What best describes the relationship between the lines with equations -x-8y=8 and -16x+2y=0?

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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: http://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. Student HeCp ► More Examples More examples " 40 ' are available at www.mcdougallittell.com 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 www.mcdougallittell.com 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 www.mcdougallittell.com 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 Student HeCp p More Examples M°r e examples are available at www.mcdougallittell.com 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. 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(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 Student HeCp ► Homework Help Extra help with problem solving in Exs. 33-34 is available at www.mcdougallittell.com 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. I Student HeCp ► Homework Help 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 ? Student HeCp ► Homework Help Extra help with problem solving in Exs. 50-52 is available at www.mcdougallittell.com 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 www.mcdougallittell.com 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 www.mcdougallittell.com 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) -] [2 -8 4 l -2 * 8 12 x x- intercept 7 x-interce Pt > 4 1 2 Zo V y y “ 2 / 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. c 'y A 1 c uaarame ii 'X uuauranie 1 ( _ , +) oio rlo lac 1/ j IT, 1 -r / i y 1 eje de la s X 1 - 7 -5 -3 O ] [ 3 5 7 9 x origen (0,0) - 1 -3 ■i *■/ i Cuadrante III Cuadrante IV < ,-> ( r5 } 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. 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(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. 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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 \ X \ y -2> c + 6 \ \ \ J \ \ 1 \ 5 -1 K X L | \ 93. x < -5 -1 I I I I + I I I I I > -10 -5 0 94. x > 16 < 1 1 1 10 I I I h 0 8 16 24 32 96. > 97. < 99. < 101. < Quiz 2 (p. 539) 1. up 2. up 3. down 4. up 5. down 6 . down X 1 ] -1 l 5 X 0, 2) /' \ 7 \ L \ 1 / \ / 7 \ 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 i i i 1 i 5 -1 / i 5 X / / / y - 4x< 0 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 \ t \ / 5 1 r \ / \ 3“ / \ / \ !v <x 2 y -3 i 3 X U - j /■ \ 5 -] / i \ 5 X 1 \ / (_ 3 _ \ / \ / \ 1 / / 5 \ \ / 7 \ \ y < -x 2 + 2x ■ y / \ J / \ / / \ Y » 5 -1 l X / \ \ / l / l i l y< —2x 2 + 6x 9.8 Practice and Applications (pp. 550-552) 15. yes 17 . no 19 . no 21 . outside 23 . inside 25 . sometimes 27 . sometimes ■ y / \ o / \ \ J / \ / \ Y / ; XZ 1 i 3 X \ / \ \ o. / ^ j / y < x 2 - 4 • y A -6 -2/ 2 6 X rz. l \ 1 \ / 6 1 1 l_ 1 \ 1 10 1 1 1 1 14 1 y> —3x 2 - 5x - 1 7 Pi 1 >/ \ l 1 X / \ / \ / 3 \ / \ / 5 \ / 7 \ y > -x 2 - 3x - 2 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 \ y \ \ J / \ / / < X 2 + 3x \ 1 1 \ [ / 1 5 X \ \ / / V ‘ 3 Selected Answers SELECTED ANSWERS SELECTED ANSWERS 5. : 13. y \ 1 J \ / \ / \ L [ \ ; ! 5 X 1 \ / 0 _ 15 w 9 \ J (2 •-Si V6 ' 2 15. 556) 1. -2 3. 10 t • ■y t < , 3 1 \ -6 -2 , 6 X y * 1 _ 1 \ 5 1 4'~ 3 81 83. y = x 2 - 5x + 4 y = 2x 2 - 3x - 2 .y 0 < y — -x 2 - 2x 2 \ -t Jr 2 \ 2 6 X \ 1 I l / 0 / \ 1 _L 9 ' 2* + 2 49. • y 7 5 J 1 - 1 1 3 5 7 X L 53. 55. 64r 57. 243 59. 4 ? 61 to X 4 >’ 8 63. 9* 3 4 y 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. \ 1 * 1 l l / l / l / 1 6\ l 1 1 * 1 -6 -2 L/2 6 X L - -3x 2 + 6x - 1 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) \ /1 > y 1 c / \ / \ / 4 \ / 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