (p. 140) If a = b, then
a + c — b + c.
addition property of inequality (p. 324) If a > b, then
a + c > b + c and if a < b, then a + c < b + c.
algebraic model (p. 36) An expression, equation, or
inequality that uses variables to represent a real-life
situation.
associative property of addition (p. 79) The way three
numbers are grouped when adding does not change
the sum. For any real numbers a , b , and c,
{a + b) + c — a + (b + c).
associative property of multiplication (p. 94) The way
three numbers are grouped when multiplying does not
change the product. For any real numbers a , b , and c,
(ab)c — a(bc).
axiom (p. 740) A rule that is accepted as true without
proof. An axiom is also called a postulate .
axis of symmetry of a parabola (p. 521) The vertical
line passing through the vertex of a parabola or the line
dividing a parabola into two symmetrical parts that are
mirror images of each other.
Q
bar graph (p. 43) A graph that represents a collection
of data by using horizontal or vertical bars whose
lengths allow the data to be compared.
base (p. 9) In exponential notation, the number or
variable that undergoes repeated multiplication.
For example, 4 is the base in the expression 4 6 .
base number of a percent equation (p. 183) The
number that is the basis for comparison in a percent
equation. The number b in the verbal model
44 a is p percent of bT
binomial (p. 569) A polynomial consisting of two terms.
closure property of real number addition (p. 78) The
sum of any two real numbers is again a real number.
closure property of real number multiplication
(p. 93) The product of any two real numbers is again a
real number.
coefficient (p. 107) If a term of an expression consists
of a number multiplied by one or more variables, the
number is the coefficient of the term.
commutative property of addition (p. 79) The order
in which two numbers are added does not change the
sum. For any real numbers a and b, a + b = b + a.
commutative property of multiplication (p. 94) The
order in which two numbers are multiplied does not
change the product. For any real numbers a and b ,
ab — ba.
completing the square (p. 716) The process of
rewriting a quadratic equation so that one side is
a perfect square trinomial.
compound inequality (p. 342) Two inequalities
connected by the word and or the word or.
conclusion (p. 120) The then part of an if-then
statement is called the conclusion.
conjecture (p. 741) A statement that is thought to be
true but has not been proved.
constant function (p. 218) A function of the form
y — b, where b is some number.
constant of variation (pp. 236,639) The constant in a
y
variation model. It is equal to - in the case of direct
variation, and xy in the case of inverse variation.
converse of a statement (p. 726) A related statement in
which the hypothesis and conclusion are interchanged.
The converse of the statement 44 If p, then q ” is 44 If q,
then pT
converse of the Pythagorean theorem (p. 726)
If a triangle has side lengths a , b , and c such that
a 2 + b 2 = c 2 , then the triangle is a right triangle.
coordinate plane (p. 203) The coordinate system
formed by two real number lines that intersect at
a right angle.
c
^y
A
Qu
/
ltl
\
Q
uaurani ii
'X
laurai
+ , +
i
r;
J
(
)
i
r ■
1
x-axis
1
-
7
-5
-3
0
]
L
3
5
7
9x
origin
0,0)-
1
-3
■i £-1
1
Quadrant III
Quadrant IV
<-, ->
(+,
-)
- 5 '
Glossary
GLOSSARY
GLOSSARY
counterexample (p. 73) An example used to show that
a given statement is false.
cross product property (p. 634) In a proportion, the
product of the extremes equals the product of the
means. If j- = then ad — be.
b a
cube root (p. 710) If b 3 = a , then b is a cube root of a.
o
data (p. 42) Information, facts, or numbers used to
describe something.
decay factor (p. 482) The expression 1 — r in the
exponential decay model where r is the decay rate.
See also exponential decay.
decay rate (p. 482) In an exponential decay model, the
proportion by which the quantity decreases each time
period. See also exponential decay.
decimal form (p. 469) A number written with place
values corresponding to powers of ten. For example,
100, 14.2, and 0.007 are in decimal form.
deductive reasoning (p. 120) Using facts, definitions,
rules, or properties to reach a conclusion.
degree of a monomial (p. 568) The sum of the
exponents of each variable in the monomial. The
degree of 5 x 2 y is 2 + 1 = 3.
degree of a polynomial in one variable (p. 569) The
largest exponent of that variable.
direct variation (p. 236) The relationship between two
variables x and y for which there is a nonzero number k
y
such that y = kx, or - = k. The variables x and y
vary directly.
discriminant (p. 540) The expression b 2 — 4 ac where
a , b, and c are coefficients of the quadratic equation
ax 2 + bx + c — 0; the expression inside the radical in
the quadratic formula.
distance formula (p. 730) The distance d between
the points (x v y x ) and (v 2 , y 2 ) is
d = V(x 2 - xy + (y 2 - y,) 2 .
distributive property (pp. 100,101) For any real
numbers a , b , and c , a(b + c) = ab + ac , (b + c)a —
ba +ca , a(b — c) — ab — ac , and (b — c)a — ba — ca.
division property of equality (p. 140) If a = b and
c A 0, then — = —.
c c
division property of inequality (pp. 330,331) If a > b
and c > 0, then — > — and if a < b, then — < —. If a > b,
c c c c
and c < 0, then — < — and if a < b, then — > —.
c c c c
domain of a function (p. 49) The collection of all input
values of a function.
o
equation (p. 24) A statement formed by placing an
equal sign between two expressions.
equivalent equations (p. 132) Equations that have the
same solution(s).
equivalent inequalities (p. 324) Inequalities that have
the same solution(s).
evaluate an expression (p. 4) Find the value of an
expression by substituting a specific numerical value
for each variable, and simplifying the result.
exponent (p. 9) In exponential notation, the number of
times the base is used as a factor. For example, 6 is the
exponent in the expression 4 6 .
exponential decay (p. 482) A quantity displays
exponential decay if it decreases by the same
proportion r in each time period t. If C is the initial
amount, the amount at time t is given by y = C(1 — r)\
where r is called the decay rate, 0 < r < 1, and (1 — r)
is called the decay factor.
exponential function (p. 455) A function of the form
y = ab x , where b > 0 and b A 1.
exponential growth (p. 476) A quantity displays
exponential growth if it increases by the same
proportion r in each unit of time. If C is the initial
amount, the amount after t units of time is given by
y = C(1 + r)\ where r is called the growth rate and
(1 + r) is called the growth factor.
extraneous solution (p. 705) A trial solution that does
not satisfy the original equation.
extremes of a proportion (p. 633) In the proportion
T = 3, a and d are the extremes.
b d
Q
factor a polynomial completely (p. 617) To write a
polynomial as the product of monomial and prime
factors.
factor a trinomial (p. 595) Write the trinomial as the
product of two binomials.
factored form of a polynomial (p. 588) A polynomial
that is written as the product of two or more factors.
formula (p. 171) An algebraic equation that relates two
or more variables.
function (p. 48) A rule that establishes a relationship
between two quantities, the input and the output. There
is exactly one output for each input.
function form (p. 211) A two-variable equation is
written in function form if one of its variables is
isolated on one side of the equation. The isolated
variable is the output and is a function of the input.
Student Resources
function notation (p. 254) A way to describe a function
by means of an equation. For the equation y = f(x ) the
symbol fix) denotes the output and is read as “the value
of/at x” or simply as “/of x.”
graph of an equation in two variables (p. 211) The set
of all points (x, y ) that are solutions of the equation.
graph of an inequality in one variable (p. 323) The set
of points on the number line that represent all the
solutions of the inequality.
x<2
-3 -2-1 0 1 2 3
graph of a number (p. 65) The point on a number line
that corresponds to a number.
graph of a quadratic inequality (p. 547) The graph of
all ordered pairs (x, y) that are solutions of the
inequality.
r
V
y
< -x 2 -
- 5x + 4
/
/
2
/
\
-
2
h
>
X
r T
grouping symbols (p. 10) Symbols such as parentheses
() and brackets [ ] that indicate the order in which
operations should be performed. Operations within the
innermost set of grouping symbols are done first.
growth factor (p. 476) The expression 1 + r in the
exponential growth model where r is the growth rate.
See also exponential growth.
growth rate (p. 476) In an exponential growth model,
the proportion by which the quantity increases each
unit of time.
©
hypotenuse (p. 724) The side opposite the right angle in
a right triangle.
hypothesis (p. 120) The if part of an if-then statement.
o
identity (p. 153) An equation that is true for all values
of the variable.
identity property of addition (p. 79) The sum of a
number and 0 is the number. For any real number a,
a + 0 = 0 + a = a.
identity property of multiplication (p. 94) The
product of a number and 1 is the number. For any real
number a, 1 • a = a.
if-then statement (p. 120) A form of statement used in
deductive reasoning where the if part is the hypothesis
and the then part is the conclusion.
indirect proof (p. 742) A type of proof in which a
statement is assumed false. If this assumption leads to
an impossibility, then the original statement has been
proved to be true.
inductive reasoning (p. 119) Making a general
statement based on several observations.
inequality (p. 26) A statement formed by placing an
inequality symbol, such as <, between two
expressions.
input (p. 48) A value in the domain of a function.
input-output table (p. 48) A table used to describe a
function by listing the outputs for several different
inputs.
integers (p. 65) The numbers ... -3, -2, -1, 0, 1, 2,
3, . . . .
inverse operations (p. 133) Two operations that undo
each other, such as addition and subtraction.
inverse property of addition (p. 79) The sum of a
number and its opposite is 0: a + (—a) = 0.
inverse variation (p. 639) The relationship between two
variables x and y for which there is a nonzero number k
k
such that xy = k, or y = - The variables x and y are
said to vary inversely.
o
leading coefficient (p. 505) For a quadratic equation in
standard form, ax 2 + bx + c — 0 where a ¥= 0, a is the
leading coefficient.
least common denominator, LCD (p. 663) The least
common multiple of the denominators of two or more
fractions.
left-to-right rule (p. 16) When operations have the
same priority, you perform them in order from left
to right.
legs of a right triangle (p. 724) The two sides of a right
triangle that are not opposite the right angle.
like terms (p. 107) Terms that have the same variables
with each variable of the same kind raised to the same
power. For example, 3 x 2 y and ~lx 2 y are like terms.
line graph (p. 44) A graph that uses line segments to
connect data points. Line graphs are especially useful
for showing changes in data over time.
linear combination of two equations (p. 402) An
equation obtained by (1) multiplying one or both
equations by a constant and (2) adding the resulting
equations.
linear equation in one variable (p. 134) An equation in
which the variable appears only to the first power.
Glossary
GLOSSARY
GLOSSARY
linear equation in x and y (p. 210) An equation that
can be written in the form Ax + By — C , where A and
B are not both zero.
linear function of x (p. 254) A function of the form
f{x) — mx + b.
linear inequality in x and y (p. 367) An inequality
that can be written in one of these forms: ax + by < c,
ax + by < c , ax + by > c , or ax + by > c.
linear model (p. 298) A linear equation or function that
is used to model a real-life situation.
linear system (p. 389) Two or more linear equations in
the same variables. This is also called a system of
linear equations.
©
means of a proportion (p. 633) In the proportion
T = b and c are the means.
b a
midpoint of a line segment (p. 736) The point on the
segment that is equidistant from its endpoints.
midpoint formula (p. 736) The midpoint between
(x v y x ) and (x 2 , y 2 ) is
modeling (p. 36) Representing real-life situations by
means of equations or inequalities.
monomial (pp. 568,569) A number, a variable, or a
product of a number and one or more variables with
whole number exponents; a polynomial with only one
term.
multiplication property of equality (p. 140) If a = b,
then ca = cb.
multiplication property of inequality (pp. 330,331)
If a > b and c > 0, then ac > be and if a < b, then
ac < be. If a > b and c < 0, then ac < be and if a < b,
then ac > be.
multiplicative property of negative one (p. 94) The
product of a number and — 1 is the opposite of the
number: — 1 • a — — a .
multiplicative property of zero (p. 94) The product of
a number and 0 is 0. That is, 0 • a = 0.
+*2 Zl± 2ij
Q
negative number (p. 65) A number less than zero.
See also real number line.
negative square root (p. 499) The negative number that
is a square root of a positive number. For example, the
negative square root of 9 is —3.
numerical expression (p. 3) An expression that
represents a particular number.
opposites (p. 71) Two numbers that are the same
distance from zero on a number line but on opposite
sides of zero.
order of operations (p. 15) The rules for evaluating an
expression involving more than one operation.
ordered pair (p. 203) A pair of numbers used to
identify a point in a coordinate plane. The first number
is the v-coordinate and the second number is the
y-coordinate. See also coordinate plane.
origin (p. 203) The point in a coordinate plane where
the horizontal axis intersects the vertical axis. The
point (0, 0). See also coordinate plane.
output (p. 48) A value in the range of a function.
©
parabola (p. 520) The U-shaped graph of a quadratic
function, y = ax 2 + bx + c where a A 0.
o
(-4,0)
Z
(4,0)
-]
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-8
4
l
-2
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8
12 x
x-
intercept
7
x-interce
Pt
>
4
1 2 Zo
V
y
y
“ 2
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axis of symmetry
parallel lines (p. 245) Two different lines in the same
plane that do not intersect. (Identical lines are
sometimes considered to be parallel.)
percent (p. 183) A ratio that compares a number to 100.
perfect square trinomials (p. 609) Trinomials of the
form a 2 + 2 ab + b 2 and a 2 — 2 ab + b 2 ; perfect square
trinomials can be factored as the squares of binomials.
perpendicular lines (p. 306) Two lines in a plane are
perpendicular if they intersect at a right, or 90°, angle.
If two nonvertical lines are perpendicular, the product
of their slopes is —1.
Student Resources
point of intersection (p. 389) A point (< a , b) that lies on
the graphs of two or more equations is a point of
intersection for the graphs.
point-slope form (p. 278) An equation of a nonvertical
line in the form y — y x — m(x — x x ) where the line
passes through a given point (x p y x ) and the line has a
slope of m.
polynomial (p. 569) A monomial or a sum of
monomials. See monomial.
positive number (p. 65) A number greater than zero.
positive square root, or principal square root (p. 499)
The square root of a positive number that is itself
positive. For example, the positive square root of 9 is 3.
postulate (p. 740) A rule that is accepted as true without
proof. A postulate is also called an axiom .
power (p. 9) An expression of the form a b or the value
of such an expression. For example, 2 4 is a power, and
since 2 4 = 16, 16 is the fourth power of 2.
power of a power property (pp. 444,445) To find a
power of a power, multiply the exponents. For any real
number a and integers m and n , ( a m ) n — a mn .
power of a product property (p. 444) To find a power
of a product, find the power of each factor and
multiply. For any real numbers a and b and integer m,
(ab) m = a m • b m .
power of a quotient property (pp. 462,463) To find a
power of a quotient, find the power of the numerator
and the power of the denominator and divide. For any
integer m and real numbers a and b , where b A 0,
prime polynomial (p. 617) A polynomial that is not the
product of factors with integer coefficients and of
lower degree.
product of powers property (pp. 443,445) To multiply
powers having the same base, add the exponents.
For any real number a and integers m and n ,
a m . a n = a m + n .
product property of radicals (p. 511) If a and b
are real numbers such that a > 0 and b > 0, then
Vab = Va • Vb.
properties of equality (p. 140) The rules of algebra
used to transform equations into equivalent equations.
proportion (p. 633) An equation stating that two ratios
are equal.
Pythagorean theorem (p. 724) If a right triangle has
legs of lengths a and b and hypotenuse of length c,
then a 2 + b 2 = c 2 .
b
o
quadrant (p. 204) One of four regions into which the
axes divide a coordinate plane.
quadratic equation (p. 505) An equation that can be
written in the standard form ax 2 + bx + c — 0, where
a A 0.
quadratic formula (p. 533) A formula used to find the
solutions of a quadratic equation ax 2 + bx + c = 0
when a 0 and b 2 — 4 ac > 0:
—b ± Vb 2 — 4ac
X ~ 2 a
quadratic function (p. 520) A function that can be
written in the standard form y = ax 2 + bx + c, where
a A 0.
quadratic inequality (p. 547) An inequality that can be
written in one of the forms
y < ax 2 + bx + c,y < ax 2 + bx + c,
y > ax 2 + bx + c, or y > ax 2 + bx + c.
quotient of powers property (pp. 462,463) To divide
powers having the same base, subtract the exponents.
For any real number a 0 and integers m and n ,
quotient property of radicals (p. 512) If a and b are
real numbers such that a > 0 and b > 0 , then
la _ Va
V b~ Vb'
o
radical expression, or radical (p. 501) An expression
written with a radical symbol.
radicand (p. 499) The number or expression inside a
radical symbol.
range of a function (p. 49) The collection of all output
values of a function.
rate of a per b (p. 177) The relationship of two
quantities a and b.
rate of change (p. 298) The quotient of two different
quantities that are changing. In a linear model, the
slope gives the rate of change of one variable with
respect to the other.
ratio of a to b (p. 177) The relationship ^ of two
quantities a and b.
rational equation (p. 670) An equation that contains
rational expressions.
rational exponent (p. 711) For any integer n and real
number a > 0, the nth root of a is denoted a lln or Wl.
Let a lln be an nth root of a, m be a positive integer and
a > 0. Then a mln = (, a l/n ) m = (Vaf = <fcT.
Glossary
GLOSSARY
GLOSSARY
rational expression (p. 646) A fraction whose
numerator and denominator are nonzero polynomials.
rational function (p. 678) A rational function is a
function that is a quotient of polynomials.
rational number (p. 646) A number that can be written
as the quotient of two integers.
real number line (p. 65) A line whose points
correspond to the real numbers.
Negative numbers Positive numbers
- 4 - 3 - 2-101234
real numbers (p. 65) The set of numbers consisting of
the positive numbers, the negative numbers, and zero.
(The real numbers can also be thought of as the set of
all decimals, finite or infinite in length.)
reciprocals (p. 113) Two numbers are reciprocals if
their product is 1. If t is a nonzero number, then its
reciprocal is K
relation (p. 252) Any set of ordered pairs.
roots of a quadratic equation (p. 527) The solutions of
a quadratic equation.
rounding error (p. 164) The error produced when a
decimal expansion is limited to a specified number of
digits to the right of the decimal point.
scatter plot (p. 205) A coordinate graph containing
points that represent a set of ordered points; used to
analyze relationships between two real-life quantities.
scientific notation (p. 469) A number expressed in the
form cX 10”, where 1 < c < 10 and n is an integer.
simplified expression (p. 108) An expression is
simplified if it has no grouping symbols and if all the
like terms have been combined.
simplest form of a radical expression (p. 511) An
expression that has no perfect square factors other than
1 in the radicand, no fractions in the radicand, and no
radicals in the denominator of a fraction.
slope (p. 229,230) The ratio of the vertical rise to the
horizontal run between any two points on a line.
, . o^-*)
The slope is m = —3—r.
v^2
slope-intercept form (p. 243) A linear equation written
in the form y — mx + b. The slope of the line is m. The
y-intercept is b. See also slope and y-intercept.
y = 2x + 3
Slope is 2.
y-intercept is 3.
solution of an equation or inequality (p. 24)
A number that, when substituted for the variable in
an equation or inequality, results in a true statement.
solution of an equation in two variables (p. 210) An
ordered pair (v, y) that makes the equation true.
solution of a linear system in two variables (p. 389)
An ordered pair (v, y) that makes each equation in the
system a true statement.
solution of a system of linear inequalities in two
variables (p. 424) An ordered pair that is a solution of
each inequality in the system.
square root (p. 499) If b 2 = a , then b is a square root
of a. Square roots can be written with a radical
symbol, \l~.
square root function (p. 692) The function defined by
the equation y = Vx, for v > 0.
standard form of an equation of a line (p. 291)
A linear equation of the form Ax + By — C, where A
and B are not both zero.
standard form of a polynomial in one variable
(p. 569) A polynomial whose terms are written in
decreasing order, from largest exponent to smallest
exponent.
standard form of a quadratic equation (p. 505) An
equation in the form ax 2 + bx + c — 0 , where a A 0 .
subtraction property of equality (p. 140) If a = b,
then a — c — b — c.
subtraction property of inequality (p. 324) If a > b,
then a — c > b — c and if a < b, then a — c <b — c.
system of linear equations (p. 389) Two or more linear
equations in the same variables. This is also called a
linear system.
system of linear inequalities (p. 424) Two or more
linear inequalities in the same variables. This is also
called a system of inequalities.
Q
terms of an expression (p. 87) The parts that are added
to form an expression. For example, in the expression
5 — x, the terms are 5 and —x.
theorem (p. 724) A statement that has been proven to
be true.
Student Resources
trinomial (p. 569) A polynomial of three terms.
unit analysis (p. 178) Using the units for each variable
in a real-life problem to determine the units for the
answer.
unit rate (p. 177) A rate expressing the amount of one
given quantity per unit of another quantity, such as
miles per gallon.
values (p. 3) The numbers a variable represents.
variable (p. 3) A letter used to represent a range of
numbers.
variable expression (p. 3) A symbolic form made up
of constants, variables, and operations.
verbal model (p. 36) An expression that uses words to
describe a real-life situation.
vertex of a vertically oriented parabola (p. 521) The
lowest point on the graph of a parabola opening up or
the highest point on the graph of a parabola opening
down. See also parabola.
vertical motion models (p. 535) Models that give the
height of an object as a function of time. They include
the case of a falling object.
whole numbers (p. 65) The positive integers together
with zero.
Jt-axis (p. 203) The horizontal axis in a coordinate
plane. See also coordinate plane.
x-coordinate (p. 203) The first number in an ordered
pair. See also ordered pair.
jc-intercept (p. 222) The v-coordinate of a point where a
graph crosses the v-axis.
o
y-axis (p. 203) The vertical axis in a coordinate plane.
See also coordinate plane.
y-coordinate (p. 203) The second number in an ordered
pair. See also ordered pair.
y-intercept (p. 222) The y-coordinate of a point where a
graph crosses the y-axis.
o
zero-product property (p. 588) If the product of
two factors is zero, then at least one of the factors
must be zero.
Glossary
GLOSSARY
ENfiLISH-TO-SPANISH GLOSSARY
English-to-Spanish Glossary
O
absolute value (p. 71) valor absoluto Distancia
existente entre el cero y el punto que representa en la
recta numerica un numero real. El simbolo | a \
representa el valor absoluto de un numero a .
absolute value equation (p. 355) ecuacion de valor
absoluto La de la forma I ax + b I = c.
absolute value inequality (p. 361) desigualdad de
valor absoluto Aquella que presenta una de estas
formas: | ax + b \ < c, | ax + b | < c, | ax + b \ > c,
6 I ax + b I > c.
addition property of equality (p. 140) propiedad de
igualdad en la suma Si a — b, entonces a + c = b + c.
addition property of inequality (p. 324) propiedad
de desigualdad en la suma Si a > b, entonces a + c
>b + cy si a<b, entonces a + c < b + c.
algebraic model (p. 36) modelo algebraico
Expresion, ecuacion o desigualdad que usa variables
para representar una situation de la vida real.
associative property of addition (p. 79) propiedad
asociativa de la suma La agrupacion que tengan tres
numeros al sumarse no altera la suma. Para todos los
numeros reales a, b, y c, (a + b) + c — a + (b + c ).
associative property of multiplication (p. 94)
propiedad asociativa de la multiplicacion La
agrupacion que tengan tres numeros al multiplicarse
no altera el producto. Para todos los numeros reales a,
b , y c , ( ab)c — a(bc).
axiom (p. 740) axioma Regia que se acepta como
cierta sin demostracion. Al axioma se le llama tambien
postulado.
axis of symmetry of a parabola (p. 521) eje de
simetria de una parabola Recta vertical que pasa por
el vertice de una parabola o la recta que divide la
parabola en dos partes simetricas, las cuales son
reflejos exactos entre si.
Q
bar graph (p. 43) grafica de barras La que
representa un conjunto de datos mediante barras
horizontales o verticales y cuya longitud permite la
comparacion de esos datos.
base (p. 9) base En notacion exponencial, el numero
o variable que sostiene multiplicacion repetida. Por
ejemplo, 4 es la base en la expresion 4 6 .
base number of a percent equation (p. 183) numero
base de una ecuacion de porcentajes El numero de
una ecuacion de porcentajes que es la base de una
comparacion. El numero b en el modelo verbal “a es
el p por ciento de V\
Student Resources
binomial (p. 569) binomio Polinomio que consiste de
dos terminos.
Q
closure property of real number addition (p. 78)
propiedad de cierre de la suma de numeros reales
La suma de dos numeros reales cualesquiera es otra
vez un numero real.
closure property of real number multiplication
(p. 93) propiedad de cierre de la multiplicacion de
numeros reales El producto de dos numeros reales
cualesquiera es otra vez un numero real.
coefficient (p. 107) coeficiente Si un termino de una
expresion consta de un numero multiplicado por una
o mas variables, entonces ese numero es el coeficiente
del termino.
commutative property of addition (p. 79) propiedad
conmutativa de la suma El orden de dos numeros al
sumarse no altera la suma. Para todos los numeros
reales ay b, a + b = b + a.
commutative property of multiplication (p. 94)
propiedad conmutativa de la multiplicacion El
orden de dos numeros al multiplicarse no altera el
producto. Para todos los numeros reales ay b,
ab — ba.
completing the square (p. 716) completar cuadrados
Proceso de escribir una ecuacion cuadratica de manera
que uno de sus miembros sea un trinomio cuadrado
perfecto.
compound inequality (p. 342) desigualdad
compuesta Dos desigualdades unidas entre si
mediante la palabra y wo.
conclusion (p. 120) conclusion La parte del entonces
en un enunciado de si-entonces.
conjecture (p. 741) conjetura Enunciado que se
considera probable sin que hay a sido demostrado.
constant function (p. 218) funcion constante Lade
la forma y — b, donde b es un numero.
constant of variation (pp. 236,639) constante de
variacion Constante de un modelo de variacion. Es
y
equivalente a - en el caso de una variacion directa y a
xy en el caso de una variacion inversa.
converse of a statement (p. 726) reciproco de un
enunciado Afirmacion relacionada en la que se
intercambian la hipotesis y la conclusion. El reciproco
del enunciado “Si p , entonces cf es “Si q , entonces p”.
converse of the Pythagorean theorem (p. 726)
reciproco del teorema de Pitagoras Si un triangulo
tiene lados de longitudes a,b,yc tales que
a 2 + b 2 = c 2 , entonces es un triangulo rectangulo.
coordinate plane (p. 203) piano de coordenadas El
sistema de coordenadas formado por dos rectas
numericas reales que al cortarse configuran un angulo
recto.
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Cuadrante III
Cuadrante IV
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counterexample (p. 73) contraejemplo Ejemplo que
sirve para mostrar la falsedad de un enunciado dado.
cross product property (p. 634) propiedad de los
productos cruzados En una proporcion, el producto
de los extremos es igual al de los medios. Si
entonces ad — be.
cube root (p. 710) raiz cubica Si b 3 = a , entonces b
es una raiz cubica de a.
data (p. 42) datos Informaciones, hechos o numeros
que sirven para describir algo.
decay factor (p. 482) factor de decrecimiento
La expresion 1 — r, en el modelo de decrecimiento
exponencial donde r es la tasa de decrecimiento.
Ver tambien decrecimiento exponencial.
decay rate (p. 482) tasa de decrecimiento La
proporcion de un modelo de decrecimiento
exponencial en la cual disminuye la cantidad durante
cada periodo de tiempo. Ver tambien decrecimiento
exponencial.
decimal form (p. 469) forma decimal Numero escrito
con valores relativos que corresponden a potencias de
diez. Por ejemplo, 100, 14.2 y 0.007 estan expresados
en forma decimal.
deductive reasoning (p. 120) razonamiento
deductivo Empleo de hechos, definiciones, reglas o
propiedades para sacar una conclusion.
degree of a monomial (p. 568) grado de un monomio
Suma de los exponentes de cada una de las variables
del monomio. El grado de 5 x 2 y es 2 + 1 =3.
degree of a polynomial in one variable (p. 569)
grado de un polinomio de una variable Mayor
exponente de esa variable.
direct variation (p. 236) variacion directa Relation
entre dos variables v e y para la cual hay un numero k
y
distinto a cero tal que y = kx, 6 - = k. Las variables
v e y varian directamente entre si.
discriminant (p. 540) discriminante La expresion
b 2 — 4 ac donde a,byc son coeficientes de laecuacion
cuadratica ax 2 + bx + c = 0; la expresion del radical
de la formula cuadratica.
distance formula (p. 730) formula de la distancia
La distancia d que hay entre los puntos (xq, qq) y
(x 2 , v 2 ) es d = V(x 2 - X x ) 2 + (y 2 - y x ) 2 .
distributive property (pp. 100,101) propiedad
distributiva Para todos los numeros reales a,byc,
a(b + c) — ab + ac , (b + c)a — ba + ca ,
a(b — c) — ab — ac y (b — c)a - ba — ca.
division property of equality (p. 140) propiedad de
igualdad en la division Si a = b y c A 0,
a b
entonces — = —.
c c
division property of inequality (pp. 330,331) propiedad
de desigualdad en la division Si a > b y c > 0,
entonces — > — y si a < b, entonces — < —. Si a > b,
cc , c c j
~ a b , a b
y c < 0, entonces — < — y si a < b, entonces — > —.
J c c J c c
domain of a function (p. 49) dominio de una funcion
Conjunto de todos los valores de entrada de una
funcion.
Q
equation (p. 24) ecuacion Enunciado formado por dos
expresiones unidas entre si mediante el signo de igual.
equivalent equations (p. 132) ecuaciones equivalentes
Las que tienen la misma solucion o soluciones.
equivalent inequalities (p. 324) desigualdades
equivalentes Aquellas que tienen la misma solucion o
soluciones.
evaluate an expression (p. 4) evaluar una expresion
Hallar el valor de una expresion mediante la
sustitucion de cada variable por un valor numerico
especifico y la simplificacion del resultado.
exponent (p. 9) exponente En notacion exponencial,
el numero de veces que la base se usa como factor. Por
ejemplo, 6 es el exponente en la expresion 4 6 .
exponential decay (p. 482) decrecimiento exponencial
Una cantidad presenta un decrecimiento exponencial
cuando disminuye en una misma proporcion r durante
cada periodo de tiempo t. Si C es la cantidad inicial, la
existente tras transcurrir el tiempo t viene dada por
y = C(1 — rY, donde r es la tasa de decrecimiento,
0<r<l,y(l — r) el factor de decrecimiento.
English-to-Spanish Glossary
ENGLISH-TO-SPANISH GLOSSARY
ENfiLISH-TO-SPANISH GLOSSARY
exponential function (p. 455) funcion exponencial
La de la forma y = ab x , donde > 0 y b A 1.
exponential growth (p. 476) crecimiento exponencial
Una cantidad presenta un crecimiento exponencial
cuando aumenta en una misma proporcion r durante
cada unidad de tiempo. Si C es la cantidad initial, la
existente despues de t unidades de tiempo viene dada
por y = C(1 + rf, donde r es la tasa de crecimiento y
(1 + r) el factor de crecimiento.
extraneous solution (p. 705) solucion extrana
Solucion de prueba que no satisface la ecuacion
original.
extremes of a proportion (p. 633) extremos de una
q r q
proporcion En la proporcion — = —, a y d son los
extremos.
■ o
factor a polynomial completely (p. 617) descomponer
un polinomio en todos sus factores Escribir un
polinomio como producto de factores monomicos y
primos.
factor a trinomial (p. 595) descomponer un trinomio
en factores Escribir el trinomio como producto de dos
binomios.
factored form of a polynomial (p. 588) forma
factorial de un polinomio Polinomio escrito como
producto de dos o mas factores.
formula (p. 171) formula Ecuacion algebraica que
relaciona dos o mas variables.
function (p. 48) funcion Regia que establece una
relation entre dos cantidades: la de entrada y la de
salida. A cada entrada le corresponde una sola salida.
function form (p. 211) forma de funcion Una
ecuacion de dos variables esta expresada en forma de
funcion si una de sus variables esta aislada en un
miembro de la ecuacion. La variable aislada es la
salida que ademas esta en funcion de la entrada.
function notation (p. 254) notation de funcion
Forma de describir una funcion por medio de una
ecuacion. Para la ecuacion y = /(x), el simbolo/(x)
indica la salida y se lee “el valor de/en x” o
simplemente “/de x”.
graph of an equation in two variables (p. 211)
grafica de una ecuacion de dos variables
Conjunto de todos los puntos (x, y) que son soluciones
de la ecuacion.
graph of an inequality in one variable (p. 323)
representacion grafica de una desigualdad de una
variable Conjunto de puntos de la recta numerica que
representan todas las soluciones de la desigualdad.
x< 2
- 3 - 2-10123
graph of a number (p. 65) representacion grafica de
un numero Punto situado en una recta numerica que
corresponde a un numero.
graph of a quadratic inequality (p. 547) grafica de
una desigualdad cuadratica Grafica de todos los
pares ordenados (x, y) que son soluciones de la
desigualdad.
A
y<-
x z - 5x + -
4
i
/
:
/
A
\
2
\2
X
grouping symbols (p. 10) signos de agrupacion
Signos como los parentesis () o los corchetes [ ]
que indican el orden en que deben realizarse las
operaciones. Se efectuan primero las operaciones de
los signos de agrupacion situados mas en el interior.
growth factor (p. 476) factor de crecimiento
La expresion 1 + r, en el modelo de crecimiento
exponencial donde r es la tasa de crecimiento.
Ver tambien crecimiento exponencial.
growth rate (p. 476) tasa de crecimiento La
proporcion de un modelo de crecimiento exponencial
en la cual aumenta la cantidad durante cada unidad de
tiempo.
Q
hypotenuse (p. 724) hipotenusa El lado opuesto al
angulo recto de un triangulo rectangulo.
hypothesis (p. 120) hipotesis La parte del si en un
enunciado de si-entonces.
o
identity (p. 153) identidad Ecuacion que es cierta para
todos los valores de la variable.
identity property of addition (p. 79) propiedad de
identidad de la suma La suma de un numero y 0
es igual a ese numero. Para todo numero real a ,
a + 0 = 0 + a = cl
Student Resources
identity property of multiplication (p. 94) propiedad
de identidad de la multiplication El producto de un
numero y 1 es igual a ese numero. Para todo numero
real a, 1 • a = a.
if-then statement (p. 120) enunciado de si-entonces
Tipo de enunciado que se emplea en el razonamiento
deductivo y en el cual la parte del si es la hipotesis y la
parte del entonces la conclusion.
indirect proof (p. 742) prueba indirecta Tipo de
pruebas en que se supone que el enunciado es falso.
Si mediante esa suposicion se da una imposibilidad,
entonces la certeza del enunciado original queda
demostrada.
inductive reasoning (p. 119) razonamiento inductivo
Formulation de un enunciado general basandose en
varias observaciones.
inequality (p. 26) desigualdad Enunciado compuesto
de dos expresiones unidas entre si mediante un signo
de desigual como <.
input (p. 48) entrada Un valor en el dominio de una
funcion.
input-output table (p. 48) tabla de entradas y salidas
La que describe una funcion mediante la presentation
de las salidas correspondientes a varias entradas
diferentes.
integers (p. 65) numeros enteros Los numeros . . .
-3, -2,-1,0, 1,2, 3,....
inverse operations (p. 133) operaciones inversas Dos
operaciones que se anulan mutuamente como son la
suma y la resta.
inverse property of addition (p. 79) propiedad del
elemento inverso de la suma La suma de un numero
y su opuesto es igual a 0: a + (— a ) = 0.
inverse variation (p. 639) variation inversa La
relation entre dos variables x e y para la cual hay un
k
numero k distinto a cero tal que xy = k 6 y — —.
Se dice que las variables x e y vanan inversamente
entre si.
o
leading coefficient (p. 505) coeficiente dominante
En una ecuacion cuadratica expresada en forma
normal, ax 2 + bx + c — 0, donde a ¥= 0, a es el
coeficiente dominante.
least common denominator, LCD (p. 663) minimo
comun denominador, mcd El menor de los multiplos
comunes a los denominadores de dos o mas fracciones.
left-to-right rule (p. 16) regia de izquierda a derecha
Las operaciones de igual prioridad se efectuan de
izquierda a derecha.
legs of a right triangle (p. 724) catetos de un
triangulo rectangulo Los dos lados de un triangulo
rectangulo que no estan opuestos al angulo recto.
like terms (p. 107) terminos semejantes Aquellos que
tienen iguales variables y en los que cada una de estas
esta elevada a igual potencia. Por ejemplo, 3 x 2 y y
-7x 2 y son terminos semejantes.
line graph (p. 44) grafica lineal La que utiliza
segmentos de recta para unir puntos de datos. Es de
mucha utilidad para indicar los cambios producidos en
los datos a lo largo del tiempo.
linear combination of two equations (p. 402)
combination lineal de dos ecuaciones Ecuacion
obtenida (1) al multiplicar una o ambas ecuaciones por
una constante y (2) al sumar las ecuaciones resultantes.
linear equation in one variable (p. 134) ecuacion
lineal con una variable Una ecuacion en que la
variable viene elevada solo a la primera potencia.
linear equation in x andy (p. 210) ecuacion lineal
con x ey La que puede escribirse en la forma
Ax + By = C, donde A y B no son ambos cero.
linear function of x (p. 254) funcion lineal de X
Funcion de la forma/(x) = mx + b.
linear inequality in x andy (p. 367) desigualdad
lineal con x ey La que puede escribirse en una de
estas formas: ax A by < c , ax + by < c , ax + by > c, 6
ax A by > c.
linear model (p. 298) modelo lineal Una ecuacion o
funcion lineal que sirve para representar una situation
de la vida real.
linear system (p. 389) sistema lineal Dos o mas
ecuaciones lineales con las mismas variables. Se le
denomina tambien sistema de ecuaciones lineales.
means of a proportion (p. 633) medios de una
a Cl c
proportion En la proportion — = —, b y c son los
medios.
midpoint of a line segment (p. 736) punto medio de
un segmento de recta El punto del segmento que es
equidistante de los extremos.
English-to-Spanish Glossary
ENGLISH-TO-SPANISH GLOSSARY
ENfiLISH-TO-SPANISH GLOSSARY
midpoint formula (p. 736) formula del punto medio
El punto medio entre (xq, >q) y (x 2 , y 2 ) es
f x i +x 2 y i + 3^
\ 2 ’ 2 /
modeling (p. 36) hacer un modelo La representation
de situaciones de la vida real por ecuaciones o
desigualdades.
monomial (pp. 568,569) monomio Numero, variable o
producto de un numero y una o mas variables con
exponentes que sean enteros positivos o cero;
polinomio de un solo termino.
multiplication property of equality (p. 140)
propiedad de igualdad en la multiplicacion Si
a = b, entonces ca = cb.
multiplication property of inequality (pp. 330,331)
propiedad de desigualdad en la multiplicacion
Si a > b y c > 0, entonces ac > be y si a < b, entonces
ac < be. Si a > b y c < 0, entonces ac < be y si a < b,
entonces ac > be.
multiplicative property of negative one (p. 94)
propiedad multiplicativa del uno negativo El
producto de un numero y — 1 es igual al opuesto de ese
numero: — 1 • a — —a.
multiplicative property of zero (p. 94) propiedad
multiplicativa del cero El producto de un numero y 0
es igual a 0. Es decir, 0 • a = 0.
negative number (p. 65) numero negativo Numero
menor que cero. Ver tambien recta numerica real.
negative square root (p. 499) raiz cuadrada negativa
Numero negativo que es una raiz cuadrada de un
numero positivo. Por ejemplo, la raiz cuadrada
negativa de 9 es —3.
numerical expression (p. 3) expresion numerica La
que representa un numero determinado.
opposites (p. 71) opuestos Dos numeros situados a
igual distancia del cero en una recta numerica pero en
lados opuestos del mismo.
order of operations (p. 15) orden de las operaciones
Reglas para evaluar una expresion relacionada con mas
de una operation.
ordered pair (p. 203) par ordenado Par de numeros
empleados para identificar un punto situado en un
piano de coordenadas. El primer numero es la
coordenada v y el segundo la coordenada y. Ver
tambien piano de coordenadas.
origin (p. 203) origen Punto de un piano de
coordenadas donde el eje horizontal corta al vertical.
El punto (0, 0). Ver tambien piano de coordenadas.
output (p. 48) salida Un valor en el recorrido de una
funcion.
o
parabola (p. 520) parabola Grafica en forma de U de
una funcion cuadratica, y = ax 2 + bx + c donde
a A 0.
eje de simetna
parallel lines (p. 245) rectas paralelas Dos rectas
diferentes del mismo piano que no se cortan. (A veces
se consideran paralelas las rectas identicas.)
N
V
k
percent (p. 183) porcentaje Razon que relaciona un
numero con 100.
perfect square trinomials (p. 609) trinomios
cuadrados perfectos Los de la forma a 2 + lab + b 2
y a 2 — lab + b 2 ; este tipo de trinomios pueden
descomponerse en factores como cuadrados de
binomios.
perpendicular lines (p. 306) rectas perpendiculares
Dos rectas situadas en un piano son perpendiculares si
al cortarse forman un angulo recto, o sea de 90°. Si dos
rectas no verticales son perpendiculares, el producto de
sus pendientes es — 1.
point of intersection (p. 389) punto de interseccion
Un punto (a, b) situado en las graficas de dos o mas
ecuaciones es un punto de interseccion de esas
graficas.
point-slope form (p. 278) ecuacion punto pendiente
de una recta Ecuacion de una recta no vertical de la
forma y — y l = m(x — x{), donde la recta pasa por un
punto dado (xq, )q) y la recta tiene pendiente m.
Student Resources
polynomial (p. 569) polinomio Monomio o suma de
monomios. Ver monomio.
positive number (p. 65) numero positivo Numero
mayor que cero.
positive square root, or principal square root (p. 499)
raiz cuadrada positiva, o raiz cuadrada principal
Raiz cuadrada de un numero positivo que resulta
tambien positiva. Por ejemplo, la raiz cuadrada positiva
de 9 es 3.
postulate (p. 740) postulado Regia que se acepta
como cierta sin demostracion. A1 postulado se le llama
tambien axioma.
power (p. 9) potencia Expresion de la forma a b o
valor de ese tipo de expresiones. Por ejemplo, 2 4 es
una potencia, y como 2 4 = 16, 16 es la cuarta potencia
de 2.
power of a power property (pp. 444,445) propiedad
de la potencia de una potencia Para hallar una
potencia de otra se multiplican los exponentes. Para
todo numero real a y para los numeros enteros my n,
(, a m ) n = a mn .
power of a product property (p. 444) propiedad de
la potencia de un producto Para hallar la potencia de
un producto se halla la potencia de cada factor y se
multiplica. Para todos los numeros reales ay by para
el numero entero m, (< ab) m — a m • b m .
power of a quotient property (pp. 462,463) propiedad
de la potencia de un cociente Para hallar la potencia
de un cociente se halla la potencia del numerador y la
del denominador y se divide. Para todo numero entero
m y todos los numeros reales ay b, donde b A 0,
prime polynomial (p. 617) polinomio primo El que
no es el producto de factores con coeficientes de
numero entero y de grado menor.
product of powers property (pp. 443,445) propiedad
del producto de potencias Para multiplicar potencias
de igual base se suman los exponentes. Para todo
numero real a y para los numeros enteros my n,
a m . a n = a m + n .
product property of radicals (p. 511) propiedad del
producto de radicales Si a y b son numeros reales
tales que a > 0 y b > 0, entonces Vab = Va • Vb.
properties of equality (p. 140) propiedades de
igualdad Reglas de algebra que sirven para
transformar ecuaciones en otras equivalentes.
proportion (p. 633) proporcion Ecuacion
estableciendo la igualdad de dos razones.
Pythagorean theorem (p. 724) teorema de Pitagoras
Si un triangulo rectangulo tiene catetos de longitudes
a y b y la hipotenusa de longitud c, entonces
a 2 + b 2 = c 2 .
quadrant (p. 204) cuadrante Una de las cuatro
regiones en que los ejes dividen al piano de
coordenadas. Ver tambien piano de coordenadas.
quadratic equation (p. 505) ecuacion cuadratica
La que puede escribirse en la forma normal
ax 2 + bx + c = 0, donde a A 0.
quadratic formula (p. 533) formula cuadratica
Aquella que sirve para hallar las soluciones de una
ecuacion cuadratica ax 2 + bx + c = 0 cuan do
a A 0 y b 2 - 4 ac > 0: x — —
J 2 a
quadratic function (p. 520) funcion cuadratica
La que puede escribirse en la forma normal
y = ax 2 + bx + c, donde a A 0.
quadratic inequality (p. 547) desigualdad cuadratica
Aquella que puede escribirse de una de estas formas:
y < ax 2 + bx + c,y < ax 2 + bx + c,
y > ax 2 + bx + c,oy> ax 2 + bx + c.
quotient of powers property (pp. 462,463) propiedad
del cociente de potencias Para dividir potencias de
igual base se restan los exponentes. Para todo numero
real a A 0 y para los numeros enteros my n.
quotient property of radicals (p. 512) propiedad del
cociente de radicales Si a y b son numeros reales
tales que a > 0 y b > 0, entonces
Q
radical expression, or radical (p. 501) expresion
radical, o radical Expresion escrita con el signo
radical.
radicand (p. 499) radicando Numero o expresion que
aparece debajo del signo radical.
range of a function (p. 49) recorrido de una funcion
Conjunto de todos los valores de salida de una funcion.
fa_ _ Vfl
- Vb'
English-to-Spanish Glossary
ENGLISH-TO-SPANISH GLOSSARY
ENfiLISH-TO-SPANISH GLOSSARY
rate of a per b (p. 177) relacion de a por b Relation
^ de dos cantidades ay b que se miden con unidades
diferentes.
rate of change (p. 298) tasa de variacion Cociente de
dos cantidades diferentes que cambian. En un modelo
lineal, la pendiente indica la tasa de variacion de una
variable con respecto a la otra.
ratio of a to (p. 177) razondeaa# Relation^
de dos cantidades ay b.
rational equation (p. 670) ecuacion racional Aquella
que contiene expresiones racionales.
rational exponent (p. 711) exponente racional Para
todo numero entero n y para el numero real a > 0,
la raiz enesima de a es denotada por a l/n 6 'Wa.
Sea a 1/n una raiz enesima de a , m un numero entero
positivo y a > 0. Entonces
a m,n = ( a l/n ) m = (' Va) m = ^fa m .
rational expression (p. 646) expresion racional
Fraccion que tiene por numerador y denominador
polinomios distintos a cero.
rational function (p. 678) funcion racional Funcion
que es el cociente de polinomios.
rational number (p. 646) numero racional El que
puede escribirse como cociente de dos numeros
enteros.
real number line (p. 65) recta numerica real Recta
cuyos puntos corresponden a los numeros reales.
Numeros negativos Numeros positivos
I-1-1-1-1-1-1-1-1—►
- 4 - 3 - 2-101234
real numbers (p. 65) numeros reales Conjunto de
numeros compuesto por los positivos, los negativos y
cero. (Se puede considerar que los numeros reales son
el conjunto de todos los decimales finitos o infinitos.)
reciprocals (p. 113) reciprocos Dos numeros cuyo
producto es 1. Si ^ es un numero distinto a cero,
entonces su reciproco es K
relation (p. 252) relacion Conjunto cualquiera de
pares ordenados.
roots of a quadratic equation (p. 527) raices de una
ecuacion cuadratica Soluciones de una ecuacion
cuadratica.
rounding error (p. 164) error de redondeo El
producido tras limitar la expansion de un decimal a un
numero especifico de enteros a la derecha del punto
decimal.
o
scatter plot (p. 205) diagrama de dispersion Grafica
de coordenadas cuyos puntos representan un conjuto
de pares ordenados; es de utilidad para analizar las
relaciones entre dos cantidades reales.
scientific notation (p. 469) notation cientifica
Numero expresado en la forma c X I0 n , donde
1 <c< lOy/iesun numero entero.
simplified expression (p. 108) expresion simplificada
Aquella que carece de signos de agrupacion y tiene
combinados todos los terminos semejantes.
simplest form of a radical expression (p. 511)
expresion radical en su minima expresion La que
no tiene en el radicando factores de raiz exacta
distintos a 1 ni fracciones, ademas de no tener
radicales en el denominador de una fraccion.
slope (p. 229,230) pendiente Razon de la distancia
vertical a la distancia horizontal existente entre dos
puntos cualesquiera de una recta. La pendiente es
(3^2 “ 3h)
slope-intercept form (p. 243) ecuacion pendiente
interception de una recta Ecuacion lineal escrita en
la forma y = mx + b. La pendiente de la recta es m y
la interception en y es b. Ver tambien pendiente e
interception en y.
y = 2x + 3
La pendiente es
La interception
eny es 3.
solution of an equation or inequality (p. 24) solution
de una ecuacion o desigualdad Numero que cumple
una ecuacion o desigualdad al sustituir a la variable de
la misma.
solution of an equation in two variables (p. 210)
solution de una ecuacion de dos variables Par
ordenado (v, y) que cumple la ecuacion.
solution of a linear system in two variables (p. 389)
solution de un sistema lineal de dos variables Par
ordenado (x, y) que satisface cada ecuacion del
sistema.
Student Resources
solution of a system of linear inequalities in two
variables (p. 424) solucion de un sistema de
desigualdades lineales de dos variables Par
ordenado que cumple cada desigualdad del sistema.
square root (p. 499) raiz cuadrada Si b 2 = a,
entonces b es una raiz cuadrada de a . Las raices
cuadradas pueden escribirse con el signo radical, .
square root function (p. 692) funcion de raiz cuadrada
La definida por la ecuacion y = Vx, para v > 0.
standard form of an equation of a line (p. 291)
forma usual de la ecuacion de una recta Ecuacion
lineal de la forma Ax + By — C, donde A y B no son
ambos cero.
standard form of a polynomial in one variable
(p. 569) forma usual de un polinomio de una variable
Polinomio cuyos terminos estan escritos en orden
descendente, del exponente mayor al menor.
standard form of a quadratic equation (p. 505)
forma usual de una ecuacion cuadratica Ecuacion
de la forma ax 2 + bx + c = 0, donde a A 0.
subtraction property of equality (p. 140)
propiedad de igualdad en la resta Si a = b,
entonces a — c = b — c.
subtraction property of inequality (p. 324)
propiedad de desigualdad en la resta Si a > b,
entonces a — c> b — c, y sia<b, entonces
a — c < b — c.
system of linear equations (p. 389) sistema de
ecuaciones lineales Dos o mas ecuaciones lineales
que tienen las mismas variables. Se le llama tambien
sistema lineal.
system of linear inequalities (p. 424) sistema de
desigualdades lineales Dos o mas desigualdades
lineales que tienen las mismas variables. Se le llama
tambien sistema de desigualdades.
o
terms of an expression (p. 87) terminos de una
expresion Partes que se unen para formar una
expresion. Por ejemplo, en la expresion 5 — x, los
terminos son 5 y —x.
theorem (p. 724) teorema Afirmacion cuya certeza ha
sido demostrada.
trinomial (p. 569) trinomio Polinomio de tres
terminos.
unit rate (p. 177) tasaunitaria Relation que expresa
la magnitud de una cantidad dada por unidad de otra
cantidad como, por ejemplo, millas por galon.
o
values (p. 3) valores Numeros que representa una
variable.
variable (p. 3) variable Letra empleada para
representar una gama de numeros.
variable expression (p. 3) expresion algebraica
Forma simbolica compuesta por constantes, variables y
operaciones.
verbal model (p. 36) modelo verbal Expresion que
emplea palabras para describir una situation de la vida
real.
vertex of a vertically oriented parabola (p. 521)
vertice de una parabola orientada verticalmente
Punto inferior de la grafica de una parabola que abre
hacia arriba o punto superior de la grafica de una
parabola que abre hacia abajo. Ver tambien parabola.
vertical motion models (p. 535) modelos de
movimiento vertical Aquellos que dan la altura de un
objeto como una funcion del tiempo. Incluyen el caso
de un objeto que cae.
©
whole numbers (p. 65) numeros naturales Numeros
enteros positivos y cero.
o
x-axis (p. 203) eje de las X Eje horizontal de un piano
de coordenadas. Ver tambien piano de coordenadas.
x-coordinate (p. 203) coordenadax Primer numero
de un par ordenado. Ver tambien par ordenado.
x-intercept (p. 222) intercepcion en x Coordenada x
de un punto donde una grafica cruza al eje de las x.
o
y-axis (p. 203) eje de lasy Eje vertical de un piano de
coordenadas. Ver tambien piano de coordenadas.
y-coordinate (p. 203) coordenada y Segundo numero
de un par ordenado. Ver tambien par ordenado.
y-intercept (p. 222) intercepcion eny Coordenada y
de un punto donde una grafica cruza al eje de las y.
unit analysis (p. 178) analisis por unidades Usar las
unidades de cada variable de un problema real para asi
determinar las unidades de la solucion.
o
zero-product property (p. 588) propiedad del
producto cero Si el producto de dos factores es cero,
entonces al menos uno de ellos debe ser cero.
English-to-Spanish Glossary
ENGLISH-TO-SPANISH GLOSSARY
Credits
Cover Photography
Ralph Mercer
Photography
i, ii Ralph Mercer; iii RMIP/Richard Haynes (all); iv Kevin Horan/Tony
Stone Images; v Stuart Westmorland/Photo Researchers, Inc.; vi Baron
Wolman/Tony Stone Images; vii Darrell Gulin/Tony Stone Images; viii
Dennis Hallinan/FPG International; ix Melissa Farlow/National
Geographic Image Collection; x Bob Daemmrich/The Image Works; xi
Billy Hustace/Tony Stone Images; xii Dean Abramson/Stock
Boston/PNI/PictureQuest; xiii Vincent Laforet/Allsport; xiv Roger
Ressmeyer/CORBIS; xv CORBIS/Phillip Gould; xvi Rex A.
Butcher/Tony Stone Images; xxvi Stuart Westmorland/Photo
Researchers, Inc.; 1 Stuart Westmorland/Photo Researchers, Inc.; 3
Lewis Portnoy/The Stock Market; 4 CORBIS/AFP; 9 Gibbs, M.
QSF/Animals Animals; 13 John Kuhn; 15 David Young-Wolff/Tony
Stone Images; 17 Eric R. Berndt/Unicorn Stock Photo;
19 RMIP/Richard Haynes; 20 Mark Gibson; 23 RMIP/Richard Haynes
(all); 24 Zane Williams/Tony Stone Images; 25 CORBIS; 26 Zigy
Kaluzny/Tony Stone Images; 28 Hurlin-Saola/Liaison Agency;
30 Martha Granger/EDGE Productions; 34 John Feingersh/Stock
Boston; 36 Ted Streshinsky/Photo 20-20; 38 David H. Frazier/Tony
Stone Images; 40 Robert Ginn/PhotoEdit; 42 Mark Stouffer/Bruce
Coleman Inc.; 44 Tom McHugh/Photo Researchers, Inc.; 46 Stephen R.
Swinburne/Stock Boston; 48 Mark Wagner/Tony Stone Images;
52 Courtesy of the Archives Division of the Oklahoma Historical
Society, negative number 1757.; 53 Stephen Frink/The Stock Market;
62 Baron Wolman/Tony Stone Images; 63 George Hall/Woodfin Camp
and Associates; 67 Charles Krebs/The Stock Market; 69 Jay M.
Pasachoff/Visuals Unlimited; 71 courtesy, NASA; 75 courtesy, NASA;
78 Bert Sagara/Tony Stone Images; 82 AP Photo/Osamu Honda;
86 Alan Schein/The Stock Market; 88 Hulton-Deutsch
Collection/CORBIS; 93 Marty Stouffer/Animals Animals; 95 Nick
Bergkessel/Photo Researchers, Inc.; 97 Galen Rowell/CORBIS;
100 Bob Daemmrich/The Image Works; 102 Phil Degginger/Animals
Animals; 105 David Joel/Tony Stone Images; 107 Richard T.
Nowitz/Corbis; 109 Alan Schein/The Stock Market; 111 Phil
Degginger/Bruce Coleman Inc.; 113 James Handkley/Tony Stone
Images; 128 Darrell Gulin/Tony Stone Images; 129 Tom & Pat
Leeson/DRK Photo; 132 Bobbi Lane/Tony Stone Images (background);
Deborah Davis/Tony Stone Images (inset); 134 Tom Bean Photography;
136 Robert Brenner/PhotoEdit; 138 Antman/The Image Works;
140 Mel Traxel/Motion Picture and Television Photo Archive; 142 Rob
Matheson/ The Stock Market; 144 Photri/The Stock Market; 151 Tom
Brakefield/DRK Photo; 157 Bob Daemmrich Photography; 161 Don
Mason/The Stock Market; 163 Shahn Kermani/Liaison Agency, Inc.;
167 Aaron Haupt/Photo Researchers, Inc.; 171 National Geographic
Image Collection (r); Mike Brown/Florida Today/Liaison Agency, Inc.
(1); 173 AP Photo/Kevork Djansezian; 175 Flip Nicklin/Minden
Pictures; 177 Dave G. Houser; 179 School Division, Houghton Mifflin
Company; 183 Martha Granger/EDGE Productions; 187 David Young
Wolff-PhotoEdit; 198 Stephen Frisch/Stock Boston; 199 Paul
Barton/The Stock Market; 200, 201 Dennis Hallinan/FPG International;
203 CC Lockwood/Animals Animals; 207 Eastcott/The Image Works;
210 Bongarts Photography/Sportschrome; 214 Thomas
Zimmerman/Tony Stone Images; 216 Pat and Tom Lesson/Photo
Researchers, Inc. (background); Calvin Larsen/Photo Researchers, Inc.
(inset); 220 Kent & Donna Dannen; 222 Bob Rowan; Progressive
Image/CORBIS; 226 Frank Fournier/The Stock Market; 228
RMIP/Richard Haynes (all); 229 Kevin Horan/Tony Stone Images;
233 Frank Siteman/Stock Boston; 234 James Lemass/Index Stock
Photography, Inc.; 235 Mark Antman/Stock Boston; 236 Michael
Nelson/FPG International; 237 Adam Tanner/The Image Works; 240
Bob Daemmrich Photography (1); Artville, LLC. (r); 243 Tony
Freeman/PhotoEdit; 247 CORBIS; 248 Matthew J. Atanian/Tony Stone
Images; 252 Stephen Dalton/Animals Animals; 257 Tim
Mosenfelder/CORBIS; 266 Melissa Farlow/National Geographic Image
Collection; 267 Mastrorillo/The Stock Market; 269 Simon
Bruty/Allsport; 271 courtesy, NASA; 274 Barbara Filet/Tony Stone
Images; 278 CORBIS/The Purcell Team; 282 Peter David/Photo
Researchers, Inc.; 285 Andrew Hourmont/Tony Stone Images; 291
Wayne Lankinen/DRK Photo; 296 Len Rue Jr./Photo Researchers, Inc.;
298 Bob Daemmrich Photography; 302 Bob Daemmrich Photos; 303
Werner Forman Archive, Maxwell Museum of Anthropology,
Albuquerque, NM, USA/Art Resource; 304 Ken Frick; 305
RMIP/Richard Haynes; 306 CORBIS/Douglas Peebles; 308 John
Darling/Tony Stone Images; 311 Stock Montage; 320, 321 Bob
Daemmrich/The Image Works; 323 Roger Ressmeyer/CORBIS;
327 James Sugar/Black Star/PNI (1); Kevin Scola (r); 330 Catherine
Karnow/Woodfin Camp and Associates; 334 Paul S. Howell/Liaison
Agency, Inc.; 336 David J. Sams/Stock Boston; 338 Charles
Thatcher/Tony Stone Images; 340 Courtesy of Chance Rides, Inc.; 342
Chuck Pefley/Tony Stone Images; 343 W. Perry Conway/CORBIS (1);
Rich Iwaski/Tony Stone Images (r); 346 Stock Montage (1); Frederica
Georgia/Photo Researchers, Inc. (r); 348 RMIP/Richard Haynes (inset);
352 Willie Hill/Stock Boston; 355 Walter Chandoha; 359 Paul
Souders/Tony Stone Images; 361 Bachmann/PhotoEdit; 365 J. F.
Towers/The Stock Market; 367 Michael Newman/PhotoEdit; 371 Yoav
Levy/Phototake; 372 A1 Giddings Images; 384 RMIP/Richard Haynes
(all); 385 Jon Riley/Tony Stone Images; 386 Billy Hustace/Tony Stone
Images; 387 Bob Daemmrich/Uniphoto; 391 Bob Daemmrich
Photography; 393 Hulton-Deutsh Collection/CORBIS;
396 RMIP/Richard Haynes; 398 Associated Press AP; 400 Bob
Daemmrich Photos; 402 Tom Evans/Photo Researchers, Inc.; 406 North
Wind Picture Archives; 407 Reproduced by Permission of the
Commercial Press (Hong Kong) Limited from the publication of
Chinese Mathematics: A Concise History ; 409 Don Smetzer/Tony
Stone Images; 410 Doug Martin/Photo Researchers, Inc.; 413 Kevin R.
Morris/CORBIS; 417 Mary Kate Denny/PhotoEdit; 421 Bill
Varie/CORBIS; 424 Charlie Westerman/International Stock Photo;
428 M. Granitsas/The Image Works; 438 Dean Abramson/Stock
Boston/PNI; 439 David Madison; 443 Mark Wagner/Tony Stone
Images; 447 Keith Wood/Tony Stone Images; 449 Archive Photos;
457 Texas Historical Commission; 459 Nancy Sheehan/PhotoEdit;
462 Jonathan Daniel/Allsport; 466 Tim Flach/Tony Stone Images;
469 Hulton Getty Collection/Tony Stone Images; 473 The Granger
Collection (1); North Wind Picture Archives (r); 476 Tom
McHugh/Photo Researchers, Inc.; 482 James Wilson/Woodfin Camp
and Associates; 486 Owen Franken/CORBIS; 496, 497 Vincent
Laforet/Allsport; 499 Catherine Karnow/CORBIS; 503 Gianni Dagli
Orti/CORBIS; 505 Richard B. Levine; 509 Ken M. Johns/Photo
Researchers, Inc. (tl); Photo Researchers, Inc. (cl); Charles D.
Winters/Photo Researchers, Inc.(tc); E.R. Degginger/Photo Researchers,
Inc. (cr, bl); Tom McHugh/Photo Researchers, Inc. (be); Biophoto
Associates/Photo Researchers, Inc. (br); 511 Stephen Munday/Allsport;
513 Mike Hewitt/Allsport; 520 Chris Cole/Allsport; 524 Tim
Davis/Photo Researchers, Inc. (1); Michel Hans/Vandystadt/Allsport (r);
526 Susan Van Etten/PhotoEdit; 530 courtesy, NASA; 533 Mike
Powell/Allsport; 537 Gordon & Cathy Illg/Animals Animals (1); 540
Bob Gurr/DRK Photo; 544 Gary A. Conner/PhotoEdit; 547 Amos
Nachoum/CORBIS; 562 Stephen Frisch/Stock Boston; 563 Phillip
Bailey/The Stock Market (tr); G. Brad Lewis/Tony Stone Images (bl);
564, 565 Roger Ressmeyer/CORBIS; 568 Tony Freeman/PhotoEdit;
572 David Lissy/Index Stock Photography; 575 Photodisc, Inc.;
579 Jeff Greenberg/PhotoEdit; 581 Ron Kimball Studios; 584 Mark E.
Gibson/The Stock Market; 586 Michael Schimpf; 588 Dave G.
Credits
CREDITS
CREDITS
Houser/CORBIS; 592 Allen E. Morton/Visuals Unlimited; 595 Orion
Press/Tony Stone Images; 598 Rachel Epstein/PhotoEdit; 600 Richard
Vogel/Liaison Agency, Inc.; 603 Charles & Josette Lenars/CORBIS;
607 Vandystadt/Allsport; 609 David Young-Wolff/PhotoEdit; 612 Tony
Freeman/PhotoEdit; 614 Mike Powell/Allsport (1); Darren
Carroll/Duomo (r); 616 Ben Klaffke; 619 Dave Schiefelbein; 621 Alan
Klehr/Tony Stone Images; 630, 631 Phillip Gould/CORBIS; 633
Wolfgang Kaehler/CORBIS; 635 Louis Mazzatenta/National
Geographic Image Collection; 637 Robert Ginn/PhotoEdit (1); Cordelia
Williams (r); 639 Line Cornell/Stock Boston; 643 Chris Arend/Alaska
Stock Images; 646 AFP/CORBIS; 650 AFP/CORBIS; 652 Raymond
Gehman/CORBIS; 658 S B Photography/Tony Stone Images; 661 Zigy
Kaluzny/Tony Stone Images; 663 Ron Dorsey/Stock Boston; 668
Joseph Pobereskin/Tony Stone Images; 670 Karl Weatherly/CORBIS;
675 John W. McDonough/Sports Illustrated Picture Collection; 688,
689 Rex A. Butcher/Tony Stone Images; 692 Louis
Mazzatenta/National Geographic Image Collection; 696 Peter
Menzel/Tony Stone Images; 698 Jeff Greenberg/Visuals Unlimited; 702
John Bova/Photo Researchers, Inc.; 704 Art Montes De Oca/FPG
International; 706 Jeff Persons/Stock Boston; 708 Mike
Powell/Allsport; 710 Tony Duffy/Allsport; 715 RMIP/Richard Haynes
(all); 716, 718 Norbert Wu; 724 Tony Duffy/Allsport; 730 Doug
Pensinger/Allsport; 734 Bob Daemmrich/PNI/PictureQuest; 736 Bob
Daemmrich/The Image Works; 737 David Young-Wolff/PhotoEdit; 740
John Neubauer/PhotoEdit; 742 Dennis MacDonald/PhotoEdit; 756
Werner Forman/CORBIS; 757 RMIP/Richard Haynes; 813
RMIP/Richard Haynes
Illustration
Steve Cowden 641, 700, 720, 732
Laurie O’Keefe 238, 357, 401, 537 (r)
School Division, Houghton Mifflin Company 666
Doug Stevens 365, 577, 605, 708 (t)
Student Resources
Selected Answers
Pre-Course Practice
Decimals (p.xx) i. 21.1 3.67.95 5.15.105 7.66.3
9.76.304 11.729.008 13.3.7 15.0.35
Factors and Multiples (p.xx) 1.1, 2, 3, 4, 6, 12
3. 1, 2, 3, 6, 9, 18, 27, 54 5. 2 • 3 3 7. 5 • 7 9. 1, 2, 4
11.1,2,7,14 13.4 15.3 17.6 19.2 21.36
23. 42 25. 48 27. 900 29. 24 31. 60 33. 28 35. 54
Fractions (p.xxi) 1.1 3.^ 5. f 1 . if 9.^
o y L J y
19 3 9 1 1 31
lr 24 13 ’l0 15 'l0 1? '2 19-6 21 ' l 2 23 13 40
25. l| 27. l|
Fractions, Decimals, and Percents (p.xxi) i. 0.08,
^ 3. 0.38, || 5. 1.35, 1^ 7. 0.064, 9.44%,
^ 11 . 13%, ^ 13. 160%, l| 15. 660%, 6|
17. 0.6, 60% 19. 0.68, 68% 21 . 5.2, 520% 23. 3.063,
306.3%
Comparing and Ordering Numbers (p. xxii)
1 . 13,458 < 14,455 3. -8344 > -8434 5. 0.58 > 0.578
Q 11 o
13. 1075,
15 9 9 3 11 2
7 —— > —- q — = — ii —2— > —3 —
16 10 24 8 16 9
1507, 1705, 1775
17 — — — — 19
7 ’ 11 ’ 2’8
15. -0.205, -0.035, -0.019, -0.013
_4 _3 _4 _2 _ 7 ,3 5 ,4
2 ’ 2 ’ 3 ’ 3
21
7 3 ^-
— 1 —— 1 —
5 ’ 5 ’ 3 ’ 5
Perimeter, Area, and Volume (p.xxii) i. 10 m
3. 22.6 km 5. 95 ft 7. 3.92 in. 2 9. 39,304 ft 3
11. 78.65 mm 3
Data Displays (p. xxiii) 1. Sample answer: 0 to 60 by
tens: 0, 10, 20, 30, 40, 50, 60 3. Sample answer: 0 to
25 by fives: 0, 5, 10, 15, 20, 25
5. Sample answer: bar graph
Measures of Central Tendency (p. xxiii) i. 4.9; 5; 7
3. 52.1; 53; no mode
Chapter!
Study Guide (p.2) i.B 2. A 3. B 4. A
1.1 Guided Practice (p.6) 7. p minus 4, subtraction
9. 8 times x, multiplication 11.1 13. -jy 15. 54
1.1 Practice and Applications (pp.6-8) 21.20
23. 2 25. 20 27. 9 29. 70 31. 6 33. 260 mi
35.40 ft 37.340 mi 39.240 ft 41.64 m 43.10 m 2
45. 6 yd 2 49. 4 h 53. 9.48 55. 15 57. | 59. 23.9
61.11.1508 63.53.55 65.13.405
1.2 Guided Practice (p. 12) 5 . B 7. A 9. 9 11. 36
1.2 Practice and Applications (pp. 12-14) 13. 2 3
15. 9 5 17. 3 4 19. 5 2 ; 25 21. 16 23. 64 25. 1 27. 0
29. 729 31. 32 33. 125 35. 371,293 37. 35,831,808
39. 531,441 41. 29 43. 9 45. 20 47. 6 49. 15,625
51.100,000 53.8 m 3 55. 2 3 , 8 cubic units 57. 4 3 ,
64 cubic units 65. 18 67. 45 69. 9 71. 28 73. 3
75.9 77.5 79. 81. | 83.9 85.3 87.7
89,91, and 93. Estimates may vary. 89. about 0.3; 0.27
91. about 5; 4.764 93. about 6; 6.325
1.3 Guided Practice (p. is) 3. 60 5. 12 7. 17
9.23 11.4 13.246 15.3
1.3 Practice and Applications (pp. 18-21) 17.34
19.1 21.82 23.300 25.42 27.11 29.16 31.48
33.14 35.46 37.3 39. | 41.128 47. 35($230 +
$300 + $40 + $15 + $100 + $200) - $2000 49. |x 2
51. 2($7) + $5 + 2($4) 59.8 61.162 63.11 65. z 6
67. 81 69. 900 71. composite; 1, 3, 9 73. composite;
1, 2, 19, 38 75. composite; 1, 2, 5, 10, 25, 50 77. prime
Quiz 1 (p.2i) i. 18 2.14 3.32 4.9 5.5 6.16
7.6 8.54 9.216 10 . 200 mi 11 . 2000 mi 12 . 20 mi
13. 6 3 14. 4 5 15. (5y) 3 16. 3 3 17. (2jc) 4 18. 8 2
19.64 ft 3 20.2 21.- 22.-
1 .4 Guided Practice (p. 27) 9. not a solution
11. solution 13. not a solution 15. solution 17. not a
solution 19. solution 21. not a solution 23. solution
25. solution
1.4 Practice and Applications (pp. 27-29) 27. not a
solution 29. solution 31. solution 33. solution 35. 5
37.8 39.9 41.21 43.2 45.5 47.6 51. solution
53. not a solution 55. solution 57. 34 boxes or more
59. 7, 2, 1 65. 16 67. 2 69. 7 2 71. 9 6 73. (8d) 3
75. 12 77. 3 79. 9 81. 9 83. 5.6 85. 0.457
87.758.95 89.0.3 91.4.10
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
1.5 Guided Practice (p. 33) 3. B 5. A
7. x + 10 = 24 9. — < 2
n
I. 5 Practice and Applications (pp. 33-35)
II. 10 - x 13. x + 9 15.-^ 17.X +18 19. X- 7
25. X + 10 > 44 27. 35 < 21 - X 29. 7x = 56
31.—= 7 33. 28 - x = 18; 10 35. — = 7; 7
37. 110 = 55 1\ 2 h 43. solution 45. not a solution
47.0.28 49.0.4 51.0.45 53.0.174
Quiz 2 (p. 35) 1. solution 2. not a solution 3. solution
4. solution 5. not a solution 6. solution 7. solution
8 . not a solution 9. solution 10. 8v = 32; 4 units
11. 1 <17 12. lOx = 50 13. y + 10 & 57
14. y — 6 = 15
1.6 Practice and Applications (pp. 39-41) 5. 20 min
7. walking speed = 4 (mi/h), time to walk home = t,
distance to home = 1 (mi) 9. t = ^ h or 15 min
11 . original length + number of days • growth rate =
total length 17. number of weeks worked = 8,
amount saved each week = m ($), price of stereo with
CD = 480 ($) 19. $60 25. 1000 27. 14 29. 12
31. solution 33. 0.25/ + 0.50(100) = 100; 200 35. 1^
37. 39. 41. 43. 4^ 45.
o 3 / Z 3
1.7 Guided Practice (p. 45) 3. false 5. false
1.7 Practice and Applications (pp. 45-47) 7. Player 4;
Player 1 9. 1990; 2000 11. about 150 ft 13. The
braking distance at that speed is about 300 ft. You need
to have time to react to any emergency and still allow
time for your car to travel that distance while stopping.
15. the 6 years from 1991 to 1996 17. 1998
19. Sample answer:
I chose a line graph
because line graphs are
useful in showing
changes over time.
23. 42 in., 98 in. 2
25. 56 ft, 84 ft 2
27. solution
29. not a solution
31. solution
33. not a solution
35. < 37. >
39. = 41. >
43. =
Year
1.8 Guided Practice (p. 51)
Cabin Rental
c
140
120
| 100
"D 80
8 60
0
40
20
°(
5 1 2 3 4 5 6 n
Number of people
1.8 Practice and Applications (pp. 51-54)
Input t
0
5
10
15
20
25
30
Output d
0
1
2
3
4
5
6
17. no 19. no
Student Resources
Time (h)
23. a . d = 11 1
Input t
1
14
28
Output d
77
154
308
c. 100 days
Cooling Water
y,
Temperature (°C)
NJ CD OO C
3 O O O O C
) 5 10 15 20 25 't
Time (min)
31.64 33.15 35.45 37. — > 7 39. f 41. 4^
X 2 3
43. | 45. 3
O
Quiz 3 (p. 54) i. 6 bottles
Attendance at Art Activities
by 18-to-24-year-olds
Activity
3. Sample answer: Attending historic parks was most
popular; attending a jazz concert is about a third as
popular as attending a historic park. Since the percents
total more than 100%, some 18-to-24-year-olds attend
more than one kind of arts activity.
4. Sample table:
Input t
0
1
2
3
4
Output h
200
225
250
275
300
6 . h > 200 and h < 300
Chapter Summary and Review (pp. 55-58) i. 20 3. 6
5.10 7. 6 miles 9. 525 miles 11. 26 m 13. 6 3
17
15.16 17.33 19.54 21.3 23. 25. solution
27. solution 29. 3 31. 16 33. 10 35. x + 30
37. x — 9
39. 48.9 + 55.1 < 53.5 + 53.3; 104 < 106.8; yes
Percent of Voting-Age Population Who Voted
Maintaining Skills (p.6i) i. 2.7 3. 12.1 5. 5.806
7.4.244 9.155.8 11.0.99
13-20.
17.
19.14.
16.13.
15.
18.
20.
0.2
0-4 \
4 9
5 10
11
10
1.7
1.9
f-*-
• + 1
1 + +
1 • 1 1
1 1 1 •
0 0.5 1.0 1.5 2.0
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
Chapter 2
Study Guide (p. 64) i. B 2. A 3. D 4. C
61. 3 63. 75 65. 3 67. x + 8 = 17 69. 9y < 6
71. -6 < - 2 , -2 > -6 73 . -3 < 0.4, 0.4 > -3
75. -10 <
10 ’
10
>-10 77 - 79.
81
2.1 Guided Practice (p. 68)
3. < l + lll + llll + ll >
-6 -4 -2 0 2 4 6
5. < I I + I l#l I I I I + I I >
-4 -2 0 2 4 6 8
7. > 9. > 11. -8, -3, -2, 1, 2 13. -9, -7,
2.1 Practice and Applications (pp. 68-70)
15. 3 9 10
< 1114111114411 >
0 2 4 6 8 10 12
19. -6 -4 -2
< 1111414141111 *
-10 -8 -6 -4-2 0 2
23. -2 < 3, 3 > -2 25. -6 < -1, -1 > -6
27. -4 < 0, 0 > -4 29. 10 < 11, 11 > 10
35. - 1.5 0.5 2.5
< I I 4 I I 4 I I 4 I >
- 3 - 2-1 0 1 2 3
39. _ 9
2 -2.8 4.3
< I !•! 4 I I I I I I 4 I I >
-6 -4 -2 0 2 4 6
43.
8
3
_ 2 _9_
5 10
I 4 I 4 I
1
3
45. -3.0, -0.3, -0.2, 0, 0.2, 2.0 47. -5.2, -5.1, —
3.4, 4.1, | 49.-|, -2.6,-|,0,|,4.8 51. > 53.-8
55.
- 2.0
CO
in
-4-
C/J
is
Q.
O
c
CD
o
I s *
o
ro c css
Q. o Q3 Q5
< COCL OGC
-4«-
57. Pollux, Altair, Spica, Regulus, Deneb 59. Regulus
63.4 ft 2 65.81 cm 2 67.4 69.5 71.3 73.65.9°,
67.5°, 69.1°, 69.9°, 72.3° 75. 64.3 < T < 72.3
11 . 5-1 79. 2 6 81. prime 83. 2 4 • 3 2
2.2 Guided Practice (p. 74) 3. -1 5. 2.4 7. 12
9. -5.1 11 . 8, -8 13. 5.5, -5.5 15. False. Sample
counterexample: if a = —2, then —a = —(—2) = 2,
which is greater than —2.
2.2 Practice and Applications (p.74-76) 17. -8
19.10 21.3.8 23. ^ 25.7 27.-3 29.0.8 31. |
33. 4, -4 35. no solution 37. 3.7, -3.7 39. —y-
41. Mercury: 1080; Mars: 288 43. negative
45. positive 47. -6 ft/sec 49. 400 ft/min 51. False:
Sample counterexample: The opposite of —a is a.
If —a = 5, then a = — 5, which is negative. 53. true
2.3 Guided Practice (p.8i) 5. -5 + 9 = 4 7.-10
9. 7 11. -10 13. 7
2.3 Practice and Applications (pp. 81-83) 19. -6
21.-11 23.-4 25.6 27.7 29.-11 31.3
33. —31 35. —35 37. commutative property
39. property of opposites 41. 10 43. 0 45. 5 47. 4
49. -2 j 51.-81.14 53. 356.773 55. two strokes
under par 59. 4 2 61. Jt 3 63. 33 65. 4 67. 24
69. solution 71. not a solution 73. not a solution
75. 9300 77. 100 79. 2900
Quiz 1 (p.83) i. -2 < 7, 7 > -2 2. -3 < -2,
—2 > —3 3. -6 <1,1 > -6 4. -10,-8,-3, 2, 9
5. -7,-5.2, 3.3, 5, 7.1 6.-1, -f, 0, 2 7.5
8.13 9. —0.56 10. no solution 11.2.7,—2.7
12. —|,j 13.-13 14.-6 15.4 16.-7 17.-2
18. 0 19. yes
2.4 Guided Practice (p. 89) 3. -7 5. 7 7.-1
9.3^ 11. 12, —5x 13. —12y, 6
2.4 Practice and Applications (pp. 89-91)
15.9 17.-11 19.39 21.36 23.9.2 25. -1.2
27.3 29. -4^ 31.-1 33.31 35.-43
37.10.2 39.1 41. 1^ 43.14,13,12,11
45. -6.5, -7.5, -8.5, -9.5 47. -2-|, -l|, -j, j
49. -X, -7 51. 9, — 28x 53. a, - 5 55. up 275 ft
57. -7301 - 662 - 1883 + 77 - 1311 + 8021; -3059
65. 35 67. 41 69. 64 71. true
73 ' • I I I ♦ I I I I 141 I I I I ♦ I I I I I I I •
-12 -8 -4 0 4 8 12
75. -4.3 2 6.5
- I •! I I I I I + I I I I • I »
-4 -2 0 2 4 6
79.0.04 81.0.0338 83.19.176
2.5 Guided Practice (p.96) 7. -35 9. -1 n. 5 1 4
13. 40
2.5 Practice and Applications (pp. 96-98) 15. yes
17. -28 19. -12.6 21. ~ 23. -216 25. -49
27. -54 29. 97.2 31. —| 33. -lx 35. -5a 3
37. -10r 2 39.-2x 2 41.-48 43.-147 45.41
47. true 49. False. Sample counterexample: 3 > 2,
but 3 • 0 = 2 • 0 51. -20 ft 53. d ~ -300 1
55. about 150 ft 63. 2 65. 4 67. 12
Student Resources
69.
71.2 73.-9 75.7.2 77.10.43 79. 12, -Z
81. 4w, —11 83. —7x, 4x 85.20 87.150 89.10,920
2.6 Guided Practice (p. 103) 5. 12(x + 5); 12x + 60
7. D 9. B 11. 4(1) + 4(0.15); 4 + 0.6; 4.6
2.6 Practice and Applications <pp. 103 - 106 )
13. 3(4 + x) = 12 + 3x 15. (x + 5)( 11) — llx + 55
17. 3x + 12 19. 7 + It 21. 12 + 6 u 23. 4y + 2
25. 12 + 18a 27. 1.3x + 2.6 29. 5y — 10 31. 63 — 9 a
33. 28 - 4m 35. 10 - 30r 37. 18x - 18
39. -9.3m - 2.4 41. -3r - 24 43. -I - s
45. -y - 9 47. -24a - 18 49. ~6y + 5
51. —13.8 + 42vi’ 53. forgot to distribute 9(3) — 9(5);
-18 55.24.44 57.27.60 59.5.80 61.-12.30
63. -22.10 65. -54.95 67. $19.96 69. $10.45
71. 200(x + 225); 200x + 45,000 73. 60,000 yd * * 3
12
79. — 81. 3 83. 5 85. identity property of addition
87. associative property of addition 89. 12 91. 3
1 1 1 41 24
93- -lj »■ i 1"-g
Quiz 2 (p. 106) 1. -15, -13, -11, -9 2. 30, 28, 26, 24
3 - _3 |, 2 | 4. 2x, -9 5. 8,-x 6. -lOx, 4
7. -0.25, 0.12, -0.12, -0.13 8. -63 9. 30
10. -2800 11. 10.8 12. -3 13. 270 14. llx + 22
15. 60 - 5y 16. -12a + 16 17. $49.90
2.7 Guided Practice (p. no) 3. 6 r 5. -8
7. 4 a 1 + 3<2-5 9. 18/+ 4 11. -11m - 20
13. 9* — 27
2.7 Practice and Applications (pp. 110-112) 15. 3a, 5a
17. m, 6m 19. — 6vv, —3 w 21. —7m 23. 2c — 5
25. 6r - 7 27. already simplified 29. 6 p 2 + 4p - 2
31. -27 - 4y 33. -11 - 6r 35. 10m + 19
37. 2c + 48 39. 7 is not a like term with 3* and — 2x;
x + 7 = 16 41. x + (* — 7) + x + (* — 7); 4* — 14
43. 2(* + 2) + (* + 4) + 2(x + 2) + (* + 4); 6* + 16
47. 15,675 tons 49. T = -45c + 480 51. 1.06* + 21.2
59. about 35% 61. 9 63.-6 65. -14.1 67. -180
69. -3 71. 29.88 73.
1 3 4 3 7
10 ’ 10 ’ 10 ’ 10 ’ 10
2 4 3 2 5
75 ‘ 4 ’ 8 ’ 8 ’ 4 ’ 8 77 ‘ 6 ’ 6 ’ 4 ’ 2 ’ 2
79 1 8 2 12 I 11 4 5 2
8’ 8’ 8’ 8 ’ 8 15’ 5’ 6 ’ 3 ’ 10
2.8 Guided Practice (p. ii6) 3. yy 5. -y or -5
7.-4 9.-2 11. 2 13. all real numbers except* = 4
15. all real numbers except * = 0
2.8 Practice and Applications (pp. 116-118)
17. multiply by reciprocal;—27 19. —3 21. —1
23. -5 25. 2 27. -12 29. 31. 12 33. -48
o
35. -y 37. 39. -y 41. 4 43. 6x - 3
45. already simplified 47. 11 + 2t 49. all real numbers
except* = — 2 51. all real numbers 53. —10.5 m/sec
57.24 59.5 61.10 63. 2* > 7 65.-21 67.-19.9
1 17
69. 4-or— 71. < 73. < 75. > 77. <
Quiz 3 (p. 118) 1. 3*, —lx 2. 6 a and 9a, -5 and 10
3. —5 p, —p 4. —26 1 5.1 + 2d 6. g 2 — 8 g 1.3a — 4
S.3p-9 9.5-3 w 10.-5 11.16 12.-32
98 1
13. — 14. -54 15. - 16. 5 - 2* 17. already
simplified 18. 3* — 2 19. all real numbers except
* = — 2 20 . all real numbers 21 . all real numbers
except * = 0
Ch. 2 Extension (pp. 119-120)
Exercises (p. 120) 1. inductive reasoning 3. inductive
reasoning 5. 64, 128, 256
Chapter Summary and Review (pp. 121-124) 1. -6,
-4, -3, 1, 2, 5 3 . - 2 , -1, -y, |, 1, 4, 6 5. 5
7.-45 9. -9.1 11. 3y 13.-12 15.5 17.-8
19.19 21.-11.2 23. —3y 25.600 27.4.2
29. -14 31. -3/ 33. —12 1 2 35. -81 b 2 37. 9y + 54
39. 6-2 w 41. -3 1 - 33 43. — 6x + 60 45. 9 a
47. 3 + / 49. 4t + 2 51. -4 53. 10 55. —| 57. -9
Maintaining Skills (p. 127) 1. 25 3 .100
Chapter 3
Study Guide (p. 130) 1. D 2. D 3. C 4. B 5. C
3.1 Guided Practice (p.135) 7. -1 9. -17 11. 4
13. 3 15. -3 17. addition
3.1 Practice and Applications (pp. 135-137)
19. subtract 28 21. add 3 23. subtract —12 25. 9
27.-5 29.10 31.8 33.-4 35.24 37.-15
3
39. -24 41. y 43. 0 45. 5 47. 6 49. 3 51. 1
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
53. 20 cm 55. B; 8 57. 6463 seats 59. 10,534 acres;
4218 + 3800 + 2764 - 248 = x 61. Simplify with
subtraction rule; subtract 2 from both sides. 65. 5x = 160
67. 36 - k = 15 69. 4x + 8 71. —5y - 20
73. —lx + 12 75. y 77.^- 79 y 81.1
3.2 Guided Practice (p. 141) 7. -1 9. 32 n. 28
13.-6 15.60mi/h
3.2 Practice and Applications (pp. 141-143)
17. divide by 5 19. divide by —4 21 . multiply by 7
23.-8 25.-6 27.11 29. -y^ 31.-10 33.30
35.84 37. | 39.0 41.12 43.18 45.-45
47. multiply by — f; —8 49 .•§■•/? = 3.30; $8.80
J O
51. 13 57. 27 - 8x 59. ~2x + 6 61. 12y + 15 63. 8
65.-19 67.2 69. A; 18 71.10 73.5 75.9 77.3
3.3 Guided Practice (p. 147) 7. 2 9. -1 11. 2
13. 25 15. -9 17. 19
3.3 Practice and Applications (pp. 147-149) 19. 2
21. 14 23. 2 25. 3 27. 5 29. -3 31. 3 33. 9
35. 14 37. 11 39. 6 41. 5 and 3x are not like terms,
so 3x cannot be subtracted from 5; — j 43. Subtract 3
from each side; multiply each side by 2; divide each side
by 5. 45. 14 months 55. a 6 57. 4 3 59. t 3 61.10
63. 47 65. 14 67. < 69. < 71. < 73. <
Quiz 1 (p. 149) 1 . 21 2.-17 3.-7 4.-1 5.282
6.5 7. B 8 . 6x = 72; $12 9.9 10.2 11.2 12.1
13. -25 14. 14 15. 9 min
3.4 Guided Practice (p. 154) 9. one solution, -1
11. one solution, 7 13. identity 15. B
3.4 Practice and Applications (pp. 154-156)
17. subtract x from each side 19. add 8x to each side
21. 3 23. 3 25. 2 27. j 29. -8 31. 4 33. -2
35. 3x — 12x = — 9x; x = — 5 37. one solution, 2
39. one solution, —1 41. one solution, —5 43. no
solution 45. one solution 47. 121 hours 49. 25 sec; the
gazelle would probably be safe since the cheetah begins to
tire after 20 seconds. 57. 144 miles 59.8 61.216
63. 144 65. 10 67. yes 69. -4 71. 12 73. -23
75.0 77.90 79.2000 81.9 83.8 85.3910
3.5 Guided Practice (p. 160) 11. -5 13. -2 15. -6
17. 2
3.5 Practice and Applications (pp. 160-162) 19. 19
21.14 23.3 25.21 27.-1 29.-4 31.-1 33.^
35. 1 37. 3x — 12 + 2x = 6 — x, 6x — 12 = 6,
6x = 18, x = 3 39. —4(3 — n) = —12 + 4n,
8(4n — 3) = 32 n — 24 ; n = j 43. C, x = 25; you will
need to use the gym more than 25 times to justify the
cost of the yearly fee. 51. 400,000 km; 700,000 km;
I, 100,000 km; 1,900,000 km 55. 36 57. -77
59. 3w 2 — w 61. s + lit 63. —6m — m 2 65. 11.5
67. 6.42 69. 22.49
3.6 Guided Practice (p. 166) 7. 23.4 9. -13.9
II. 56.1 13.8.8 15.6.82 17.4.22 19. $12
3.6 Practice and Applications (pp. 166-169) 21. 5.78
23. 7.57 25. 4.33 27. 0.77 29. 2.22 31. 0.94
33. 0.42 35. -2.63 37. M = 150 + 0.38x 39. 1.0
41. 1.9 43. 162 + 30 = 71 n, where n is the number of
buses needed 45. Round up to 3 buses; you need enough
buses to seat all the students and adults. 51. $697.45
Input t
2
3
4
5
6
Output A
18
23
28
33
38
55. 3 57. 59. -5.6 61. 16 63. 14 65. 13y
67.25-^ 69.11
to
Quiz 2 (p. 169) 1. no solution 2. one solution 3. identity
4. no solution 5. —3 6. —7 7. 10 8. 1 9. 5 10. 4
11.—1 12.19 13.8 14.13 15. You need to use the
bike for at least 10 hours to justify the cost of the helmet.
16.-1.14 17.-0.68 18.1.63 19.0.36 20.-5.03
21. -2.23 22. 7
3.6 Technology (p. 170) 1. 12.3 3. 5.3
3.7 Guided Practice (p. 174) 3. r = s + t 5. y = |
7. y = 2x — 4 9 . w = j-
3.7 Practice and Applications (pp. 174-176)
11. C = |(F - 32) 13. w = j;w = 4
15 ./ = —;/= 16 17. 18 cm 2 19. 16.67 cm 3
W
21. 6 min 23. 30 ft 27. solution 29. not a solution
31. not a solution 33. not a solution 35. solution
37.28% 39. | 41. | 43. ^ 45. ^
4 ?
3.8 Guided Practice (p. iso) 5. - 7. -
9. 0.05 mi/min 11. 231 miles
3.8 Practice and Applications (pp. iso-182) 13. j
15. 4 17. 4r- 19. 4 21. -§ 23. 15mi/day 25. $.40/can
J J J O
27. 8 oz/serving 29. miles 33. 24 months 35. 21.2 hours
37. 2 km 39. 21 mi/hr 41. 12 min 43. $91
49. 4 >-3;-3 <4 51. -6 <3; 3 >-6 53.1.43
55. 75 ft 57. 18 59. 21 61. 162 63. 490
3.9 Guided Practice (p. 186) 7. 175% 9. 72
11. a = 0.06(10)
Student Resources
3.9 Practice and Applications (pp. 186 - 188 )
17.20 19.30.8 ft 21.10 23.84 ft 25. $1000
27. 200 29. 480% 31. 30% 33. 20%
35. no; A: 30%(60) = $18 discount, cost = $42;
B: 20%(60) = $12, cost = $48, 10%(48) = 4.8,
final cost = $48 - $4.80 = $43.20 37. 21% 39. 27%
41. a = 3 b\ Sample answer: a = 30, b = 10 ,p = 300
45. 21* = 105; x = 5 47. 32 49. -16 51. 217, 270,
2017, 2170, 2701 53. 5.09, 5.1, 5.19, 5.9, 5.91
Quiz 3 (p. 188) 1 . t = - 2. h = +- 3. v = -^
r b a
4. 1 ^ ay f 5. + 6 . 300 students/school 7. 240 hours
1 week 12 in.
8. $5.75 9. 5.75 = p( 23); p = 0.25, or 25%
Chapter 4
Study Guide (p.202) 1. B 2. B 3. D
4.1 Guided Practice (p. 206)
J
Y
-3
-]
[
1
5
X
B(
-3, -2)
1
o.
A(- 2 -3)
j
Chapter Summary and Review (pp. 189-192)
1.11 3.-8 5.-9 7.-3 9.1 11.2 13. one
solution; 2 15. one solution; 5 17. one solution, —2
19. 12 + n = 6 + 2n\ n = 6; the plants will be the
y
same height after 6 weeks. 21. 1.08 23 . I = —r
25. b = P - a - c 27. 85 mi
Maintaining Skills (p. 195)
Cumulative Practice (pp. 196-197) 1. 8 3. 41 5. 216
7. 5 9. 7 11. 57 13. not a solution 15. solution
17. not a solution 19. v 3 - 8 21. —3x < 12
25. < 27. > 29. < 31. < 33. ~5y 3 35. ~4 + 2 1
37. 4x — 6 39. 43x + 25 41.-30 ft; negative; downward
velocity is negative. 43. 15(v + 6) = 15v + 90
45. -9 47. 18 49. 0 51. | 53. 3(50) + 2 n = 750;
300 rolls 55.-20.33 57.-2.30 59.-1.22
61.10 cm 63. $45.50
7. always 9. always
4.1 Practice and Applications (pp.206-208)
11.A(2, 4), B( 0, -1), C(—1, 0), D(—2, -1)
ZE
C(3, 3)
J
>4(4, 1)
1
S
-J
l
L
5
X
1
8(0,
-3
)
3 '
.y
A( 0, 3)
1
C(2, 0)
-3
L
:
L
5
X
B(
-2, -
1)
1
3
19. IV 21.1 23. Ill 25. Ill 27. pounds; inches
31. Gas mileage decreases as weight increases.
33.
•
i_
Wing-beat rate (beats/sec)
35. As wing-beat rate increases, the wing length
decreases. 41. 7 43. 3 45. 39 47. -13 49. 1.07
2 14 5
51. — 53.5 55.1 57.5 59.2- 61.7^ 63. yy
4.2 Guided Practice (p. 213) 3. solution 5. solution
7. y = — x — 2 9. y = ~2x + 4 11. Sample answer:
>y
( 0 ,
6 )
5
3
-1
i
L
3
\7 ■*
JE
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
4.2 Practice and Applications (pp. 213 - 215 )
17. not solution 19. solution 21. not solution
2 19
23. y = -~x + 2 25. y = ~x + -y 27. y = ~x - 5
3 3
29. y = — 2 X _ 2 Sample answers given for 31-39
31. (0, -5), (1, -2), (-1, -8) 33. (0, -6), (1, -8),
(-1, -4) 35. (0, 3), (3, 1), (-3, 5) 37. (0, 5), (2, 0),
(4,-5) 39. (0,-4), (l,-
/
/
/
1
5
i j
1 1
5
X
fT
o
1
ro
7
/I
f), (3, -9)
49. 7.lx + 10. ly = 800 51. about 48 minutes 53. The
boiling temperature of water decreases as altitude increases.
61. -12 63. 6 65. — 14x + 6 y 67. ~5t 3 - 9r
69. - 3k 3 + h 71. -15 73. 63 75. 63% 77. 2%
79. 127% 81. 860%
4.3 Guided Practice (p.219)
10
6
X
10
-1
4
-(
>
> x
n
T7
9. x = 3 11. sometimes 13. always
4.3 Practice and Applications (pp. 219 - 221 )
15. not solution 17. not solution Sample answers
given for 19-23: 19. (f o), (f 2 ), (f - 2 )
21. (0, -5), (3, -5), (-3, -5) 23. (0, 7), (-2, 7),
(-3, 7)
6
K =
= 8
-6
-2
>
6
X
6
31. x = -4 33. a. H = 110; domain: 0-5; range: 110
b. H = 160; domain: 0-10; range: 160 37. 7 39. 8
15 14 15 4
41.10 43.5 45.15 47.21;^,^ 49.21;^,^
51. 26;
24 _5_
26’ 26
„ rr\. _9_ 28
53. 60, 60 , 60
Quiz 1 (p. 221 )
X
6(0,,
2)
A (-4,1)
T7
c
(-3,0
)-!
1
5
X
3
X |
C (1, 6)
0
2
-(
-2
2
(
X
A(-
1,
-5).
u
•6 (0, -7)
X
6(1,
3)
C(-1, 1).
5
-1
:
L
5
X
3
5
Ai
-1
1, -6)
X
J
1
6 (5, 0)
-1
]
.
5
5
7 x
—
3
C(0,
-4
)
X
A (2, -6)
5.1 6. Ill 7. IV 8. II 9. y = ~2x 10. y = f x - 10
1
11. y = ~ 2 X ~ ^ 12. Sample answer: (0, —6),
13. Sample answer: 14. Sample answer:
(0, 1), (1, 5), (-1, -3) (0, 2), (1, -4), (-1, 8)
\
• y
\
A
\
-(
)
\ -
>
(
X
1
6
3
Student Resources
4.4 Practice and Applications (pp. 225-227)
15. x-intercept = 2, ^-intercept = 3
17. x-intercept = — 4, y-intercept = — 1 19. —2 21. 19
23. 6 25. -12 27. -2 29. 26 31. -4
15. Sample answer:
17. Sample answer:
(0, -2), (3, 6), (-3, -10)
- 6
/
- 2 -
/
/
-6
-2
=0
/ 2
6 *
/
A
16. Sample answer:
4.4 Guided Practice (p.225) 3. 6 5 . -3 7. -2
9. x-intercept = — 2, 11. v-intercept = 2,
y-intercept = 2 y-intercept = — 4
13. v-intercept = — 3, y-intercept = 3
49. 7.5; if students get in free, the adult ticket price needs
to be $7.50. 53. about 189,000 57. -4 59. -5
61. | 63. -17 65. 6 67. -2 69. -60 71. j
75. $1.65 77. $8.36 79. $3.15 81. $5.11
4.5 Guided Practice (p. 233) 5. positive 7. negative
9. zero 11. undefined
4.5 Practice and Applications (pp. 233-235)
3 13 11
13. —^ 15.^ 17-4 19.-1 21.1 23 .- 25.-4
3 3
27.—— 29. neither 31. zero 33. neither 35. ^
39. it represents how the rise changes with respect to
the run. 41. 6% 45. 5 47. 4 49. y = 2x + 9
51. y = 4x + 5 53. v = — x — ^ 55. true 57. false
59. true
4.6 Guided Practice (p. 239) 3. ^ 5. j i.y = 5x
3
f k=
5x
/
1
5
-i i
1
5
/
/
/
4.6 Practice and Applications (pp. 239-241) 13. 12
15. 25 17. y = 5x 19 .y = 6x 21. y = — jjc
23. y = — lOv 25. yes, direct variation
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
31. yes; line through origin 33. no; line does not pass
through origin 35. 17 min 37. about 16 in. 41. 2
2 12
43. -5 45. -3 47. y = ~~^x + -y 49. solution
51. solution 53. solution 55. 66 57. 56 59. 3570
Quiz 2 (p.24i)
7.f 8.f 9- g 10.2
14. y = 8x 15. y = 4x
y i
/
y
/
1
5
1 ,i
j
L
5
/
/
/
/1
. 0 12.-1 13. y = 3x
r
j
\
\
2\
-(
)
f
X
Z"
6 _
\v
-6
)X
\
\
19. 10,500 bolts
4.7 Guided Practice (p. 246 ) 3. m = 2, b = 1
5. m = 5, b = — 3 7. m = —1, b = 15 9. B
4.7 Practice and Applications (pp. 246 - 249 )
11 . y = x + 9 13. y = 2x — 10 15. y = ^x — 6
17. m = 6, b = 4 19. m = 2, b = —9 21 . m = 9,
b = 0 23 . m = —3, b = 6 25. m = 2, Z? = 4
7
/
L
1
K =
= 3x + 7/
,
7
/
/
1
'/
-l
i
3
X
/
L
Student Resources
4
47. m = ~—,b = 4 49. parallel; same slope, m = — 3
51. parallel; same slope, m = 1 53. not parallel;
9 1
different slopes 55. (1) — (2) — 63. line a and
line b 71. 5 73. 12 75. 6 77. -5
13
79. Atomic weight ~ 2 X Atomic number 81. —
23 13 1
“- 1 :i 85A i 81A 21
4.8 Guided Practice (p. 255) 3. -22 5. 8
7. function; domain: 10, 20, 30, 40, 50; range: 100, 200,
300, 400, 500 9. not a function 11. not a function
4.8 Practice and Applications (pp. 255-258)
13. function; domain: 1, 2, 3, 4; range: 2, 3, 4, 5
15. not a function 17. function; domain: 0, 2, 3, 4;
range: 1, 2, 3, 4 19. function 21. function 23. function
25. 6, 0,-6 27. 1,-5,-11 29.11,1,-9
31. 23, 7, -9 33. 4, -6, -16
47.—1 49.-3 51. not a function
53. function; domain: —2, 0, 1,2; range: —2, 0, 1, 2
_ y
■3 450
s 400
0 350
f 300
- 250
| 200
I 150
« 100
I 50
0 :
270 274 278 282 286 *
Score
Yes. Sample explanation: For each input, there is exactly
one output. (The score 285 occurs twice, but the prize
money is the same each time.) Domain: 270, 282, 283,
284, 285, 286; range: 486,000, 291,600, 183,600,
129,600, 102,600, 78,570
57. 1500 miles 59. /(0 = 5.88t 63. 6 65. y
3
67. no solution 69. -1 71. -4 73. 0 75. —
Quiz 3 (p. 258 ) 1. y = 3x + 4; m = 3, b = 4
2. y = — x + 2; m = — 1, b = 2 3. y = ~2x + 6;
m = —2, b ~ 6 4. y = ~^x + 4 \m = — 77 , b = 4
/ \ ° O
5. y = — 8 ; m = y, Z? = — 8
\
J
\
\
l
:
L
5
JC
\
\
\
X
\
X
v-
6x
3/
10 . not parallel 11. not parallel 12 . -24, 0, 32
13. 6, -9, -29 14. -9, 3, 19 15. -21, -12, 0
16.4.2,0, -5.6 17. |, 0, -1
■ y
/
6
/
/
i
-f
-2
t
X
/
A»(x)
= 4x-
- 7
o
i
1
1
y
l
\
2 \
-(
-2
(
X
\
\<
f (x)
p
\x
\
\
■ y
\
5
3
\g(x)
= :
-6x +
\
\
-3
-1
i
5
X
L
\
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
37. 11
Chapter Summary and Review (pp. 259-262)
1 . Quadrant I 3. Quadrant II
Graph for Ex. 1 and 3 5. I
Resources
35. -9
)
6
-f
-2
>
X
'ft
X) =
= X
7
39. function; domain: —1, 0, 1; range: 2, 4, 6
41. function; domain: —2, 0, 2; range: 6
Maintaining Skills (p.265) i. y 3.7 5. j 7. -7-
9. 0 11.-11 13.-11
Chapter 5
Study Guide (p. 268) i. C 2. C 3. B
5.1 Guided Practice (p.272) 5 . no l.y = x
9. y = —x + 3 11. y = 5x + 5
5.1 Practice and Applications (pp. 272-275)
2
13. y = 3x + 2 15. y = 6 17. y = —x + 7
2 12 1
19. y = —x — — 21. y = ——x + — 23 . m = ——;
b = 1 25. m = = 2 27 . m = — j; b = — 1
29. y = — 3x - 1 31. y = -x + 1 33. y = 2x - 1
41. 13.16 sec 43. Sample answer : The prediction may
be unrealistic because athletes may be unable to continue
the downward trend. 45. All three lines have the same
slope, 47. y = x + 63.64, y = — x $ 63.64,
y = x — 63.64 49. y = -x + 63.64 53. 92 min
57.3 59. —1 61. —1 63. Sample answer: (—1,-3),
(0, —4), (1, —5) 65. Sample answer: (0, 7), (—1, 12),
(1,2) 67. Sample answer: (—3, 4), (0, 3), (3, 2)
69. 3; 5 71. -2; 3
\
\
\ 5
\
y
= -
2x+ 3
\
\
1
'-1
5
5
7 *
\
7
7
/)
= 3x + 5
c
/
/
/
t
/
/
-3
/-i
j
3
X
u
L
73. 5; -6
t
/
1
/
5
-1
1
3
X
/
A
5x
- (
/
->
1
75 -I
-I
79.
200
8,. l£
”4
5.2 Guided Practice (p.28i) 3. y - 4 = 4(x - 3)
5. y - 4 = ±(x - 3) 7. y — 2 = 3(x — 2)
9 -y
11. y
13. y
1 _L 9
4 X + 4
5.2 Practice and Applications <pp. 281-284)
l, „ l
15. y
2 = -{x
1) 17. y + 3 = + 1)
19. y + 4 = ~(x - 4) 2t.y - 2 = -5(x + 6)
23. y + 2 = 2(x +8) 25. y - 4 = 6(x + 3)
27. y + 1 = 0(x - 8); y = -1 29. y - 4 = 2(x - 1);
y = 2x + 2 31. y + 5 = —2(x + 5); y = — 2x — 15
33. y — 1 = — j(x + 1); y = —^x + j 35. y = 2x — 2
1 8
37. y = — x — - 39. y = —9x — 5 41. y = 2x — 1
43. y = —x — 4 45. 55.25 psi 53. yes 55. yes 57. no
Quiz 1 (p. 284) i. y = -2x +1 2. y = 5x
3. y = —jx +1 4. y = x — 2 5. y = 2x + 3
6. y — 1 = — 2(x — 1) 7. y + 2 = 3(x + 8)
S.y= ~\x 9. y = x + 1 10. y = 4 11. y = 4x
12. y = 4x — 4 13. y = — -^x — 4 14. y = —2x + 5
4
5.3 Guided Practice (p. 288) 3. y = -^x + 2
c 5 1 2
5 ' y= 3 x "3 7 ^ = “3 X
5.3 Practice and Applications (pp. 288-290)
9. y — 3 — ~ 2) or y - 4 = ~x
11 . y + 10 = -^x or y - 4 = -^(x - 12)
13. y — 1 = — {x — 1) or y — 2 = — x
15. y — 6 = x + 8ory— l=x+13 17. y — 5 = 0
19. y = -^-x +16 21. y = 4x + 1 23. y = 2
25. y = *r-|x + 3 27. y = -|x + j 29. y = —2x + 1
31. y = — 3x + 14 33. point-slope form; y = x — 2
4 1
35. point-slope form; y = jx — j 43.-5 45. 4
17 1 19 23 5 17
47 — 49 — 51 7 — 53 8— 55 26— 57 6 —
2 3 24 24 6 18
5.4 Guided Practice (p. 294) 3. 2x - y = 9 or
— 2x + y = -9 5. 3x - 4y = 0 1.5x~ y = 1
9. 3x + y = 10 11. 3x + 5y = 15 13. x = —2
5.4 Practice and Applications (pp. 294-297)
15. 5x + y = 2 17. — 4x + y = — 9 or 4x — y = 9
19. 3x + 8y = 0 21. 2x — y = -19 23. 3x + y = 1
25. 5x — y = 17 27. 2x — 5y = -41 29. x + 3y = 16
31. 2x — 3y = —6 33. 2x + y = 1 35. x + y = -3
37. x + lOy = 27 39. y = -2 41.x = 4 43. x = -3^
45. x = 9 47. y = 10 49. -x + y = 4 51. x + y = 7
53. 4x + 3y = — 8 55. Only the right side was
multiplied by 3.
73. $908 75. $14,098 77. $0 79. $12,346
Quiz 2 (p. 297) 1. y = ~x - 1 2 . y = 3x - 16
3. y = 4 4. y = — 4x + 3 5. y = ^x — ^
6. 3x + y = 9 7. —x + 2y = 8 8. — 2x + 5y = — 5
9. 2x — y = 4 10 . x + 2y = 6 11 . 2x — 5y = —23
12. 2x — y = —2 13. x + 2y = 2 14. y = 3
5.5 Guided Practice (p.30i) 3. C; the slope, 1.5,
represents the amount paid for each unit produced per
hour. 5. B; the slope, 0.32, represents the amount paid
per day for each mile driven.
5.5 Practice and Applications (pp. 301-304) 7. 124
9. y = 124* 11. about 3.2 hours 13. 10
Selected Answers
SELECTED ANSWERS
17. 2 days 19. (1, 48.9) 21. about 67 cents
23. Sample answer: about 51 cents 25. 5v + 7y = 315
27. 2x + y = 102 29. 62; 52; 42; 32; 22
31. 2C + 1.25 B = 10 33. Sample answer: 4500 years
3
37. 0 39. —50 41. 3 feet 43. — \ Sample answer: The
slope is the rise divided by the run of the ramp.
4
45. y = ~2x + 3 47. y = ~x ~ 3 49. y = 2 51. >
53. < 55. = 57. =
5.6 Guided Practice (p. 309) 3. yes 5. no
7. y = x + 3; the product of the slopes of the lines is
(1)( — 1) = — 1, so the lines are perpendicular.
9. y = 2x — 8
5.6 Practice and Applications (pp. 309 - 312 ) 11 .no
13. yes 15. yes 17. y = — x — 2, y = x — 3; yes
1 8
19. y = —3, x = —2; yes 21. y = ——x — —; the
4. yes 5. yes 6. y = x + 1 ; The product of the slopes of
the lines is (1)(— 1) = — 1, so the lines are perpendicular.
4
7. y = ——x — 4; The product of the slopes of the lines is
( —= — 1, so the lines are perpendicular.
8. y = —2x +11
Chapter Summary and Review (pp. 313-316)
3
l.y = 6x - 4 3. y = Sx +8 5. y = -x
7. y = 2x — 2 9. y = — x — 4 11. y + 1 = -|(x + 3);
y = 2 X + ^2 13 - y — 3 = 5(v + 2) or y = 5x + 13
15. y = 3x + 5 17. y = — 8v + 12 19. y = ^x
21.y= — 1 23. y = 1, x = — 1 25. y = — 6, x = —8
27. 2x + y = 7 29. $1,489,200 31. 6; 4; 2; 0 33. yes
35. y = ~2x
product of the slopes of the lines is ( — — )(3) = —1, so Maintaining Skills (p. 319)
the lines are perpendicular. 23. y = 4x — 23; the
product of the slopes of the lines is (4)^—^ = —1, so
2
the lines are perpendicular. 25. y = jx; the product of
55. horizontal
57. vertical
y= -2
x = 4
H-1-1-1-1-1-1-b
-21 -14 -7
14 21 28
the slopes of the lines is ^ —J = — 1, so the lines
are perpendicular. 21.y = —x — 2 29. y = v — 1
31. y = —2x + 5 33. y = —jx + 3 35. x = ~2
3 1
37. y = ——x + 2 39. y = —v — 6 41. always
4 3 3
43. always 45. y = jx + 3, y = ~^x + ^ 49. —6k — 8
51. 6x + 12y + 2 53. 4^-
3 - < 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I l >
0 30 60 90 120 150
5. < 7. > 9. > 11. > 13. >
Chapter 6
^ 1
Study Guide (p.322) 1. C 2. B 3. C 4. B
6.1 Guided Practice (p. 326) 3. open 5. solid
7. solid 9. left 11. left 13. left
6.1 Practice and Applications (pp. 326-328)
15. all real numbers less than 8 17. all real numbers
greater than or equal to 21 19. solution 21. solution
29. subtract 11 31. subtract 6 33. add 3
41. jc < 2
45. p > 11
I I I I I I I 0 I I I
-4 -2 0 2 4
49. -2 > c
I I I I I I I I 4 I I
8 10 12
55. C < 14
I I I 0 I I I I I I I
-4 -2 0 2 4
I I I I I I I 0 I I I
8 10 12 14 16
“■ra 6, -f 63 4 65 - 7 6, -ii 69 'yr
Quiz 3 (p.312) 1. lx + 3y = 42 2 . y = -^x + 14;
14, 7, 0
3.
\\
V
\
\
\
\
6
\
2
\
)
1
0
1
4
X
57. r > 0.11
59. d > 16.3
H-h
4-
61. subtract 4 from each side; x < -3 65. 6 67. 14
69.32 71.3 73.-1 75. y=-v + 3 77. y = ~x + 2
79. y = 2x — 1 81. y = ~\ x + 4^ 83. y = jjc + ^
32 15
85.-3 87 .- 4 + 89 .- 4 + 91.-1
93+ 95. f
6.2 Guided Practice (p.333) 3. multiply by 5; do not
reverse 5. divide by 4; do not reverse 7. multiply
by —6; reverse 9. not equivalent 11. equivalent
13. not equivalent
Student Resources
6.2 Practice and Applications (pp. 333-335)
15. multiply by 3; do not reverse 17. multiply by 2;
do not reverse 19. divide by —7; reverse 21 . divide
by —3; reverse 23. solution 25. solution 27. Not
equivalent; 12y > —24 is equivalent to y > —2.
29. equivalent 31. equivalent 33. Reverse the inequality
sign when dividing by — 3; x < —5.
35. p < 4 37. j < -18
■ I I I I I I I I I —»- « I I I + I I I I I I I >
-4 -2 0 2 4 -20 -18 -16 -14 -12
39. n > -60
—HO I I I I I I I I I
-60 -40 -20 0 20
43. a > 20
I I I I I I I + I I I
-10 0 10 20 30
47. d< 5; 1.999 - 2
■ I I I I I I I I I I I I I I 0 I I I I
-8 -6 -4 -2 0 2 4 6 8
49. a < -18; 5.91 ~ 6
■ I 111111*111111111111 I
-25 -20 -15 -10 -5
51. always 53. never 55. 20 n > 25,000; n > 1250
57. 31 or fewer rides 63. —14 65.0 67.2 69.-4
9 A
71.-27 73.9 75.-1 77.-2 79 . b =—
h
81. A(4, -2), 5(2, 1), C( —3, -3), D(0, 0) 83. 1, 2, 4, 5,
7, 10, 14, 20, 28, 35, 70, 140 85. 1, 2, 3, 4, 6, 8, 9, 12,
16, 18, 24, 36, 48, 72, 144 87. 1, 5, 17, 25, 85, 425
89. 1, 3, 9, 13, 19, 39, 57, 117, 171, 247, 741, 2223
6.3 Guided Practice (p. 339) 3. not multistep;
subtract 2 5. not multistep; divide by —4 7. multistep;
subtract 12, divide by 5 9. multistep; subtract 2,
multiply by 2 11. multistep; subtract 2w, subtract 2,
divide by 4
6.3 Practice and Applications (pp. 339-341)
13. 14, 14; -7; -7; 7 15. subtract 11, divide by -2 and
reverse inequality 17. subtract 22, divide by 3
19. divide by 6, add 2; or distribute 6, add 12, divide by 6
7
21.x < 5 23. — < x 25. X>-8 27. x>-3
o
33. x < 12 35. 6 < x 37. x < -1 39. x > -
14
41. x > —— 43. In line 2, distribute the 4 over — 1 and
7
distribute 3 over 1 ;/>——. 45. ft < 16; you may
purchase up to 16 tickets. 47. 0.75f + 14 < 18.50
49. 2x + 18 > 26; x > 4 m 51. ^(8x) < 12, x < 3 ft
55. 3 57. 3 59. h = 4 + a 61. $77.48 63. | 65. ^
Quiz 1 (p.34i)
i. 2.
< 1 11111 10 111 * ■ 110 111 11111 *
6 8 10 12 14 -10 -8 -6 -4 -2
3.
« I I I 0 I I I I I I I
-10 -8 -6 -4 -2
4. a < 5
« I I I I I I 0 I I I I
0 2 4 6 8
5. in < -8
- I I I + I I I I I I I
-10 -8 -6 -4 -2
7.Z> -21
r 1*1111111
-22 -20 -18 -16 -14
9. —7 < k
< 11110 111111
-10 -8 -6 -4 -2
6. -12 > b
- I I I I I 0 I I I I I
-16 -14 -12 -10 -8
8. x > 36
< 1111111 + 111
33 34 35 36 37
10 . h > 52 11 . 8 or fewer plays 12 . -2 > x
13. x<-3 14. x<2 15. x>-2 16. 7>x
17. 17 < x
6.4 Guided Practice (p. 345) 3. A 5. (4 + x) is
greater than 7 and less than 8. 7. ( — 8 — x) is greater
than or equal to 4 and less than 7. 9. — 4 < x < 4
6.4 Practice and Applications (pp. 345-347)
11. x is greater than or equal to —23 and less than or
equal to —7. 13. x is greater than or equal to —4 and
less than 19. 15. 2 < x < 3 17. —2 < x < 2
19. 0 < X < 5 21. -4 < X < -2
- I I I + I I I I 0 I I > < I I I 0 I + I I I I I ■
-2 0 2 4 6 -6 -4 -2 0 2
23. 85 </< 1100 25. 15 =
27. 85,000 < c < 2,600,000
29. 12 < x < 14
< I I I I I 0 I + I I I
8 10 12 14 16
37. 4 < x < 7
< 11111
0 2 4 6 8
/< 50,000
35. -4 < x < 3
■ I + I I I I I I + I I
-4 -2 0 2 4
39. -2 < x < 10
—40 I I I I I I I I I I I + I
-2 0 2 4 6 8 10
41. -16 < X < -14
- I I I + I 0 I I I I I
-18 -16 -14 -12 -10
43. 2 < x < 4
<1111111 0+n+-4
-4 -2 0 2 4
51. 24 53. 2 55. 20 57. -6 59. -8 61. 16
63. -12 65. -7 67. more than 25 times
69. 262 million 71.37.5% 73. 3375.75%
77. 84%
6.5 Guided Practice (p.35i) 3. B 5. A 7. all real
numbers less than 10 or greater than 13
9. x < —6 or x > —1
^40 I I I I 0 I I I I *
-6 - 4-2 0 2
6.5 Practice and Applications (pp.351-353) n.all
real numbers less than or equal to 15 or greater than or
equal to 31 13. all real numbers less than or equal to
—7 or greater than 11 15. x < — 3 or x > 0
17. x < 7 or x > 8
19. x > 7 or x <0 21 . x < —2 or x > 5
4+ I I I I I I
-2 0 2 4
6
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
23. x > -1 orx < -4
*1 + 11 + 111111
25 . x < —6 or x> —2
53.
I I I 0 I I I + I I I
-8 -6 -4 -2 0
29. x < 10 or x > 12; solution 31. x < — 3 or x > 2;
not a solution
33. x < -2 orx > 4
35. x > 6 or x < — 3
I I I 0 I I I I I + I
-4 -2 0 2 4
37. x < -2 6>r x > 1
*1110 11 + 1111
-4 -2 0 2 4
I I I I 0 I I I I I I I I
- 6-3 0 3
39. x < — 8 or x > — 2
<>11+111111
-8 -4 0 4 8
41.
t (sec)
0
0.5
1
1.5
2
v (ft/sec)
-4
-2
0
2
4
The velocity of the yo-yo decreases until it reaches the
bottom of the string and then as the yo-yo ascends, the
velocity increases. At 1 second, the yo-yo has reached
the bottom and has a velocity of 0. From then, it rises
and gains speed.
43. t < 32 or t > 212 45. y < 11 or y > 65
49.
51.
53.
X
0
1
2
3
4
y
2
5
8
11
14
X
0
1
2
3
4
y
5
4
3
2
1
X
0
1
2
3
4
y
-4
-2
0
2
4
55.
57.
-5-4 6
I I ++ I I I I I I I I I + I
-6 -4 -2 0 2 4 6
I I + I + !• I I I
n 1 1 3 1 5 3 7 ,
59. 1.20 61. 6.65 63. -0.29
65. 9 < x 67. x>25
I I I I I I I 0 I I I
6 7 8 9 10
I I I I I I I + I I I
22 23 24 25 26
71. X <
-*^0 I I I I I I I I I *
-8 -4 0 4 8
73.28 75.221 77.28,000 79.5400 81.11,000
6.6 Guided Practice (p. 358) 3. 2 5. none
7. x — 4 = 10, x — 4 = —10
9. 3x + 2 = 6, 3x + 2 = —6
6.6 Practice and Applications (pp. 358-360)
11. 9, —9 13. no solution 15. 100, —100 17. 7, —3
19. 12, -12 21. 10, -2 23. 3.5, -3.5 25. 10, -4
27. 18, -18 29. always 31. always 33. 6,-5
35. -1, -4 37. 8, -1 39. 11, -7 45. Sample
answer: |x — 21 = 8 47. midpoint: 92.95 million miles;
distance: 1.55 million miles
7
3y
15
J
1
5
-1
l
3
X
L
-
55. y = — 5x + 20 57. y = 4x — 12 59. y = ~2x 1
61.48,000 63.47,500 65.47,509.13
Quiz 2 (p. 360)
i. 3 < x < 12
111111110 1111 *
2 4 6 8 10 12 14
3. -4 < x < -2
- I + I + I I I I I I I
5. x < — 9 or x > —4
-h-K > I I I I 0 I I I
-10 -8 -6 -4 -2
2. -9 < X < 7
4. x > 5 6>r x < —5
I I I I I I I I
- 6 - 4-2 0 2 4 6
6. x < — 1 or x > 5
I I I I I
-2 0 2 4 6
7. —128.6 < T < 136 8. 14, —14 9. no solution
10.33,-15 11 . -9,-21 12.18,-6 13.7,-11
14. Sample answer: \x — 7.51 = 10.5 or |2x — 151 = 21
6.7 Guided Practice (p. 364) 5. not a solution
7. solution
6.7 Practice and Applications (pp. 364-366)
9. x > 1, x < — 1; or 11. x — 1 < 9, x — 1 > —9; and
13. 10 + lx > 11, 10 + lx < —11; or
15. —15 < x < 15
KH-
-h
21 . x > 12 or x < —4
- I I I I I CD | | | | | | | dH -
-12 -8 -4 0 4 8 12
27. never 29. always
31. -2 < x < 1
I I I CH —\-0 I I I I
-4 -2 0 2 4
19. x >30 or x < —10
*1 + 1111111 + 1
-10 0 10 20 30
25. -16 < X < -2
- I + I I I I I I + I I
-16 -12 -8 -4 0
37. x > 1 orx < — ^
1—h oi I I I 0 I I I I
-4 -2 0 2 4
39. x > 8 or x < -2
41. 0 :
I I I I + I I I I + I
I I I + I I I I I + I
43. t < 3 or t > 7 45. orange 51. all real numbers
except 4 53. $38 55. Sample answers: (—12, 0),
f 2
(—12, 3), (—12,—4) 51 . Sample answers: (^-,0),
(i- 1 )'(i- 5 )
59. function 61. not a function
63.5jf 65.11-1 67.19f
Student Resources
6.8 Guided Practice (p. 370) 5. B 7. to the right
9. solution 11. not a solution 13. not a solution
6.8 Practice and Applications (pp. 370-373)
15. Both (0, 0) and (— 1, -1) are solutions. 17. (0, 0) is
a solution; (2, 0) is not a solution. 19. Neither (0, 0) nor
(2, —4) are solutions. 23. solid 25. dashed 27. y = x;
solid 29. y = ^ x — 8; solid 31. y = —2x — 3; dashed
33. solid 35. yes
.y
/
/
-i-
/
2
5 -1
L
7
/>
[
5
2x -
4
/
/
.y
/
/
/
-l
/
5 -]
L
7
/>
[
5
2x-
/<1
4
/
j
6
-10'i
-2
>
JC
s s
s
V
6
x + 2y <
-10
s
1 II II
51. Sample answer: (1, 3), (2, 2), (3, 1)
53. y < ~2x + 3200
57. 15 59. 69 61. 30°C 63. m = b = -2
65. m = -3, b = 7 67. m = 0, 6 = 5 69. 52%
Quiz 3 (p. 373)
1. x > 18 or x ^ —18
-18 18
■ I l + l 111111111111111 W -4
-20 -10 0 10 20
3. -9 < JC < -5
« I I 0 I I I 0 I I I I
-10 -8 -6 -4 -2
5. -16 < X < 9
-16 9
- I 11 l + l 111 I I 111 I I W I I I 11
-20 -10 0 10 20
2. x > 5 or x < 3
-2 0 2
4. 1 < X < 7
4 6
0 2 4
6 8
6.1 > 5 ori <
-6
H-K) 1 1 1 1 1 1 1
-6 -4 -2 0 2 4 6
7. t < 0.75 or t > 2.25 8. Both (0, -1) and (2, 2) are
solutions. 9. (0, 0) is not a solution; (—4, 1) is a
solution. 10 . (2, 1) is not a solution; (—1, 2) is a
solution. 11 . (1, —1) is a solution; (2, —3) is not a
solution.
\
y
\
V
\
J
/< - 2x
\
\
\
5
-1
\ l
5
X
\
\
O .
\
5
\
\
\
1
y
\j
\
\
5
i i
\1
3
X
\
3 x
• +
K >1
3
\
\
\
1
6.8 Technology (p. 374)
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
51.
13. y > x
Chapter Summary and Review (pp. 375-378)
i. x < 2
"IMI
-4 -2
•
0 2 4
5. 8 < x
I I I I
1 1 1 1 1 -
* 1 1 1 1
2 4
6 8 10
9. n < -6
1 1 1 1 1 1 1 >
-8 -4 0 4 8
3. 2 < x
<1111 1
-4 -2
m
0
2
1 1 >
4
7. 27 < p
x i i i i i
1 1
A. 1
i i
* i i i i I
24 25
1 1
26
¥ 1
27
1 1 ■
28
11. / < 56
i i
1 1 1 r
32 40
1 1
48
56
III*
64
y
O
x<2
J
-3
-1
l
5
X
o _
5
y
O
J
5
-1
' 3
X
o.
x -
3 K >
3
J
>y
s s
3“
s (
s
3a
• +
6/
<
12
s <
S
s _
X
5
-1
]
L
5
V
3
Maintaining Skills (p.381) i. 70 3.37 5.69 7.51
9.-17 11.7 13.3 15.7
Cumulative Practice (pp.382-383) i. 7 3.45 5.3
Input n
0
1
2
3
4
5
6
Output C
65
66
67
68
69
70
71
13. x>2 15. x>5 17. x < -4- 19. x>l
21.x < 13
23. -1 < X < 5
■ 1*11111*1
25. 2 < x < 4
27. 8 < x < 40
20 30
29. x < -5 or x > —2
- I I + I I 0 I I I I I
-6 - 4-2 0 2
31. x < 1 or x > 1
* I I 4 I I I I I
33. x < 2 or x > 10
— >11 110 1 11
6 8 -4 0 4 8 12
9.2.5 11.-18 13.4.6 15. 83°F 17. 18 + 3x
19.-15 + 5/ 21.116 + 7 23. 6y + 6 25.-18
27. 24 29. -4 31. 10
39. 52 mi/h 41. 25 ft/sec
J
J
1
C( 2, 0)
3
-1
1
3
X
A(0,
—2)
B(-3
, -3)
3
33. 75 35. 1 37. 12.5 cm-
B(—2,
' 4) «
J
J
1
5
-1
1
1
5
X
11
C(0, -1)
3
A( 1, -
-4)
35. no solution 37. 9, —9
41. -2 < x < 2
■ I I ♦ I I I ♦ I I I •
45. -3 < x < 5
• I I I 0 I I I I I I
-6 - 4-2 0 2
AO / 13 ^ 11
49. X < —y or x >
HHol I I I I I I I I I I loHHK-
-6 -4 -2 0 2 4 6
39. no solution
43. 2 < x < 18
- I I + I I I I I I I + I I
0 4 8 12 16 20
47. —2 < x < 10
< I I 0 I I I I I
-4 0 4 8 12
47. The sales of catfish have increased since 1990,
although not consistently. There are points clustered
around sales of $370 to $380 million.
Student Resources
47.
49. —19 < X < 9
I » I I I I I 4 I I I
-20 -10 0 10 20
55. y = — 2x + 5 57. y = 4x + 1 59. y = x + 1
61 . y = ^x - ^ 63. y = ~3x — 4
65. y — 4 = x — l or y — 6=x — 3
67. y + 7 = — 8(x + 1) or y — 1 = — 8(x + 2)
69. y — 7 = J(x - 4) ory - 10 = J(x - 8)
71.x < 2 73. x > -7 75. x> 4 77. -5 < x < 2
79. x > 4 orx ^ — 2 81. — 6 < x < 1
Chapter 7
Study Guide (p.388) i.C 2. B 3. A 4. D
7.1 Guided Practice (p. 392) 3. y = x - 2;
y = — 2x + 10 5. (4, 2)
7.1 Practice and Applications (pp. 392-394)
7. solution 9. not a solution 11. not a solution 13. (4, 5)
15. (3,0) 17. (6,-6) 19. (-3,-5) 21. (-4,-5)
23. (1, 4) 25. 125,000 miles 27. 14 years 33. 4
35.5 37.-2 39. y = x + 7 41. y = -2x — 9
43. y = — 3x + 2 45. 4.764 47. 2 49. 10
7.1 Technology (p.395) i. (-3.5, 2.5)
3. (-0.8, -2.05)
7.2 Guided Practice (p. 399) 3. Equation 2; y has a
coefficient of — 1 5. x=l 7. (—5, 18) 9. (1, 3)
51. x < 1 or x > 3 <111111 OH-0 I I ■
-4 -2 0 2 4
53. 1, 3 55. 1, 3 57. 1, 2, 3, 6 59. 1, 3
7.3 Guided Practice (p.405) 3. 9x + lx = 16x;
24 + 8 = 32; Solution: (2, 2) 5. Sample answer:
multiply equation 2 by —4, then add and solve for x.
Solution: (1, -1)
7.3 Practice and Applications (pp. 405-408)
7. (-3, 7) 9. (2,0) 11.(3, 5) 13. (-8, 6) 15. (3, 0)
17. (3, 2) 19. (2,0) 21. (j, 5 ) 23. (21,-3)
25. (8,-1) 27. (3, -4) 29. 1-79, -y) 31. (1,2)
33. (1, 0) 35. (2, 1) 37. (2, 0) 39. (3, 2) 41. (2, 0)
43. about 3 cubic centimeters 45. There are 15,120 men
and 20,000 rolls of cotton. 49. y = 3x + 10
51. y = ~3x + 30 53. y = x - 1 55. (1, 3) is a
solution; (2, 0) is not a solution. 57. ( — 3, —2)
59. (10, -2) 61. true 63. true 65. false
Quiz 1 (p.408) i. (3, -4) 2. (0, 0)
3. (6, 8) 4. (1, 9) 5. (-1, 3) 6. (-6, 10) 7. (6, 8)
8. (5, 1) 9. ( —,£) 10.(2, -1) 11.(0, 1) 12.(2, 1)
13. Four compact discs were bought at $10.50 each and
6 were bought at $8.50 each.
7.2 Practice and Applications (pp. 399-401)
11. Equation 2; m has a coefficient of 1, no constant.
13. Equation 2; x and y have coefficients of 1.
15. Equation 2; x has a coefficient of 1. 17. (9, 5)
19. (4, -2) 21. (-1,5) 23. (0,0) 25. (-7, 4)
27. 29 - 30 11-inch softballs and 50 12-inch
softballs 31. $3375 in ABC and $1125 in XYZ
33. 1200 meters uphill, 1000 meters downhill 39. -2x
41.26
\
\
\ 0
\ j
\
Y-\*
+ 1
>
*\
-3
\ 3
X
\
—3 —
\
\
7.4 Guided Practice (p.412) 3. (7.5, 0.5)
5. You would have to sell $600,000 of merchandise.
7. 10 d
7.4 Practice and Applications (pp. 412-414)
9 . (0,2) 11 .( 3 , 6 )
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
13. Sample answer: Multiplication and addition. No
variable can be easily isolated. 15. Sample answer:
Substitution. Equation 2 can be solved for v or y.
17. Sample answer: Substitution. Equations 1 or 2 can
be solved for x. 19. (3, 3) 21. (^, 23. (-2, 1)
25. ( — 3, 2) 31. 6 pea plants, 7 broccoli plants
33. about (1.6, 6474) 39. parallel; m = 4 for both lines
41. not parallel; different slopes
/o
:)
\
\
J
5
-J
L i
\>
L
5
X
\
3
\
f(x) =
4x
\
\
1 11 23 25
49 1— 51 1— 53 — 55 —
5 72 30 32
• y
O
J
2 >
3 k
<
6 ^
1 "
/ '
-3
-1
1
A
/ :
5
X
1 “
/
O
✓
/
J
>
/
47. 20; 49. 20; 20
7.6 Guided Practice (p. 427 )
7.5 Guided Practice (p. 420 )
5. No solution; the two
equations represent
parallel lines.
7. no solution 9. one solution; (5, 12)
7.5 Practice and Applications (pp. 420-422)
17. no solution 19. no solution 21. one solution
23. Infinitely many solutions; multiplying Equation 1
by 4 yields Equation 2. 25. infinitely many solutions;
one line 27. infinitely many solutions; one line
29. no solution; parallel lines 31. No; there are infinitely
many solutions for the system. 33. Yes, $14.98. Sample
explanation: The solution of the system 4x + 2y = 99.62
and 8* + y = 139.69 is (14.98, 19.85).
39. about 4:27 P.M.
5. The student graphed y > 1, instead of y > — 1;
graphed v < 2, instead of v > 2; graphed y < v — 4,
instead of y > x — 4. 7. y < — x, x > —2
7.6 Practice and Applications (pp. 427-430)
\
p 1
\ 2x+ K>
2
\
\ X <3
--
F
-3
-1
K
X
2y< 1
\
\
\
3
\
'
k
25. Sample answer: 2y — x < 4, 2y — x > —4
27. Sample answer: y > 0, y < — v + 2, y < x + 2
29. Sample answer: y > 0, 3y < — 5v, 4y < 5v + 35
31. b + c > 240; b < c\ 5b + 3c < 1200
Student Resources
37. b + c < 20, 5Z? + 6 c > 90 39. Sample answer:
5 hours babysitting and 15 hours as a cashier; 15 hours
babysitting and 5 hours as a cashier 41. y > 0, x > 0,
y<-x + 4 45.243 47.137 49.62 51.49
53.-60 55. 38 5-point questions and 30 2-point
questions 57.9.25 59.2.8 61.3.8 63.6.875
Chapter Z
Study Guide (p. 440) i. A 2. C 3. D 4. D 5. A
8.1 Guided Practice (p. 446) 5. (-5 ) 6 7. 2 12 9. y 20
11. 16ft 4
Quiz 2 (p. 430) 1 . / = 8 ft, w = 3 ft 2. premium gas costs
$ 1.57/gallon, regular gas costs $ 1.35/gallon 3. no
solution 4. one solution; (0, 1) 5. infinitely many
solutions
-
y<-x +3
\
ON
J
\
\
-3
-1
1
k
X
\
\
'J _
J
>y
3
-x
r +
y-
s 1
/> 0
-3
j
X
3
X
:+ y< 1
: ■
/
5x
-3k
<
/
/
5
/
/
/x
-2 y<-
6^
-3'
/
/
/
/
1
-3 /
f -l
1
3
X
L
9. Sample answer: x + 2y < 4, —x + y > — 1
Chapter Summary and Review (pp. 431-434)
1.(9, -3) 3.(0, 1) 5.(4,-^) 7. (0,3) 9. (§,§)
11- (|,o) 13. (-ff.ff) 15. (3, -5) 17. (- 1 , 1 )
19. 2 regular movies; 3 new releases 21. no solution
Maintaining Skills (p.437) i. 125 3.9 5.0.47
7.0.035 9.61% 11.200%
8.1 Practice and Applications (pp. 446-448) 13. 5
15.18 17.7 19. 4 9 21. (-2 ) 6 23. x 9 25.3 27.12
29. 9 31. 2 6 33. (—4 ) 15 35. c 80 37. 441 39. 576
41. 64 d 6 43. 64m 6 rt 6 45. -r 5 s 5 t 5 47. < 49. <
51. > 53. -Ax 1 55. r 8 ? 12 57. 18x 5 59. a 4 b 4 c 6
61. V= 36Trn 3 ~ 113.1a 3 63. 8 , or 8 to 1 65. 2 1 = 2 ,
2 2 = 4, 2 3 = 8 67. 2 30 = 1,073,741,824 pennies
87. x< 7 89. x< 1 91.x
97. false; 1
8.2 Guided Practice (p. 452) 3. 1 5. 64 7. 2
9.n.0.0016
to
13.0.0156 15. 17.3c 5
8.2 Practice and Applications (pp. 452-454) 19.
1 1
2’ 5’
21. 1
1
25. -
o 23 -T6
33. M 35. | 37. 400
l
343
27. 256 29.
i si 4
1 39.-^ 41.0.0313 43.0.0016
to
45. 0.0625 47. 0.0714 49. The 5 should not be raised
5 1 y^
to a negative power; — 51. — 53. — 55. x 2
1 ^ 916 ^" X
57. x 10 v 4 59.-r 61.—r- 63. about 5.31 million
7 64x 3 x 9
people 73. 4 75. 2
81. 15 83. -15
85. -13 < x < -5
-H-0 I I I I I I I CH-t-
-15 -13 -11 -9 -7 -5 -3
27
77. Y = 13.5
79. -9
87. -6 < X < 1
11 * 111111*1
9 -7 -5 - 3-11 3
20 _20
89. x > 2 or x < — -z- 3
3 -HcH—h
I I I I
91. (—|, -lj 93. (5, 0) 95. (2, 3) 97. Sample
6 9 12 „„ „ , 2 3 4
answer: — » Y 5 ’ 20 99 ' ^ am P^ e answer: "15 » 24 » 32
101. Sample answer:
103. Sample answer:
30 45 60
32 ’ 48 ’ 64
50 75 100
64 ’ 96 ’ 128
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
8.3 Guided Practice (p. 458)
y
y
= L
\ x
14
1
10'
1
1
6
/
/
/
3
'l
X
5. domain: all real numbers; range: all positive real
numbers
8.3 Practice and Applications (pp. 458-460) 7. yes;
2° = 1 9. no; 2(3)° = 2
7
71 L 0 = 7
1 \°
11. yes; I —) = 1 13. no;
X
-2
-1
0
1
2
3
y = 3 x
1
9
1
3
1
3
9
27
-2
-1
0
1
2
3
y = 5(4) x
5
16
5
4
5
20
80
320
-2
-1
0
1
2
3
>-($
36
6
1
1
6
1
36
1
216
-2
-1
0
1
2
3
y = 21
(1)
f
98
14
2
2
7
2
49
2
343
23. 55.90 25. 45.25 27. 0.00 29. 1.06
y
4
3
Tn
5
X
— 4 -
\
-12
\
-3((
3 )*
K :
-20'
-28-
1 :
43. domain: all real numbers; range: all negative real
numbers 45. domain: all real numbers; range: all
negative real numbers 47. domain: all real numbers;
range: all positive real numbers 49. domain: all real
numbers; range: all positive real numbers
55.0.38 57.-0.46 59.-1.91 61. 8* + y = 4
63. lx — 8y = 0 65. 3x + 16 y = 9 67. 1 solution
69. no solution 71. infinitely many solutions
73. -5, -4, 6 75. -3^, -2|, —2j 77. 3.001, 3.01, 3.25
Quiz 1 (p. 460)1. 59,049 2.64 3.1600 4.36 5.^
6.1 7. r 13 8. k 8 9.9 d 2 10. TL 11-At”
12. — — 13. about $1008; about $2177
m n
• y
K
= 1
0 X
"90'
70'
50'
30'
/
10
y
3
■ i
[
5
X
Student Resources
8.3 Technology (p.46i)
7. Sample answer: For a > 1, the graph of y = a x is a
curve that passes through (0, 1) and increases to the
right. The graph of y = ~a x passes through (0, — 1) and
decreases to the right. Both graphs approach the v-axis to
the left.
8.4 Guided Practice (p.465) 3. 125 5. -32 l.x
3
9.
1
11.4 13 - 256
m
17.
m yj 32 81 n
8.4 Practice and Applications (pp. 465-468) 19. 4
21.11 23.6 25.125 27.1
1 „ 8
r — 4b.
y
49. 5x 3 y 3 51. 6a% 3
35. 3 37. 10 39.
47
53.
6a \ 3
b V
96x 4
6 3 a 3
b 6
29.- 31. 33.1296
Jt r 3
41 43 — 45 —
625 27 *4 a >5
216a 3
55.
V
57
b 6
9 x 2 y 2
59. —0.437 61. 200,
y 3x 3 2
160, 128, 102, 82, 66, 52 63. product of powers property;
quotient of powers property; product of powers property;
canceling a common factor 69. 100,000 71. 1
1
73. y = 2 * + 4 75. y = — x —
77. y = ~x + 3
79. solution 81. not a solution 83. (8, 4) 85. (4, 3)
87. (9, —1) 89-93. Estimates may vary. 89. 450
91. 80.5 93. 1750
8.5 Guided Practice (p. 472) 3. 430 5. 0.05
7. 0.245 9. 6.9 X 10 6 11. 9.9 X 10 _1
13. 2.05 X 10“ 2 15. 2 X 10“ n
8.5 Practice and Applications (pp. 472-474)
17. right, 2 19. left, 7 21. 8000 23. 21,000
25. 433,000,000 27. 0.009 29. 0.098
31. 0.00000000011 33. in scientific notation
35. 9 X 10 2 37. 8.8 X 10 7 39. 9.52 X 10 1
41. 1 X 10 _1 43. 6 X 10“ 6 45. 8.5 X 10“ 3
47. 1.23 X 10 9 49. 1.5 X 10 5 51. 7.0 X 10“ 4
53. 2.7 X 10 7 55. 4.0 X 10“ 2 57. 1.09926 X 10 6 ;
1,099,260 59. 1.5 X 10“ n ; 0.000000000015
61. =7.9626 X 10“ 19 ; =0.00000000000000000079626
63. 0.00098 65. 2 X 10 -23 67. about $18.12 per
square mile 69. about (4.87 X 10 I4 )tt km 3 or about
1.53 X 10 15 km 3 73. no solution
\
y
\
!x -
t- y
>:
?
V
(<
2
\
2
\
\
4
-
2
\
X
\
—2
\
\
—4
\
\
79. 212% 81. 67.4% 83. 7.567
Quiz 2 (p. 474) 1 . 7776
6. 3Ox 2 7. sJr|^ 8.
a J
16m 4
81ft 4
4 x 3 y 7
5
10 .
243
wV 1
4
3y 9
11.5,000,000,000 12.4,800 13.33,500 14.0.000007
15. 0.011 16. 0.0000208 17. 1.05 X 10 2
18. 9.9 X 10 4 19. 3.07 X 10 7 20. 2.5 X 1CT 1
21. 4 X 1(T 4 22. 6.7 X 1(T 6
8.6 Guided Practice (p. 479) 3. 0.04 5. about $608
8.6 Practice and Applications (pp. 479-481)
7. C — 100, r = 0.5 9. C = 7.5, r = 0.75
11. y = 310,000(1.15)*; y = population, t = number of
years 13. y = 10,000(1.25) 10 ; y = profit, t = number
of years after 1990 (t < 10) 15. y = 15,000(1.3) 15 ;
y = profit, t = number of years after 1990 {t < 15)
17. $2231.39 19. $4489.99 21. $382.88 23. $510.51
25. $1466.01 27. $1770.44 29.3,4 31.2 33.3
35. about 13.2 L/min, 46.3 L/min, 86.5 L/min 45. 5
47. -2 49. -7 51. 4 53. 2 4 55. 3 7 57. r 6
59. | 61. |
8.7 Guided Practice (p. 485) 3. $6185.20 5. $4266.98
7. C 9. exponential decay 11. exponential decay
8.7 Practice and Applications (pp. 485-488) 13. 18;
0.11 15. 0.5; 0.625 17. y = 100,000(0.98)'
19. y = 100(0.91)' 21. y = 70(0.99)' 23. about $11,192
25. about $8372 27. about 229 mg 29. v = 64(0.5)'
• y
4 j
\
y
= i
°(:
\Y
~
\
\
\
V
\
■3
■1
t
1
35. y = 22,000(0.91)'; about $10,300
37. y = 10,500(0.9)'; about $3700
39. 302, 239, 189, 150, 119 41. about 106 miles
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
2
45. exponential decay; 0.98 47. exponential decay; j
49. exponential growth; ^
51. Sample answer: As Z? increases, the curve becomes
steeper or more vertical. 57. 24 59. 72 61. —0.92
63. -0.64 65. y - 5 = 3(jc - 2) 67. y + 4 = 4(jc + 1)
69. y — 7 = -6 (jc + 1) 71. 2.5 73. 0.2 75. 5.5
Quiz 3 (p.488) 1. $270 2. $314.93 3. $367.33
4. $462.73 5. 1600 raccoons 6. about $12,422
7. about $10,286 8. about $9360 9. about $5841
10 . y = 20,000(0.92)*; about $13,200
11 . exponential decay; 0.1 12 . exponential growth; 1.2
Chapter Summary and Review (pp.489-492) i. 128
3.4096 5. 81x 4 7.8 p 4 9.1 11. T 13. L 15. L
49 y 6 a 2
21. 1 23. L 25. 9y 27. AA 29. 70 31. 0.0002
3 ol IZj
33. 5.2 X 10 7 35. 9 X 1(T 3 37. 1.5 X 10 7
39. 1.44 X 10 7 41. 7 X 10 8 43. y = 2(1.05)'
45 . y= 125(0.97)'
o 5
Maintaining Skills (p. 495) i. 2 3 3 3-5-7 5. 2-
n o
Chapter
Study Guide (p.498) i. B 2 . D 3 . C
9.1 Guided Practice (p. 502) 5. ±11 7.-2
9. irrational 11. rational 13. 14.66, —2.66
15. 13.31, -9.31
9.1 Practice and Applications (pp. 502-504) 17. The
positive and negative square roots of 16 are 4 and —4.
19. The positive square root of 225 is 15. 21. The
negative square root of 289 is —17. 23. The positive
square root of 1 is 1. 25. 12 27. 14 29. ±7 31. —16
33.20 35.11 37.-1 39.13 41. no 43.no
45. yes 47. no 49. no 51. no 53. 2.24 55. 3.61
57.-7 59. ±1 61. ±3.87 63.-4.47 65.3 67.0
69.6 71.7 73.7 75.10.24,5.76 77.-0.34,-11.66
79. -11.24, -2.76 81. 5.13, -1.80 83. -2.90, 0.57
85. m is a perfect square 87. False. Sample
counterexample: the square root of 0 is 0. 95. (2, —2)
97. 116 adult tickets and 208 student tickets
99. (-4, -19) 101. (5, -6) _103. 0.53 105. 0.875
107.0.3125 109.0.4 111.0.8 113.0.9
9.2 Guided Practice (p. 508) 3. 2 5. 0 7. 2 9. ±7
11. ±V7 13. no real solution 15. 1.7 sec 17. 3.5 sec
9.2 Practice and Applications (pp. 508-510) 19. ±1
21. no real solution 23. ±15 25. ±11 27. ±16
29. ±7 31. ±8 33. ±4 35. ±V2 37. ±3 39. no
real solution 41. ±5 43. ±V3 45. ±6 47. ±Vl4
49. The equation has no real solution. 51. ±1.41
53. ±2.83 55. ±1.84 57. True; the solutions of x 1 = c
are Vc and ~Vc. 59. h = -16 1 2 + 96 61. 0.40 mm
63. 0.15 mm 65. 0.12 mm 67. 5,500,400; 22,582,900;
73,830,400 71.-18 73.12 75. 5; 6 77. 8; 2
79. v>-2 81. v < 2 83. 8 X 10 -7 85. 8.721 X10 3
Student Resources
9.3 Guided Practice (p.514) 5. D 7. B 9. 6
11.2V15 13.^1 15 . M
9.3 Practice and Applications (pp. 514-517)
17. no; radical in the denominator 19. yes 21. 2VTI
23. 3V2 25. 3V3 27. 10V2 29. 5V5 31. 12 33. - r
35 f
37 f
39.
VTT
3 V5
41. — 43. —
4 9
Vs
45. V20 = V4T5 = 2V5 47.^ 49. A 51.^-
5 2 3
V3
53. ^ 55. — 57. — 59. 20 61. -6V3
5 113
63. -12 65. -1 67. 3V6 69. -3VT0 71.
73. 2 V 5 75. 70V2 m/sec 77. No; ratio of speeds is
the square root of the ratio of depths. 79. 98
81. Multiplication; square
\ -
>
\
2x + y
'= 6
2
\
6
- 2 o
2 \ (
) X
\
\
6
\
\
97. 8lx 4 99. 144 101 . 64 x 2 y 2 103. - a 3 b 3 c 3
105. domain: all real numbers; range: all negative real
13 2 1
numbers 107. — 109.^ HI-Ton 113 .777
4 o loV 1U
Quiz 1 (p. 517) 1 . 9 2 . -5 3. 4 4.-2 5 . ±1 6. 10
7. ±7 8.11 9. ±8 10. ±V63 or ±3V7 11. ±V6
12. no real solution 13. ±4 14. ±5 15. 3A/2
16 . 2V15 17. V3 18 . -9 19. 4V30 20 . —
3
2 ,.f 22. i 23. f 24.2V2 25. f 26.^
9.4 Guided Practice (p. 523) 3. up 5. down 7. up
9. axis of symmetry: 11. axis of symmetry:
x = 0 v = 0
13. axis of symmetry:
x = l
..y
\(1,5)
/
\
5
-E
1 1 \
3
X
/
1
1
/
\
/
\
1
y = ~3x 2 + 6x + 2
9.4 Practice and Applications (pp. 523-525) 15. up
17. down 19. down 21. down 23. down
25.
( 0 , 0 )
X
-2
-1
0
1
2
3
y
24
6
0
6
24
54
27.
5 _ 25
2 ’ 2
0
1
2
5
2
3
4
5
y
0
-8
-12
25
2
-12
-8
0
29.
_I 3I
6 ’ 6
-2
-1
1
6
0
1
2
y
24
8
3-
0
4
12
32
-3
-2
-1
1
2
0
1
2
y
-16
0
8
9
8
0
-16
■ y
1
\
\
1
1 y = 4x 2
\
/
*2
1
(0,0)
1
J
-3 -1
1
3
X
1
y
c
J
x 2 -
- 2x
- 1
\
\
/
5
-
>,\
J
3
X
r
*4i
-4
y.
6
X.
00
+
3
-(
)
f
X
\\
V
(
— i/ —
1 1
7)
A
1
\
\
1
-3
1
3
X
■■
.
y= -4x 2 + 4x+7
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
^15 609 \
45. ——); this point represents the highest point on
32’ 64
the path of the basketball. At
609
15
32
0.47 sec the ball
reaches its high point of ~ 9.52 ft. 47. 10 ft
55.
6
x -
3>
/ >
3^
*
-]
f
14 x
x- 3
y<
; 1
2
±2.
6
57.
<
X
I 4 '
x - y < 2
: / y
s\
1$
/
/
v
x+ y< 10
/
/
6
\
\
\
\
/
/
\
\
\
/
/
V
\
/
/
\
s
/
\
\
-2
/!
\
i()\^
14 x
i
L
59. (-5) 9 61.X 8 63. m 8 65. 2 5 67.
M 3 7 9
69 ‘ 4’ 8’ 10
2x + y> 10
4_ 1 2
15’ 3’ 5
9.5 Guided Practice (p. 529) 3. B 5. A 7. ± 1
9. ±4 11 . 2, 5
9.5 Practice and Applications (pp. 529-531)
13. x 2 — 6x + 6 = 0 15. 3x 2 — x — 5 = 0
17. 6x 2 - 12x = 0 19. -3, 1 23. -3, 1 25. -3, 1
27. -1,5 29. -5, 1 31. -3, 2 33. -1,2 35. ±5
37. ±5 39. ±4 41. ±9 43. ±2 45. ±3 47. -4, 1
49. —4, 8 51. 10 sec 55. pasta: $5.95; salad: $1.95
57. (3, 2), one solution 59. | —y, oj; one solution
61. no solution 63. 4 65. 0 67. 2 69. -9 71. 2V6
73. 10V2 75. V3 79. > 81. > 83. < 85. <
9.5 Technology (p.532) i. —1,2 3. -0.77, 2.27 5. 2
9.6 Guided Practice (p. 536) 5. 2x 2 - 16x + 32 = 0;
a = 2, b = — 16, c = 32 7. -7, 1 9.-6
11 -1 ± VT3 13 2 x 2 + x- 6 = 0; -2,f
6 l
15. x 2 — x — 2 = 0; — 1, 2 17. x 2 — 4x + 3 = 0; 1, 3
9.6 Practice and Applications (pp. 536-539)
19. 3x 2 — 3x — 6 = 0; a = 3, b = — 3, c = —6
21 . x 2 — 5x + 6 = 0; a = 1, b = —5, c = 6
23. 3x 2 — 24x + 45 = 0; a = 3, Z? = —24, c = 45
25. k 2 — ^ = 0; <2 = l, b = 0, c = — ^
27. |-x 2 + 2x = 0; a = j , b = 2, c = —29. 9
31. 1 33. 169 35. 148 37. 21 39. 39 41. -1,-10
43. 2 45. -1.30, -0.26 47. -1.87, 13.87
49. lx 2 — 4x — 30 = 0; —3, 5 51. x 2 + 6x — 5 = 0;
—3 ± Vf4 53. 2x 2 — 5x — 1 = 0; -1, |
55. x 2 — 2x — 3 = 0; — 1, 3 57. 2x 2 — 2x — 12 = 0;
-2, 3 59. -2, -3 61. -2, -8 63. -1, 4 65. -3, 1
67. 2.30 sec 69. 2.21 sec 71. 0.92 sec 73. 0.4 sec
83. D 85. -27
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6
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96. > 97. < 99. < 101. <
Quiz 2 (p. 539) 1. up 2. up 3. down 4. up 5. down
6 . down
X
1
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2)
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y = —x 2 + 2x — 3 y — 3x 2 + '\2x 10
10. -2,5 11.6 12. -1,-3 13.-3 14.-6,“
15. -2, 8 16. |, 2 17. 1.17, -2.84 18. 1.55, -0.22
9.7 Guided Practice (p. 543) 5. one solution 7. B
9. A 11.2
9.7 Practice and Applications (pp. 543-545)
13.49 15.-40 17.0 19.-111 21.-40
23. no solution 25. two solutions 27. no solution
29. two solutions 31. two solutions 33. one solution
35. 60 37. It crosses the x-axis at two distinct points.
39.0 41.0 43.1 47. domain: 0 < t < 5;
range: 9.29 < P < 161.49 49. about 8.5 years
53. 1 < x < 4 55. -3 < x < 5
- I I I I I I + I I CH— « I I I 6 I I I I I I I
-4 -2 0 2 4 -5 -3 0 5
75. 1.4 sec 77. 5.7 sec 79. about 5.04 sec 81. A
Student Resources
3
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t
59.0.06 61.0.0 1 63.0.0018
9.8 Guided Practice (p. 550) 3. inside 5. outside
7. (0, 0), yes; (1, -2), no
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9.8 Practice and Applications (pp. 550-552) 15. yes
17 . no 19 . no 21 . outside 23 . inside 25 . sometimes
27 . sometimes
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41 . y = lx
43 . y = 2 X
45 . y
—2x
53 — 55 A
25 55 ‘ 20
5, -ilo 6 '-4 es i l
65.
91
100
67.
Quiz 3 (p.552) 1. two solutions 2 . one solution 3. no
solution 4. No. Sample answer: the vertical motion
model is h(t) = -16 1 2 + 50 1 + 5. If you let h(t) = 45
and solve for t , you have the quadratic equation
—16^ 2 + 50 1 — 40 = 0. The discriminant has a value of
2500 — 2560 = —60, so there are no solutions.
5. A 7. B
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Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
5. :
13.
y
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y = x 2 - 5x + 4
y = 2x 2 - 3x - 2
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0
< y —
-x 2 -
2x
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57. 243 59. 4 ? 61
to
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65. 1.5 X 10 3 67. 6 X 10 10 69. 8 X 10° 71. 2V10
73.6V2 75.^
77. 3V2 79.^ 81.
85.
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f < 2x 2 - 5x + 2
87. No. Discriminant for —16^ 2 + 100^ —180 = 0 is
— 1520, so there is no real solution.
1 13
17. 5, 1 19. 1, — 21. —, j 23. two solutions 25. 2 27. 1
29.
Chapter 10
Maintaining Skills (p. 559) i. 16* - 96 3. 5m - 65
5. 30a + 80 7. 21m + 6 n 9. 2x + 10
Chapters 1-9 Cumulative Practice (p. 560-561)
1 . No. Each input value can only have one output value.
5 has two. 3 . 3x — 6 5. 9 + 2 h 7. 1.25* = 60;
48 pretzels 9. 360 11. 400% 13. —3; ^ 15. -y; 14
17. -2; -28 19. Yes; slope of both lines is 4.
21. y = —x + 10 23. y = ~2x — 2 25. y = ~
27. 3* — 5y = -6 29. -2x + 7y = 15 31. * + 4y = 24
33. m < 9 35. f<-8 37. y > 14 39. y > -4
41. k > 7 43. (2, 6) 45. (-5, -4) 47. (l, --
Study Guide (p. 566) i. B 2. C 3. D
10.1 Guided Practice (p.57i) 3. linear binomial
5. quadratic binomial 7. cubic trinomial 9. — 3x 2
and — 5* are not like terms; 9x 3 — 3x 2 — 5x — 2
11. 3x — 16 13. 4x 2 — lx — 2
10.1 Practice and Applications (pp. 571-573)
15. always 17. sometimes 19. always 21. 4
23. 4 25. 20m 3 ; cubic monomial 27. —16; constant
monomial 29. lly 3 — 14; cubic binomial
31. lb 3 - 4 b 2 \ cubic binomial 33. ~6x 3 + 4x 2 - 6
35. —7m 2 + 7m — 3 37. —6 39. 3x 2 — 5
41. z 3 + 1 43. ~n 5 + 3n 2 + 3n — 5
45. 25x 3 + 8v + 2 47. jt 2 + 2x + 2 49. -3jc 2 + 6
51. 1.5* 2 + 60* 53. A = 1.38 It 2 + 3.494f + 235.325
59. 5* - 2 61. -15* + 9 63. -lx ~ 55 65.32
67. 256 69. 256 71. 1.295 73. 4
75. 1“
77. 3-^ 79. 12f 81.3-i 83.15§
10.2 Guided Practice (p. 578) 3. (* + 3), (* + 3)
5. 3 7. 20 9. — 8* 2 - 14* 11. —12* 4 - 8 * 3 + 24* 2
13. y 2 + 6y — 16 15. w 2 + 2w — 15
17. 8* 2 - 29* - 12 19. * 2 + * - 56
10.2 Practice and Applications (pp. 578-580)
21. -8* 2 + 20* 23. 2* 3 - 16* 2 + 2*
25. 12w 5 - 8w 4 - 4w 3 27. t 2 + 13f + 40
29. d 2 — 2d — 15 31. 2y 2 + 5y + 2 33. 3 s 2 4- 5s — 2
35. 8y 2 - 18y + 7 37. y 2 - 3y - 40
39. 2w 2 + 5w - 25 41. 2* 2 - 3* - 135
43 . 6z 2 + 25z + 14 45 . 10 1 2 + 9t - 9
47. 63w 2 - 143w + 60 49. d 3 - Id 2 + 4d + 30
51. 6* 3 + * 2 — 8* + 6 53. a 3 + 4a 2 — 19a + 14
55. 4y 3 + 45y 2 - 38y - 24 57. 21* 2 + 100* + 100
59. R = —3.15 1 2 — 6.2 It + 989.12, in millions of dollars
61. 2* 2 + 7* + 3 65. 49* 2 67. ^-y 2 69.9 s 71. b 1
73. 432f 4 75. — 108* 3 y 5 77. two solutions 79. two
solutions 81. one solution 83. two solutions 85. two
solutions
Student Resources
10.3 Guided Practice (p. 585) 3.x 2 - 12x + 36
5. p 2 + 12 p + 36 7. t 2 - 36 9. F; 9x 2 + 24x + 16
11. T
10.3 Practice and Applications (pp. 585-587) 13. yes
15. no 17. yes 19. yes 21. yes 23. x 2 — 25
25. 4m 2 - 4 27. 9 - 4x 2 29. x 2 + lOx + 25
9x 2 + 6x + 1 33. 16 b 2 - 24b + 9 35. x 2 - 16
2 _ oc /M „ 2 _ at ^2
31
37. 9x z - 6x + 1 39. 4y z - 25 41. a z - 4b
2
43. 9x 2 - 16y 2 45. 81 - 16? 2 47. false;
a 2 + 4ab + 4b 2 49. true 51. (x + 3) 2 = x 2 + 6x + 9;
53. (2x + 4) 2 = 4x 2 + 16x + 16;
55. 9x 2 - 24x + 16 in. 2
57. 25% normal feathers; 50% mildly frizzled;
15x^
25% extremely frizzled 61. jc 63. ^
square of a binomial
square of a binomial
67 l
69.
71 -^7
73 2Z
64
Quiz 1 (p.587) i. 2 2.0 3.3 4.5 5. 3x 2 + 5x + 9
6 . — 6x 3 - 14x 2 + 2x - 2 7. 3t 2 - 13? + 14
8 . 6x 3 + 3x 2 + 4x + 3 9. x 2 + 7x - 8
10 . y 2 + lly + 18 11 . -12x 5 + llx 4 - 3x 2
12 . 4x 2 - 49y 2 13. 16r? 2 - 49 14. 2x 3 - 3x 2 - 6x + 8
15. x 2 - 36 16. 16x 2 - 9 17. 25 - 9 b 2
18. 4x 2 - 49y 2 19. 9x 2 + 36x + 36
20. 64x 2 + 96x + 36
10.4 Guided Practice (p.59i) 3. No; 2 and -5 are
solutions, 3 is not. 5 . no 7 . yes 9 . -1,-3 11.7
10.4 Practice and Applications (pp. 591-593)
15 . - 8,6 17.-3 19.-7 21 . - 2 , -3 23.17
25.-9 27.20,-15 29. -1,-2, 4 31. -5, 6
33. - 8 , -9, 12 35. 8 , - 2 , -2
41.x-intercepts: (—5, —3); 45.x-intercepts: (—4, —3);
vertex: (—4, —1) vertex: (—3.5, —0.25)
\
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> y
1 c
/
\
/
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/
4
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/
V
2
(—4, C
E
/(-3, 0)
| -
Y
-2
X
| (-3.5, -0.25)
47. (0,-14) 49.630 ft 51.200 m 55.0.04443
57. 1,250,000 59. 9,960,000 61. 81,700,000
63. x 2 — 64 65. 6x 2 + 19x — 7 67. 24x 2 — x — 3
69. x 2 + 20x + 100 71. exponential decay;
y = P(0.84)' where P = the average price of the computer
in 1996, and ? is the number of years since 1996.
73. exponential decay; y = MO. 97)' where N is the
number of members in 1996 and ? is the number of years
since 1996. 75. 1, 2, 3, 4, 6, 12 77. 1, 2, 3, 6, 9, 18
79. 1, 3, 17, 51 81. 1, 2, 3, 4, 6, 9, 12, 18, 36
83. 1, 2, 4, 8, 16, 32, 64 85. 1, 2, 3, 4, 6, 7, 12, 14, 21,
28, 42, 84
10.5 Guided Practice (p. 599) 3. A 5. C 7. 5, -1
9. always 11. never
10.5 Practice and Applications (pp. 599-601)
15. (z + l)(z + 5) 17. ( b + 8 )(b — 3) 19. (r + 4)(r + 4)
21. (m - 10)(m + 3) 23. ( b + 8 ){b - 5) 25. 2, -7
27. -1,-15 29. 6,-9 31.4,11 33.5,-13
35. 8, -7 37. -4, -8 39. 2, 15 41. 3, -6
43. base: 8 ft, height: 5 ft 45. 305 m by 550 m 51. 15
53. 1 55. 18 57. y 2 + 5y - 36 59. -3w 2 + 3w + 60
61. 20? 2 - 62 1 + 30 63. -2, -3 65. 6, 9 67. 1, ~
69. -4, 3, \ 71. 11.056 73. 11.86 75. 20.9204
77. 114.8106
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
10.6 Guided Practice (p. 606) 3. -6 5. 10x 2 7. D
9. C 11.(2* + 1)(* - 2) 13. (3* - 4)(4* - 1)
1
15. (3* - 4)(* + 2) 17.
2’
-8
10.6 Practice and Applications (pp. 606-608)
23. (3 1 + 1 )(t + 5) 25. (2a + 1)(3 a + 1)
27. (6b + 1 )(b - 2) 29. 3(x + 1)(2* - 5)
31. (2 z ~ 1 )(z + 10) 33. (Ax + l)(x + 5)
35. (3c - 4)(c - 11) 37. (2 1 + 7)(3f - 10)
39. (2y — 5)(4 y — 3) 41 . Incorrectly factored:
3y 2 — 16y — 35 = (3 y + 5)(y — 7); solutions are —y, 7.
3 1 13 13 5 7
43- f, 5 47. f, -2 49. if 51. f, f
53. y, — — 55. a. h = — 16^ 2 + 8^+8 b. 1 sec; yes
3
57. 2 sec; the other solution of — second is the time it
takes for the T-shirt to leave the cannon and go up to a
height of 30 feet. You would probably catch the T-shirt
as it fell. 61. ( — tt, — 63. I6t 2 — 8f + 1
65. 9x 2 + 3Ox + 25 67. 121 — 132* + 36x 2 69. yy
71 -i 73 f 75 f
Quiz 2 (p.608) i. -5
5. 5, ~ 6. 0, -4, 7
7.x-intercepts: 2,-2;
vertex: (0, -4)
2.-4 3.-y,4
4. 0,
8 . v-intercepts: —3, —5;
vertex: (—4, —1)
10. (y + 4)(y — 1) 11. (w + ll)(w + 2)
12. (n + I9)(n — 3) 13. cannot be factored
14. (b - 8 )(b + 2) 15. (r - 7)(r + 4)
16. (m — 9)(m + 5) 17. (x + 6)(x +11)
18. (r - 43)(r + 2) 19. 1,-6 20.-1,-25 21.5,9
22. -9, -2 23. -f , -5 24. J, 1 25. -f , 4
3 7 2
26 -2 27 “ 3’-5
10.7 Guided Practice {p. 613) 3. (b + 5) 2
5. (w - 8) 2 7. 6(y - 2)(y + 2) 9. (2x - l) 2
13. 7 15. 3 17. 3 sec
11 .
10.7 Practice and Applications (pp. 613-615)
19. (q - 8 )(q + 8) 21. (3c - l)(3c + 1)
23. (9 — x)(9 + x ) 25. (w — 3y)(w + 3y) 27. (x + 4) 2
29. (b - l) 2 31. (3a + l) 2 33. (5 n - 2 ) 2
35. 4(2w - 5) 2 37. (a - 2b) 1 39. 4 (n - 3 )(n + 3)
41. 5(c + 2) 2 43. 9(3? 2 + 27+1) 45. 3(Jk - 10)(Jt - 3)
47. 4(b - 5) 2
7
57. — 59. 5 sec
1
51. -y 53.4
55. 6
49. 4(2w + 5) 2
61. S = 2Z) 2 ; about 2.12 in.
63. 1 sec 65. 16 ft 69. solution 71. not a solution
73. (1,-1) 75. (0,0) 77. (2, 2) 79. 6V6__81. 10V2
„„ 2V7 „ „ „ 7 ± 4+ „ 9 ± V557
83. ~y~ 85. 8 87.
9
91. 2 2 • 5 93. 3 • 19 95. 2 4 •
99. 3 • 5 • 23 101. 2 3 • 3 • 5 2
89 ' 14
5 97. 2 3 • 3 • 5
10.8 Guided Practice (p.620) 3. When factoring
out —2b, the remaining factor is (b 2 — 6b + 7);
answer is -2 b(b — l)(b + 1). 5. 3x 2 (x 2 + 2)
7. (x — l)(x 2 + x + 1) 9. (3x + l)(9x 2 » 3x + 1)
11. 2 b(b - 3)(b + 3) 13. 3 t(t + 3) 2 15. x(x - A)(x + 4)
10.8 Practice and Applications (pp. 620-622)
17. 6v(v 2 — 3) 19. 3x(\ — 3x) 21. Aa 2 (\ — 2a 3 )
23. 5x(3x 2 — x — 2) 25. 3d(6d 5 - 2d + 1)
27. (a + b)(a + 3) 29. (5* + l)(2x — 3)
31. (10* - 7)(* - 1) 33. (c - 2 )(c 2 + 2c + 4)
35. (m — 5 )(m 2 + 5m + 25) 37. 2y(y — 6)(y + 1)
39. At(t - 6)(t + 6) 41. (c 3 - 12)(c + 1)
43. 3(* + 10)(* 2 - 10* + 100) 45. -3, -4 47. 9, —3
1 j. 3 ^ -5 ± Vl7 __ 2 ± 2V43
"2
49. 0,
51.
f, 3 53.
2 9 2 ^" 4’ ^ 4 55> 12
57. 3 sec 59. h, 1 = h — 3, w = h ~ 9 61. /z = 12 in.,
I = 9 in., w = 3 in. 65. * < 1 67. -3, 3 69. 7, -19
72.
y
/
y
3x
>:
/
/
/
5
/
/
3
/
/
/
3
-1
/.
5
x
r~ 1
14 8 4
75 -7 77 -y 79 -- 2 i 81- —35
Student Resources
Quiz 3 (p. 622) 1. (lx — 8)(7x + 8); difference of squares
2. (11 — 3x)(ll + 3x)\ difference of squares
3. (2 1 + 5) 2 ; perfect square trinomial
4. 2(6 — 5y)(6 + 5y); difference of squares
5. (3 y + 7) 2 ; perfect square trinomial 6. 3 (n — 6) 2 ;
perfect square trinomial 7. 4 8.-4 9. 0, 3, —12
10 . 3x 2 (x + 4) 11 . 3x(2x + 1) 12 . 9x 3 (2x — 1)
13. 2x(4x 4 + 2x — 1) 14. 2x(x — 2)(x — 1)
15. (x 2 + 4)(x + 3) 16. 4(x — 5)(x 2 + 5x + 25)
17. 0, |,-f 18.2
o o
Chapter Summary and Review (pp. 623-626)
I . 3x — 5 3. 2x 2 + 5x + 7 5. x 3 + 2x 2 + 2x - 2
7. 6a 3 - 15 a 2 + 3(2 9. a 2 4- 3a — 40
II. d 3 - d 2 - 16 d ~ 20 13. Jt 2 - 225
15. x 2 + 4x + 4 17. (2x + 2) 2 = 4x 2 + 8x + 4; square
3 + V29
of a binomial 19. 2, 3 21 . -—- 23. 0, -9, 12
25. — 4 27. (x + 6)(x + 4) 29. (m — 10)(m + 2)
31. -8, 4 33. (3jc + l)(4x + 1) 35. (4r - 3)(r + 2)
37. 1 39. j, ~4 41. y, -y 43. 10, -10
45. 47. 5j 2 (y 2 - 4y + 2) 49. (y 2 - 2)(3y - 4)
51. (3b + 1)(% 2 — 3b + 1) 53. 5, -5
12 115 3
Maintaining Skills (p.629) i. - , - 3 -j> 2’6 5 ' lo’
13 3 55.1 . 1 . 29 11
20’ 4 7 '6’4’ 3 9,1 12 "'35 13 ' 30
15 ' 5 3
Chapter 11
Study Guide (p.632) i. B 2. B 3. D
11.1 Guided Practice (p. 636) 3. 3 5. y 7.6 9. no
11 . yes (assuming a, c ¥= 0)
11.1 Practice and Applications (pp. 636-638)
32 45 5 1
13.-pr 15.35 17.3 19. -5- 21. ^ 23. -z 25. ±8
y o Z d
27. 10 29. 31. -5, 2 33. 2, 5 35. 4, |
37. about 7.5 ft high and 5.4 ft wide 39. 6.875 in.
45. y + 3 = — 4(x — 5) 47. 2x + y = 26
49. 3x — 4y = -29 51. \2x + y = 84 53. 8 55. 100
57. 3V2 59. 4V5 61.54 63. V7
Decimal
0.78
0.2
0.6
0.073
0.03
0.48
Percent
78%
20 %
66 -|%
7.3%
3%
48%
Fraction
39
1
2
73
3
12
50
5
3
1000
100
25
11.2 Guided Practice (p. 642) 3. Direct variation; the
graph is a line passing through the origin. 5. Inverse
4
variation; the graph represents y = —. 7. neither
24 ^
9. inverse variation 11. y = —
11.2 Practice and
= 4x 15. y
13. y
22
21. v — — 23. y
1
= 3x 17. y = —x 19. y
21
13
x
27
25. y =
29. directly
2
/
31. inverse variation 33. inverse variation 35. 116 lb
37. about 0.36 pounds per square inch 39. 2.2° 45. 4.5
47. 1 49. 5.5 51. yes 53. yes 55. (x + 7)(x - 2)
57. (5x - 6)(x - 9) 59. 5x 2 (3x + 2)(x - 4)
2 7 28
61. about 1.36 to 1 63. j 65. 2 67. 9-j-g 69 - I 55
11.2 Technology (p. 645) 1. directly; 0.825; y = 0.825.V
11.3 Guided Practice (p. 649) 5.^ 7. already in
2/2 A
simplest form 9. 2 + 11. y 4 ~ 1 13. 3y + 1
15. Jt + 1
11.3 Practice and Applications (pp. 649-651) 17. 3x
4
19.-t 21.- 23.-—^r 25.
25x 2 3 t+ 2
29.-1 31. ——4 33. X+1
39> ~T
2y
x — 3
41.
1
y + 3
x + 6
43. a - 2
12 + x
35. 2x ~ 1
27.
1 - x z
37. - J
45. X - 8
_ 4(1 lx — 738) 0 . 2
47 ' “ 5 (;t+ 40) ; 3.6 lb per in . 2
55. 57. 4m 3 59.
51 '-3
53 ^
bd ' 49
67.2.387 69.111.4 71.0.02
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
Quiz 1 (p.65i) i. 8
3 _ 3
2
2. y 3.4 4. 1, -| 5.y = 4x
24
16
6- y = 5 ^ i-y=2 x *-y = — 9 -y = — ™y = —
„ 3x „ JC-4
11 .— 12 . 7
2 x + 6
13. -
1
x + 4
14.
11 + X
7.5
x
15. x + 4
1
2 x 2
(x + 5 ) 2
16. 2x + 3
11.4 Guided Practice (p. 655) 3.
7. ~ 9. The solver should have multiplied the first
expression by the reciprocal of the second expression;
x + 3 ^ 4x _ x + 3 x 2 — 9 _
x — 3 ' x 2 — 9 x — 3 4x
(x + 3)(x + 3)(x — 3) _ (x + 3 ) 2
(x - 3)(4x)
4x
11.4 Practice and Applications (pp.655-657) n.x
4- 15.- + H
35x
c —
13.
23.
17.
19.3 21.x
4c 2 (c + 1)
4(x — 7) 10(z — 7)
25. 9x 27. 2 (y ~ 3) 29. x + 3 31. 5x
_5 x + 2
33 ‘ 6 x 35 ‘ 2 (x - 2 )
41.
45.
x + 6
" 5x 2
43.
37.
1
x(x — 6 )
x + 3
39.
(2y + 3)(y ~ 2 )
4x + 3
(x — l)(4x — 3)(x + 1)
approaches 1.
47.
x + 3
x + 1
x( 2 x + 1 )
2(x - l ) 2
49. The ratio
55.
Input x
2
3
4
5
6
Output y
11
12
13
14
15
57. —19 < x < 5 59. x < —46 or x > 20
61. x < -22 or x> 12 63. -3 ± 2V3 65. —|, -2
18
67y,-| 69. 2x 2 +11x
71. 16 p 3 + Up 2 - Sp +
77. 1.125 79. 1.12
11.5 Guided Practice (p.660) 3.
5 _ 12 „ 2
73. 0.85 75. 1.74
5(y + 2)
y + 3
5 '37
7. —
c 2 - 4
y-2
ii.
r + 4
11.5 Practice and Applications (pp. 660-662)
13.
25.
x + l
a — l
15.2 17.1
19.
t + 14
27.
x + 5
29.
3 1
2x + 3
21 .
2 — 5x
31.
3x — 1
y-3
23. 2
33. The
a — 5 'x + 2 'x+l ' y — 7
solver multiplied the rational expressions rather than
y + 2 y - 4 2y - 2 3x + 9
adding them; , - + , 0 0 . 35.
37. -
45.
x — 3
x
y + 3 y + 3
39. 2(3 *: 4)2
41.
y+ 3 •
14x
r + 1
-— joules 49. — 51. —- 53.
x - 10 J y 6 2 x 8
x — 9
43. 1 joule
1
57. —r 59. —7" 61. m 16
p z a D
67. 8.1 X 10" 7
2x 8
,,9
1296c 4
55. c 2 d
63. ——“7 65. 1.6 X 10 2
8 u 3
69. 1.6 X 10" 3 71.9,11,13 73.42,
35,27 75. 8, y, 11
Student Resources
11.6 Guided Practice (p. 667) 3. f 5. p- * T S
3 (x + l)(2x + 3)
11.6 Practice and Applications (pp. 667-669) 9. 15
11.7c 5 13.5 b 15. 90x 3 17. 24y 2 19. 21a 7 + la 6
21. 8oT - 12 a 5 23.
155
_ 63x — 4 _
29.-7— 31. -
78x
3x + 1
25.
4x + 5
35.
39. —
14x 2
2 (x 2 - 20 )
(x — 10 )(x + 6 )
x 2 + 14x - 2
33.
4
19x - 11
27.
I n 2 + 1
30 n
6 x 2 53x
4x 2 + 17x + 5
37.
(3x - l)(x - 2)
41.
(3x — l)(x + 1)
5x(x - 3)
(x — l)(x + 4)
g_ £
43. T = — H-, where x is the number of miles in
the woods.
45.
Distance (woods), x
0
2
4
6
8
Total time, T
0.4
0.5
0.6
0.7
0.8
47. T =
48x
(x - 2 )(x + 2 )
49.
2 ( 2 x 2
ii
5, -b
( 2 x + l)( 2 x - 1 )
55. y + 2 = 2(x + 3) 57. y — 6 = ~-(x + 3)
3 ill 3p 2
59. y = j(x~ 7) 61.^ 63. ^ 65.-^ 67. ^
69. 6x 2 — 5x + 7 = 0 71. 3y 2 — y — 4 = 0
73. 12x 2 + 5x - 7 = 0 75. jx - 85001 < 1000
77. 0.315 79. 0.296 81. -0.708 83. -0.545 85. 0.104
87. -0.514
11.7 Guided Practice (p. 674) 3. 3x 5. 3x 3 7 . 6, -1
9.7 11.2
11.7 Practice and Applications (pp. 674-677) 13. 28
15.13 17y 19.-7 21.10,-2 23.0,16 25y
27. 2 29. 2 31. -j 33. j 35. 3, ~ 37. 0, 3
39.-12 41. 3, -2 43. 7, -6 45.-4 47.3,6
49. about 6.43 hours, or 6 hours 26 minutes; about
128.57 hours, or 128 hours 34 minutes 51. $1.00 per
pound 53. 7 dimes, 5 quarters 61. 9, 8, 7, 6, 5
63. 0, -1, -4, -9, -16 65. 0,2, 8 67. 36 69. 1
71. 125 73. 6 V 2 75. Vl3 77.
3Vl0
79.
2lVl7
81. 3V3
93. 1 95.
83. 3V3
89
220
85.
87. g 89.1
91
24
Quiz 2 (p.677) 2. 10 3. |
4.
x — 3
x + 2
x - 7
1
7.
lx 2 — 7x + 6
x+l (x + l)(x — 1 )
10. -2 11. 3 12. 130 13
xpc 2 + 2x - 2)
(x - 3)(* + 2)
_J5_. 15
x + 2 ’ x — 2
15
15
30x
x + 2 ' x — 2 (x + 2 )(x — 2 )
16. about 1.71 hours, or 1 hour 43 minutes
15. 18 hours
Extension Exercises (p.680) i.
5.
11 .
x(llx + 7)
(x - 7)(x + 7)
18
7.
(x + 9)(x - 9)
3x + 5
3x + 7
13.
9. --
10
x - 9
x + 3
3.
2 * - 1
17. (x ~ 2Y 19.
x(x + 3)
x + 1
x(x +15)
15.
25. f(x) =
1
x 9
-2
fix) =
( 8 , - 1 )
21 .
(2jc + l) 2
3(x ~ 2?
2x
23.
(x + 5) 2
(x + 2) 2
g
Chapter Summary and Review (pp. 681-684) i. —
4 36
3. 4 5. + = 5x 7. y = jx 9 .y= 14x 11 . y = —
7x(x — 2)
450 22
13.+ = - 15.+ = — 17.
3x 2 + 1
19.
3x + 2
21. ^±1 23.^4 25. 27. 20x 2
29.
x + 7 ' x — 1
8(2x + 7)
2+-3
„ x + 2 or . 0 _, x + 2
31. 9x 33.- 35.1 37.—- A —
x - 1
4x - 5 1
39. ——— 41. t 43. -4, 2 45. no solution
4V2 V5
5.
llV3
x — 2
Maintaining Skills {p. 687) i.
7. 2V2 9. (a - 9) 2 11 . (y - ll) 2 13. (15 + r) 2 or
(r + 15) 2 15. (2x + 5) 2 17. (4 - lx) 2 or (lx - 4) 2
12
17. all real numbers > — 1
■ > y
— 1
4
— 1
0
_
6
+
-1
1
3
5
7
9 x
2
19. 1200 gal/min
12.1 Practice and Applications (pp. 695-697)
21.-10 23.6 25.6 27.4
29. All nonnegative real numbers. Sample table:
X
0
1
4
9
y = 6Vx
0
6
12
18
31. All real numbers > —. Sample table:
X
10
3
11
3
4
5
6
II
<
UJ
X
1
o
0
1
— 1.4
—2.2
—2.8
33. All nonnegative real numbers. Sample table:
X
0
1
4
9
16
y = 4 + \Tx
4
5
6
7
8
Chapter 12
Study Guide (p. 690) i.D 2. A 3. B
12.1 Guided Practice (p. 695) 3. 0, 4, 5.7, 6.9, 8
5. 4, 7, 8.2, 9.2, 10 7. 1.4, 1.7, 2, 2.2, 2.4 9. domain:
all nonnegative real numbers; range: all nonnegative
real numbers 11. domain: all nonnegative real numbers;
range: all real numbers > —10 13. domain: all real
numbers > — 5; range: all nonnegative real numbers
15. all nonnegative real numbers
35. All real numbers > — 9. Sample table:
X
-9
-8
-5
0
7
+ = Vx + 9
0
1
2
3
4
37. All nonnegative real numbers. Sample table:
X
0
1
4
9
II
0
1
8
27
39. incorrect statement; S = 42 mph
41. domain: all nonnegative 47. domain: all nonnegative
real numbers; range: all real numbers; range: all
nonnegative real numbers real numbers > — 3
y
3 '
1-
-i
1
i
— 1
a
-
- 3 <
y =
--y/x- 3
Selected Answers
SELECTED ANSWERS
49. domain: all nonnegative 55. domain: all real numbers
real numbers; range: all
real numbers < 6
■ y
-7
5
y
=
v>
(
3
- 1
1
l
3
7
X
— 1
1
L
> — range: all
nonnegative real numbers
57. twice as fast 63. 2Vk5 65. 6VlO 67. 2A/5
69. VTT 71. -1.24, 3.24 73. -3.30, 0.30
75.0.19,1.31 77. 3x 2 + 5x-28
79. 10x 2 - 33x + 27 81. 2x 3 + x 2 + x - 1 83. |
85. —-~~r 87. 4 89.24 91.35 93.0.2635
x + 1 2
12.2 Guided Practice (p. 70i) 3. 4 + 6V5 5. 5V6
7.16 + 6V7 11.^5
12.2 Practice and Applications (pp. 701-703)
13. 7V7 15. — V3 17. V3 + 5V5 19. 5V2
21 . V5 23. 12V5 25. 15 27. 3 VK) 29. 6 - V6
31 . 4 V 5 + 5 33. 5V6 + 3 35.-12 37.5 39.33
5V7 „ V3 „ V30 „ 12 - 2V3
— 43. -j- 45.— 47.—^
49 . 1^2 51 . 3V5 + 5 53 . Vl2 and Vl3 are not
41
like terms; Vl2 + Vk3 = 2\f3 + Vl3 55. You ran
16V5 - 32 - 3.78 ft/sec faster. 61. 43.75% 63. 147
65. -5, 3 67. 13, -2 69. -j, -1 71. -30
73. All nonnegative real numbers. Sample table:
X
0
1
4
9
16
y = Vx — 3
-3
-2
-1
0
1
75. All nonnegative real numbers. Sample table:
0
1
4
9
16
S2
VO
II
0
6
12
18
24
12.3 Practice and Applications (pp. 707-709)
19.1 21.100 23.256 25.6 27.3 29. | 31.48
33. Line 2 should be (Vx) 2 = 7 2 ; x = 49. 35. 75
37. about 28.4 lb/in. 2 39. Sample answer: V2x — 20 = 4
41. 36 43. no solution 45. no solution 47. 7
49. no solution 51. 3 53. 3 55. 270 m/sec 2
57. false; V36 ¥= —6 61. ±VTT 63. ±2 65. ±V3
67. 4x 2 - 12* + 9 69. 9x 2 - 25y 2
71. 4<2 2 - 36 ab + 81 b 2 73. (x - 6) 2 75. ~ 77.
79 —
/S - 57
si 4
Quiz 1 (p.709)
1. domain: all nonnegative
real numbers; range: all
nonnegative real numbers
2. domain: all real numbers
> 9; range: all nonnegative
real numbers
y
-3-
y
= Vx
- 9
1
(
2
<
i
0
l
4
1
8*
— i
— 3
y
/
/
14
/
/
10
/
/
y
= 1
OVx
- 6
/
/
2
1
i
!
7
X
-2
□I
3. domain: all real
numbers > Y range: all
nonnegative real numbers
- y
-7
5
/ 2 >
r- 1
-3
- 1
/
1
3
7
“I
1
4. domain: all nonnegative
real numbers; range: all
real numbers > — 2
, y
3
i
II
1
2
s
f
10
1
4
(
5. 18VTo 6. 3V6 + 3 7. 4V7 + V5 8.4 9.64
10 . 11. 6 12 . 7 13. 3 14. 1.78 lb/in. 2
12.4 Guided Practice (p. 713) 3.7 5.125 7.27
9. 729
77. All real numbers > —3. Sample table:
X
-3
-2
1
6
13
y = Vx + 3
0
1
2
3
4
79. > 81. = 83. > 85. < 87. < 89. = 91. <
93. > 95. <
12.3 Guided Practice (p. 707) 3.64 5.196 7.36
9. no solution 11 . 4 13. 25 15. 3 17. 3
12.4 Practice and Applications (pp. 713-714)
11 . 11 1/3 13. 16 5/2 15. V7 17. (W) 7 19.100 21.2
23. 16 25. 81 27. 25 29. 256 31. 16 33. 36 35. 20
37.64 39. x 5/6 or (Vr) 5 41. x 1/2 y m or Vxy 43. y 2
45. sometimes 49. ±2\f\A 51. ±6 53. ±-^
55. —4, 8 57. prime 59. composite; 3 • 5 2
61. composite; 2 • 3 2 63. composite; 3 • 23
Student Resources
12.5 Guided Practice (p.719) 3. 100 5.25 7. 121
9. 3 ± — 11. -5±V35 13. -13,-1
2
15. “Hr, —5 17. 5 ± ^ 19. ±^~
3 6 3
12.5 Practice and Applications (pp. 719-721)
21.16 23.121 25.400 27.9 29.2,6 31. 2, -8
33. -5+V37 35.3,-13 37.2,22 39. 1 ± V6
41. 2 ± V5 43 . -7 ± V5l 45 . -5 ± 2 V 7
47. -II+ 2 V 30 49 . -4 ± V 22 51. -10±7V2
53. 6 ± V 39 55. 1 ± V5 57. about 12.25 ft by 12.25 ft
59. Base is about 6.8 ft; height is about 17.6 ft. 61. ±3
63. -7, 2 65. -3, | 67. 3 ± V2 69. ~3
71. 12 ± 5V6 73. no solution 75. about 8.6 ft
81. (4, 0) 83. ±7 85. ±9 87. ±4Vl() 89. no
solution 91. no solution 93. 3,-1 95. —4, 8
97. -5, -6 99. 3 101. (x + 5)0 - 4) 103. 0 + 2) 2
105. (2x - 3)0 + 1) 107. | 109.^
12.6 Guided Practice (p. 727) 3. c = 25 5. a = 8
7. a = 60 9. b = 16 11. 6, 8
12.6 Practice and Applications (pp. 727-729)
13. b = V7 = 2.65 15. a = 2VT0 = 6.32
17. b = 5V3 ~ 8.66 19. c = 2Vl7 = 8.25
21. a = V 9 T =“ 9.54 23. b = V33 = 5.74
25. x — 6 = 18, x = 24 27. x = 5, x + 5 = 10
29. x = 1, \flx = \fl 31. about 127.3 ft
33. about 12.2 in. 35. about 4.9 ft 37. right triangle;
5 2 + 12 2 = 13 2 39. right triangle; ll 2 + 60 2 = 61 2
41. not a right triangle; 3 2 + 9 2 ^ 10 2 43. not a right
triangle; 6 2 + 9 2 =£ ll 2
y
A
3
C
1
-
3
-
1
1
X
r
I
B
— 3
51. zero 53. two 55. two 57. 35 59. 50 61. 51
Quiz 2 (p. 729) i. 2 2. 42 3. 9 4. 3 ± V2
5. -2 ± V5 6. — 1 ± V3 7. not a right triangle;
6 2 + 9 2 =f= ll 2 8. right triangle; 12 2 + 35 2 = 37 2
9. right triangle; l 2 + l 2 = (V 2) 2 10. 2000 ft
12.7 Guided Practice (p. 733) 3.7.62 5. right
triangle 7. not a right triangle 9. 25 yd
21.16.16 23.12.73 25. right triangle 27. not a right
triangle 29. right triangle 31. AB = 4 V 2 ~ 5.66,
BC = Vl7 « 4.12, CA = 5 33. 269 mi
35. about 670 mi 37. about 457 mi
43. 9(3x - 4)(3jc + 4) 45. (jc + 6) 2
47. (3jc + l) 2 49. 2(6 - 5p){6 + 5 p)
51. 3 y(y + 6 )(y - 1) 53. 2x 2 (x - 2)(x + 2)
55.4 57. 4x 59.^^ 61.x + 6 63. 9a — 36
7 4x
r + 19 n 43 -
65. —- 67.— 69.— 71.40% 73.33.3%
x x I2x
75. 62.5% 77. 4%
12.8 Guided Practice (p. 738) 3. (|, 3^ 5. (-4, 0)
7. (0,5) 9. (2, 1); d = Vl7 = 4.12 11. (3, 4);
d = V5 ~ 2.24 13. ( — 1, 7); d = V5 ~ 2.24
12.8 Practice and Applications (pp. 738-739)
15.0.3) 17.0.3) 19. (44) 21.(-f.-f)
23. (-f,-4) 26. (i. l),(= 1.12 27. 0.-2);
d - vT) 3.61 29. §-1 , |); d = — = 4.92
31. (-2, 5); d = Vl93 « 13.89 33. (|, 8 ^, (7, 1),
(-y, l) 35. (39.95° N, 115.35° W) 37. (1, 1)
or 1 mi east and 1 mi north of the starting point, Vl3, or
3.61 mi 39. Q-, 41. (2, -1) 43. 0^; one
45. = 47. <
12.9 Guided Practice (p. 743) 3. identity property
of multiplication 5. distributive property 7. identity
property of addition
12.9 Practice and Applications (p. 744-746)
9. inverse property of addition; identity property of
addition 13. Sample answer: a = 3, b = 2
15. Sample answer: a = 3, b = 2 17. Yes; the map
cannot be colored with three different colors so that no
two countries that share a border have the same color.
27. 10,000 29. 20 31. 2 solutions 33. 1 solution
35. no real solution 37. not a solution 39. solution
3 11 7
41 ' 2 4 43 ‘l2 45 -“8
Quiz 3 (p. 746) 1. right triangle 2. right triangle
3. 13.42; (4, -3) 4. 7.21; (4, -8) 5. 16.12; (4, -7)
6. 16; (-8,0) 7. 6 ; (0,4) 8 . 10.30; (-|, |)
9. Sample answer: a = 2, b = 3, c = — 5 10. Sample
answer: a = 2, b = 3
12.7 Practice and Applications (pp. 733-735)
11.12.08 13.8.60 15.4.24 17.9 19.21.26
Selected Answers
SELECTED ANSWERS
Chapter Summary and Review (pp. 747 -750}
1. domain: all nonnegative 3. domain: all nonnegative
real numbers; range: all real numbers; range: all
nonnegative real numbers real numbers > 3
■ y
14 -
10 '
y =
= v
- 3
6
i
r"
2'
-
2
>
6
i
0
1
4
X
—2
5 .3V5-V3 7.6V2-8V3 9.48 + 8V7
29
11 . no solution 13.26 15.9 17.16 19.16 21.22
23. 2 ± 2V3 25. 8 ± 2Vl4 27. 1 ± ^ 13
2
29. c = 2 VT 3 31. b = 5, 2b + 2 = 12 33. not a right
triangle; 10 2 + 14 2 ¥= 17 2 35. 9.49 37. 10.77
39. (2,-1)
43. Sample answer:
0 c)(-b ) = (c)[(-l)(fe)]
= t(c)(-i )m
= [(-1 mm
= (-1 )mm
= — cb
Multiplication property
of-1
Associative property of
multiplication
Commutative property of
multiplication
Associative property of
multiplication
Multiplication property
of-1
Chapters 1-12 Cumulative Practice (pp. 754-755)
1.y>16;m>112 3. t = 3d; t = 9 mi 5.25
7.-63 9.-27 11 . ^ 13.-49 15.-1.64
J 2 10
17. —5.56 19. Sample answer: y = — x —j-
21 . function; domain: —1, 1, 3, 5; range: — 1, 1, 3
23. function; domain: —2, —1, 0, 1, 2;
range: —2, —1, 0, 1 25. 4x — 5y = 15
27. -7 < X < 3 - I HI I I I I I I I 0 I I »
4 -2-101234
29. v > 3 or x < 2
I I I I I I C>+0 | | | |
-1 0 1 2 3 4 5
31. (24, 21) 33. b 6 \ 64 35. -8 a 3 b 6 ; -512 37
4/r 5
128 39. 2 solutions;
V 39
41. 1 solution; —1
43. (jc - 28)(jc + 4) 45. (2x + 3) 2 47. (jc - 7) 2
3 5 ^ 0 ^ „ 2
49
2’ 3
51. -2 53. 0, -3, -6 55.
2x 2 — lx
57 — 59
3x (.x + 4)(jc - 1)
61
-15V2
x — 3
77 + llV3
63
46
65. 6±V55 67. 3 ±V22 69. -11 ±2V29
71. 4.47; (3, 5) 73. 9.43; (l, 75. 12.21; ( 5 , 7f)
/ 1 \ V 2/ V 4/
77. 11.18; (2,4y)
Skills Review Handbook
Decimals (p.760) 1. 14.42 3. 122.312 5.25.72
7.1.02 9.2.458 11.7.07 13.40.625 15.3.6
17. 520.37908 19. 16.7 21. 18.4 23. 4220
25. $62.44; $7.56
Factors and Multiples (p. 762) 1. 1, 2, 3, 6, 9, 18
3. 1, 7, 11, 77 5. 1, 3, 9, 27 7. 1, 2, 3, 6, 7, 14, 21, 42
9.3 3 11. 2 5 13.5 - 11 15. 2 2 • 37 17.1 19.1,5
21.1,3,9 23.1,5 25.5 27.1 29.14 31.51
33. 35 35. 208 37. 45 39. 42 41. 12 43. 30
45. 140 47. 51
1 12 5 13
Fractions (p. 766) 1. - 3. — , or \- 5. 20 7. -y, or
2 ~5 9 ‘ 6 11-3 13 '32 15 '2 ,0r3 2 17 ‘6 19 ‘ 3
21 '8 23 ‘ J 30 25 ‘ 2 8 27 ‘ l 24 29 ‘ 8 5 31 ‘ 3 l6
112 2 1 1
33. j 35. g 37. j 39. 7 j 41. 1^ 43. 1- 45. 6
47 -S 49 -i 511 53 - 5 l 55 -ts 57 - 2 ! 59 - 1 !
61 ■ *40 63 ‘ 40
Writing Fractions and Decimals (p. 768) 1. 0.25
3.0.08 5.0.3 7.0.90 9.| 11.^ 13. | 15. ||
Fractions, Decimals, and Percents (p.769) 1. 0.63;
tI °-°- 2i 4 5 - o i7; to 7 -°- 45 4
11.0.625;| 13.0.052;!^ 15. 0.0012; 17.8%;
o zjU zjUU
Z- 19. 150%; | 21.5%;^ 23. 480%; 4^
25. 375%; 3| 27. 52%; || 29. 0.5%; 31.0.7;
70% 33. 0.44; 44% 35. 0.375; 37.5% 37. 5.125;
512.5% 39. 0.875; 87.5%
Comparing and Ordering Numbers (p. 771)
1. 12,428 < 15,116 3. -140,999 > -142,109
5. 0.40506 > 0.00456 7. 1005.2 < 1050.7
9. -0.058 > -0.102 11. 17-t = 17| 13. ~
4 o y Z/
15. !>! 17. 42-1 > 41 ! 19. 32,227 > 32,226.5
o y Jo
21. -n| < -n| 23. -45,617; -45,242; -40,099;
o /
-40,071 25.9.003,9.027,9.10,9.27,9.3 27.|,|,
55 „ 15 ,1 .2 5 7 .1 5 7 5
6 ’4 16’ 8’ 5’ 3’4 3’ 4’ 8’ 12
Student Resources
Perimeter, Area, and Volume (p. 773) i. 34
3. 84 ft 5. 72 ft 7. 841 yd 2 9. 12.25 in. 2 11 . 20 in. 2
13. 15,625 ft 3 15. 420 yd 3 17. 212 in. 3
Estimation (p. 776) 1-53. Estimates may vary. 1. 50
3.2400 5.500 7.20 9.1600 11.700 13.22.5
15. 481 17. 1340 19. 41 21 . 209 23. 267 25. 2500
27. 30,000 29. 30 31. 3 33. 40 35. 4 37. 750
39. 80,000 41. 7000 43. 23 45. 10 47. 50 49. 16
51. 19 53. 18
Data Displays (p. 779) l-io. Sample answers are given.
1. 0 to 25 by fives 3. 0 to 20 by fives
Company Stock
Value per share
(dollars)
3 -F* 00 Ro 03 O
u
qO)^
f^P f^p f^p f^p
Year
Extra Practice
Chapter 1 (p.783) 1.105 3.8 5.512 7.76 9.49
11.31 13.3 15.1 17.7 19.12 21.24 23. solution
25. not a solution 27. solution 29. 16 = 20 — x
Cellular Telephone Subscribers
Year
Input V
0
1
2
3
4
5
Output y
1
8
15
22
29
36
Measures of Central Tendency (p. 780)
1 . 1.3; 0.5; 0 3. 30; 30; no mode 5. —550.1; 487; 376
Problem Solving (p. 782) 1 . 5 salads, 3 cartons of milk
3. $26.25 5. no later than 6:25 A.M. 7. 10 groups
9. The problem cannot be solved; not enough information
is given.
Chapter 2 (p. 784)
I. -7 < 8, 8 > -7 3. -4 > -7, -7 < -4
« 11 • I I I I 111 I I 111 • 11 > < 111 • • I 111 I I 111 I I 111 •
-10 -5 0 5 10 -10 -5 0 5 10
5.3 7.8.5 9.5 11.-2 13.3 15.-3 17.-13
19. -2.2 21. -7 23. -6.5 25. ~1 27. 15
29.-450 31.6 33.-81 35.48 37. -90 ft
39. 4 a - 24 41. 8x + 6 43. -2 - f 45. 1.5y - 4.5
47. already simplified 49. 7w — 4 51. —m 2 + 2m
53. -4 55. 66
Chapter 3 (p.785) i. 14 3.17 5.7 7.3 9.5
II. ^ 13.-20 15.84 17.4 19.2 21.-14 23.1
25. 2 27. -2 29. 3 31. -| 33. -7.46 35. -0.25
37. 12 mi/h 39. $7.25/h 41. 2 g/bar 43. 48 m
45. 45
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
Chapter 4 (p. 786)
41.
B
J
c
1
5
-1
j
5
X
3
A
J
A
J
B
1
\
-1
1
\
X
C*
3
.y
-3
l
]
[
)
X
\
\
3
\
\
4x +
y
-8
19.-1 21.6 23.-5 25.6 27.0 29.-7
7
31. undefined 33. y = 3x 35. y = ——x 37. y = x
39. y = — 3jc
47. function; domain is 1, 3, 5, 7 and range is 1, 2, 3
49. not a function
Chapter 5 (p. 787) i. y = 2x + 1 3. y = - 3
5. >’ = 3(x +1) 7. y — 6 = 0(x — 3)
9.y+l = 4(x + 3) 11 . y + 1 = ^-(x — 2)
13. j = 3x — 11 15. y = —- 2 17. y = -x + 2
19. 3x - y = 17 21. 5x + 6y = -2
23. lx + y = 8 25. 2x + y = 12
27. 4A + 5R = 50, 0.80A + 1.007? = 10.00
31. not perpendicular 33. not perpendicular
Chapter 6 (p. 788)
1. x < 1
- 1 1 1 1 1 1 1 1 1 1
-10 -5
5.x>4
x 1 1 1 1 1 1 1 1 1 1
4401 1 1 1 1
0 5
. i
fhhh-
10
III.
-10 -5
1 1 1 1 1 W1 ! !
0 5
1111 *
10
9. x > -1
11. X >
— 2—
z 2
15. -5 < X < 2
* 1 1 1 1 1 101 1 1 11
-10 -5 0 5 10
3.1 >y
* I I I I I 111 I I 11 I I I I I l + l I 11 -
-10 -5 0 5 10
l.k< 18
■ I I I I I I I I I I 11 I I I I I I I 141 I -
-20 -10 0 10 20
13. x < 9
17. -3 < X < 5
- I I I I I I I 101 I I I I Il + lI I I I I
-10 -5 0 5 10
Student Resources
21. 1 < X < 7
43.
19. -1 < x < 4
- I I I I I + I I I I O-H—
-5 0 5
23. x < 1 orx > 4
■ I I I I I I I I I I Il + l101 I I I I I I »
-10 -5 0 5 10
27. -14, 14 29. -12, 12
35. 1, 7
37. -8 < x < 8
- I I • I 11 I I 11 I I I 11 Il + lI I -
-10 -5 0 5 10
41. x < —2 or x> — i
-* -i
■i i i i + i*i i i i i i i '
-5 0 5
- I I I I I I I I I I 1101 I I I l + l I I I
-10 -5 0 5 10
25. x < 3 or x > 4
■ I I I I I I I I I
-5 0 3 4
31. -14, -6 33. j, 1
39. -5 < x < 5
1 I I I I I l + l I I 111 I Il + lI I I I •
-10 -5 0 5 10
43. -- 1 < X < 2
-I 2
• I I I I I lct-K> I I I I -
-5 0 5
Chapter 7 (p. 789) i. (-2, 5) 3. (4, 6) 5. (-5, -1)
7. (6, 3) 9. (4,1) 11.(11,-15) 13. (4, 2)
15. (0, —2) 17. (13, —2) 19. (2, 1) 21. Sample answer:
substitution, because it is easy to solve for x; (0, 5)
23. Sample answer: substitution, because the equations
are already solved for y; (— 1, —4) 25. Sample answer:
linear combinations, because it is easy to eliminate y\
(3, 0) 27. Sample answer: linear combinations, because
it is easy to eliminate y\ (2, 0) 29. 12 adult tickets and
8 student tickets 31. none 33. one 35. infinitely many
37. none
X
7
/
/
/
/
/
y< x+ 3
J /
/
/
/
/'
/
/
/
y> x+ 1
A
V
/
A
^ 1 ,
1
5
X
/
/
/
/
/
/
3
/
t
\
J
/2 k-
- 4 >
3x
N
/
4
\
\
-3
]
\
5
X
/
/
\
/
/
\
y + 2 c
-x
/
/
3
\
/
\
Chapter 8 (p.790) i. 16,807 3. 1728x 3 5. m 6 7. 98x 7
9. ~ 11.-R 13.4xy 5 15. -y
x 4 y 4
17.
19.
21 . 8
31.
23 ^
' 81
21x 3 z 9
4 b 1
27 '3
29.
3x 4 y 2
33. ^ 35. 0.000004813
9 a 5
37. 0.084162
39. 50.645 41. 0.0000000234 43. 5.28 X 10 3
45. 1.138 X 10 1 47. 8.2766 X 10 2 49. 1.6354 X 10 1
51. 3.95 X 10° 53. 8 X 10“ 3 55. $1155 57. $2286.82
59. y = 120,000(0.90)'
Chapter 9 (p.79i) i. 1.73 3. -10 5. 3.87 7. 14.83
9. ±5 11. no real solution 13. ±4 15. ±V3
17. 2.2 sec 19. 2 V 22 21. 4V7 23. 2 25. V3
■ y
\
/
\
J
/
\
/
\
j
/
-3
\-i
[ ]
1 /
3
jc
\
7
\
/
(0,
-4)
Selected Answers
SELECTED ANSWERS
SELECTED ANSWERS
35. -3, -2 37. -5, 2 39. -1,3
41. x 2 — 4x — 12 = 0; —2, 6
43. x 2 — 5x + A = 0; 1, 4 45. x 2 + 5x + 6 = 0;
-3, -2 47. 2x 2 - x - 10 = 0; -2, ^ 49. one solution
51. one solution 53. no real solution 55. no real solution
y < -x 2 + 4x + 5
Chapter 10 (p. 792) i. 8x 2 + 1 3. 14x 2 - lx + 8
5. x 2 + 9x - A 7. 4x 3 — 8x 2 + lx
9. 15b 5 - 10 b 4 + 5 b 2 11 . d 2 + Ad - 5
13. x 3 + x 2 + 18 15. x 2 + 18x + 81 17. a 2 - 4
19. 16x 2 + 40x + 25 21 . Aa 2 — 9b 2 23. -6,-3
25. -5, 1 27. |, 7 29.1,2
63. x 2 (x + 3)(x — 3) 65. x 2 (x + 9)(x — 5)
67. -3y(y + l)(y + 4) 69. lx\x 2 - 3)
71. 4 ft by 2 ft by 12 ft
Chapter 11 (p. 793) i. 6
1 15
X 11. V = -
J X
9- >’ = 4 X
3 ‘
13. y = -
5. 4
15.
1 10x 2
23. x 25. —x
2 x + 5
7. y = 3x
2x 3 1
— 1
5
X +1
27.
x — 3
29.
31. -
37.
2(jc + 1)
x — 1
2(5x + 14)
33.
5 — 3x
x 2
39. 33
35.
• + 35
1
(x — 3)(x + 8)
41.-2 43.4
(x + 5)(x — 5)
Chapter 12 (p. 794)
1. domain: all nonnegative 3. domain: all nonnegative
real numbers; range: all real numbers; range: all
nonnegative real numbers real numbers > — 5
5. domain: all real
numbers > 2; range: all
nonnegative real numbers
-7-
-5-
-3-
y =
Vx -
2
1
- j
3
5
7
X
7. domain: all real numbers
> — range: all nonnegative
real numbers
9. 5V5 11. I6V2 13. 7V3 - 3V2 15. -y- 17. 121
19. no solution 21.1,3 23.16 25.64 27. x 3/4
29. x 6 y 512 31.-14,4 33. -4,-2 35.-2, 8 37. \fl
39. 8 41. 20 43. 8 45. 18.36 47. 12.08 49. 4.47
51. (2, 4.5); d = V4225 53. (2.5, -2); d = 2.5
55. (0, 4); d = 2V5 57. (5, 6); d = Vl7 59. Sample
answer: Assume xy = 0 and both x + 0 and y + 0. If
xy = 0 and x + 0, then y = ® = 0, but this is impossible
since y ¥= 0. Therefore if xy = 0, either x = 0 or y = 0.
39. -3 41. 6 43. 6, 9 45. -4, 6 47. -2, | 49.
51. 2 53. -j, 8 55. (x + l)(x - 1)
57. (11 +x)(ll -jc) 59. (t+1) 2 61. (8)> + 3) 2
Student Resources