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

4. 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.

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

6. Additional Example 1B: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions.
|x – 1| ≤ 2
x ≥ –1
x ≤ 3
AND
-1≤x≤3 –2 –1 1 2 3 –3

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

8. Check It Out! Example 1a 2|x| ≤ 6 x ≥ –3 AND x ≤ 3 -3≤x≤3
Solve the inequality and graph the solutions.
2|x| ≤ 6
x ≥ –3 AND x ≤ 3
-3≤x≤3 –2 –1 1 2
3 units –3 3

9. Check It Out! Example 1b |x + 3|– 4.5 ≤ 7.5
Solve each inequality and graph the solutions.
|x + 3|– 4.5 ≤ 7.5
–20
–15
–10 –5 5 10 15

10. 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.

11. 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
|x| ≥ 5
x ≤ –5 OR x ≥ 5
5 units
–10 –8 –6 –4 –2 2 4 6 8 10

12. Solve the inequality and graph the solutions.
Additional Example 2B: Solving Absolute-Value Inequalities Involving >
Solve the inequality and graph the solutions.
3 + |x + 2| > 5
x < –4 OR x > 0
–10 –8 –6 –4 –2 2 4 6 8 10

13. Solve each inequality and graph the solutions.
Check It Out! Example 2a
Solve each inequality and graph the solutions.
|x| + 10 ≥ 12
x ≤ –2 OR x ≥ 2
2 units –5 –4 –3 –2 –1 1 2 3 4 5

14. Solve the inequality and graph the solutions.
Check It Out! Example 2b
Solve the inequality and graph the solutions.
x ≤ –6 or
x ≥ 1 –7 –6 –5 –4 –3 1 2 3 –2 –1

15. Additional Example 3: Application
A pediatrician recommends that a baby’s bath water be 95°F, but it is acceptable for the temperature to vary from this amount by as much as 3°F. Write and solve an absolute-value inequality to find the range of acceptable temperatures. Graph the solutions. 98
100 96 94 92 90
The range of acceptable temperature is 92 ≤ t ≤ 98.

16. The range of pressure is 50 ≤ p ≤ 200.
Check It Out! Example 3
A dry-chemical fire extinguisher should be pressurized to 125 psi, but it is acceptable for the pressure to differ from this value by at most 75 psi. Write and solve an absolute-value inequality to find the range of acceptable pressures. Graph the solution.
200
225
175
150
125
100 75 50 25
The range of pressure is 50 ≤ p ≤ 200.

17. 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.

18. Additional Example 4A: Special Cases of Absolute-Value Inequalities
Solve the inequality.
|x + 4|– 5 > – 8
|x + 4| > –3
All real numbers are solutions.

19. Additional Example 4B: Special Cases of Absolute-Value Inequalities
Solve the inequality.
|x – 2| + 9 < 7
The inequality has no solutions.

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

21. Check It Out! Example 4a
Solve the inequality.
|x| – 9 ≥ –11
All real numbers are solutions.

22. Check It Out! Example 4b
Solve the inequality.
4|x – 3.5| ≤ –8
The inequality has no solutions.

23. 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
3. A number, n, is no more than 7 units away from 5. Write and solve an inequality to show the range of possible values for n.
|n– 5| ≤ 7; –2 ≤ n ≤ 12

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