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

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

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

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

7. Check It Out! Example 1b |x + 3|– 4.5 ≤ 7.5
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. Solve the inequality and graph the solutions.
Additional Example 2A: Solving Absolute-Value Inequalities Involving >
Solve the inequality and graph the solutions.
|x| + 14 ≥ 19
– 14 –14
|x| + 14 ≥ 19
|x| ≥ 5
x ≤ –5 OR x ≥ 5
5 units
–10 –8 –6 –4 –2 2 4 6 8 10

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

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

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

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

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