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

2. Warm Up Solve each inequality and graph the solution. 1. x + 7 < 42.
x < –3 –5 –4 –3 –2 –1 1 2 3 4 5
14x ≥ 28
x ≥ 2 –5 –4 –3 –2 –1 1 2 3 4 5
x > 1
x > –2 –5 –4 –3 –2 –1 1 2 3 4 5

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
x – 1 ≥ –2 AND x – 1 ≤ 2
+1 +1
x ≥ –1
x ≤ 3
AND –2 –1 1 2 3 –3

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 2|x| ≤ 6 2 2 |x| ≤ 3
Solve the inequality and graph the solutions.
2|x| ≤ 6
2|x| ≤ 6
|x| ≤ 3
x ≥ –3 AND x ≤ 3 –2 –1 1 2
3 units –3 3

7. Check It Out! Example 1b |x + 3|– 4.5 ≤ 7.5 |x + 3|– 4.5 ≤ 7.5
Solve each inequality and graph the solutions.
|x + 3|– 4.5 ≤ 7.5
|x + 3| ≤ 12
|x + 3|– 4.5 ≤ 7.5
x + 3 ≥ –12 AND x + 3 ≤ 12
–3 –3
–3 –3
x ≥ –15 AND x ≤ 9
–20
–15
–10 –5 5 10 15

8. 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

9. 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 + 2| > 2
– – 3
3 + |x + 2| > 5
x + 2 < –2 OR x + 2 > 2
–2 –2
–2 –2
x < –4 OR x > 0
–10 –8 –6 –4 –2 2 4 6 8 10

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

11. Check It Out! Example 2b
Solve the inequality and graph the solutions. OR
x ≤ –6
x ≥ 1

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

13. 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
The inequality has no solutions.

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

15. Check It Out! Example 4a
Solve the inequality.
|x| – 9 ≥ –11
|x| – 9 ≥ –11
+9 ≥ +9
|x| ≥ –2
All real numbers are solutions.

16. Check It Out! Example 4b
Solve the inequality.
4|x – 3.5| ≤ –8
4|x – 3.5| ≤ –8 4
|x – 3.5| ≤ –2
The inequality has no solutions.

17. 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

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