Solve compound inequalities in one variable involving absolute-value expressions.

1. Objectives Solve compound inequalities in one variable involving absolute-value expressions. When an inequality contains an absolute-value expression, it can be written as a compound inequality. The inequality |x| < 5 describes all real numbers whose distance from 0 is less than 5 units. The solutions are all numbers between –5 and 5, so |x|< 5 can be rewritten as –5 < x < 5, or as x > –5 AND x < 5.

2. LESS THAN = LESS “AND”

3. |x|– 3 < –1 +3 +3 |x| < 2 2 units 2 units –1 0 1 2 –2 Additional Example 1A: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. |x|– 3 < –1 x > –2 AND x < 2

4. Additional Example 1B: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. |x – 1| ≤ 2

5. Helpful Hint Just as you do when solving absolute-value equations, you first isolate the absolute-value expression when solving absolute-value inequalities.

6. Check It Out! Example 1a Solve the inequality and graph the solutions. 2|x| ≤ 6

7. Check It Out! Example 1b Solve each inequality and graph the solutions. |x + 3|– 4.5≤ 7.5

8. The inequality |x| > 5 describes all real numbers whose distance from 0 is greater than 5 units. The solutions are all numbers less than –5 or greater than 5. The inequality |x| > 5 can be rewritten as the compound inequality x < –5 OR x > 5.

9. |x| + 14 ≥ 19 – 14 –14 5 units 5 units –8 –2 –10 –6 –4 0 2 4 6 8 10 Additional Example 2A: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. |x| + 14 ≥ 19 |x| ≥ 5 x ≤ –5 OR x ≥ 5

10. Additional Example 2B: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. 3 + |x + 2| > 5

11. Check It Out! Example 2a Solve each inequality and graph the solutions. |x| + 10 ≥ 12

12. Check It Out! Example 2b Solve the inequality and graph the solutions.

13. When solving an absolute-value inequality, you may get a statement that is true for all values of the variable. In this case, all real numbers are solutions of the original inequality. If you get a false statement when solving an absolute-value inequality, the original inequality has no solutions.

14. |x + 4|– 5 > – 8 + 5 + 5 |x + 4| > –3 Additional Example 4A: Special Cases of Absolute-Value Inequalities Solve the inequality. |x + 4|– 5 > – 8 Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers. All real numbers are solutions.

15. |x – 2| + 9 < 7 – 9 – 9 |x – 2| < –2 Additional Example 4B: Special Cases of Absolute-Value Inequalities Solve the inequality. |x – 2| + 9 < 7 Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x. The inequality has no solutions.

16. Remember! An absolute value represents a distance, and distance cannot be less than 0.

17. Check It Out! Example 4a Solve the inequality. |x| – 9 ≥ –11

18. Check It Out! Example 4b Solve the inequality. 4|x – 3.5| ≤ –8

19. –10 –5 5 0 10 –6 –5 –4 –3 –2 –1 0 Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. 3|x| > 15 x < –5 or x > 5 2. |x + 3| + 1 < 3 –5 < x < –1

20. Lesson Quiz: Part II Solve each inequality. no solutions 3. |3x| + 1 < 1 4. |x + 2| – 3 ≥ – 6 all real numbers