1. 1.7 Solving Absolute Value Equations and Inequalities ©2001 by R. Villar All Rights Reserved

2. Absolute Value Equations and Inequalities To solve absolute value equations, recall that absolute value means “distance away from zero”. Example: Solve | x + 3 | = 5 This means that x + 3 is 5 units away from 0, so there are two possibilities: x + 3 = 5 or x + 3 = –5 –3 –3 –3 –3 x = 2 or x = –8 What does absolute value mean? How do you solve absolute value equations?

3. What about inequalities? Example: Solve |x| < 3 This means that the distance on number line must be less than or equal to 3. -3 0 3 x –3 –3 < x < 3

4. Example: Solve |x | > 3 This means that the distance on number line is greater than or equal to 3. –3 0 3 x > 3 or x < –3

5. So, if the inequality is < (with respect to x) then the compound inequality is an and. If the inequality is > (with respect to x) then the compound inequality is an or. Here’s a trick you may use to remind you which is which: Since the < sign looks like the letter L crunched up, remember L for Land. So is an or.

6. Solve: |x – 3| < 5 Land x – 3 –5 +3 +3 +3 +3 x –2 –2 < x < 8

7. Solve: |x – 3| < 5 Land x – 3 –5 +3 +3 +3 +3 x –2 –2 < x < 8 –2 8

8. Solve: |x – 3| < 5 Land x – 3 –5 +3 +3 +3 +3 x –2 –2 < x < 8 –2 8

9. Solve: |2x + 3| > 13 or 2x + 3 > 13 or 2x + 3 < –13 –3 –3 –3 –3 2x > 10 2x < –16 x > 5 or x < –8

10. Solve: |2x + 3| > 13 or 2x + 3 > 13 or 2x + 3 < –13 –3 –3 –3 –3 2x > 10 2x < –16 x > 5 or x < –8 –8 5

11. Solve: |2x + 3| > 13 or 2x + 3 > 13 or 2x + 3 < –13 –3 –3 –3 –3 2x > 10 2x < –16 x > 5 or x < –8 –8 5