You have already worked with inequality statements. Let's refresh those skills.
| Inequality Notations: (see other notation forms at Notations for Solutions) | |
a > b ; a is strictly greater than b | a |
a < b ; a is striclty less than b | a |
a ≠ b ; a is not equal to b | Hint: The "open" (larger) part of the inequality symbol always faces the larger quantity. |
If you can solve a linear equation, you can solve a linear inequality. The process is the same, with one exception ... |
... when you multiply (or divide) an inequality by a negative value,
you must change the direction of the inequality.
Let's see why this "exception" is actually needed.
We know that 3 is less than 7. | ... written 3 < 7. ... written (-1)(3) ? (-1)(7) ... written -3 ? -7 |
On a number line, -3 is to the right of -7, making -3 greater than -7. -3 > -7 We have to reverse the direction of the inequality, when we multiply by a negative value, in order to maintain a "true" statement. |
When graphing a linear inequality on a number line, use an open circle for "less than" or "greater than", and a closed circle for "less than or equal to" or "greater than or equal to". |
Graph the solution set of: -3 < x < 4 | |
The solution set for this problem will be all values that satisfy both -3 < x and x < 4. Look for where the two inequalities overlap. Graph using open circles for -3 and 4 (since x can not equal -3 nor 4), and a bar to show the overlapping section. |
Graph the solution set of: x < -3 or x | |
The solution set for this problem will be the full graph of both inequalities, since the two inequalities do not overlap. Notice that there is one open circle (for -3) and one closed circle (for 1). |
Solve and graph the solution set of: 4x < 24 | |
Proceed as you would when solving a linear equation: Note: The direction of the inequality stays the same since we did NOT divide by a negative value. Graph using an open circle for 6 (since x can not equal 6) and an arrow to the left (since our symbol is less than). |
Solve and graph the solution set of: -5x | |
Divide both sides by -5. Note: The direction of the inequality was reversed since we divided by a negative value. Graph using a closed circle for -5 (since x can equal -5) and an arrow to the left (since our final symbol is less than or equal to). |
Solve and graph the solution set of: 3x + 4 > 13 | |
Proceed as you would when solving a linear equation: Note: The direction of the inequality stays the same since we did NOT divide by a negative value. Graph using an open circle for 3 (since x can not equal 3) and an arrow to the right (since our symbol is greater than). |
Solve and graph the solution set of: 9 - 2x | |
Subtract 9 from both sides. Note: The direction of the inequality was reversed since we divided by a negative value. Graph using a closed circle for 3 (since x can equal 3) and an arrow to the right (since our symbol is greater than or equal to). |