1. Solving Absolute-Value Inequalities
3-Ext
Solving Absolute-Value Inequalities
Lesson Presentation
Holt Algebra 1

2. 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 written as –5 < x < 5, which is the compound inequality x > –5 AND x < 5.

4. Example 1A: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| + 2 ≤ 6
Since 2 is added to |x|, subtract 2 from both sides to undo the addition.
|x| + 2 ≤ 6
–2 –2
|x| ≤ 4
Think, “The distance from x to 0 is less than or equal to 4 units.” –5 –4 –3 –2 –1 1 2 3 4 5
4 units
x ≥ –4 AND x ≤ 4
Write as a compound inequality.
–4 ≤ x ≤ 4

5. Example 1B: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| – 5 < –4
|x| – 5 < –4
Since 5 is subtracted from |x|, add 5 to both sides to undo the subtraction.
|x| < 1
Think, “The distance from x to 0 is less than 1unit.” –2 –1 1 2
unit
x is between –1 and 1.
–1 < x AND x < 1
Write as a compound inequality.
–1 < x < 1

6. Example 1C: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x + 4| – 1.5 < 3.5
|x + 4| – 1.5 < 3.5
Since 1.5 is subtracted from |x + 4|, add 1.5 to both sides to undo the subtraction.
|x + 4| < 5
5 units
Think, “The distance from x to –4 is less than 5 units.” –5 –4 –3 –2 –1 1 2 3 4 5
x + 4 > –5 AND x + 4 < 5
x + 4 is between –5 and 5.
–4 – –4 –4
x > –9 AND x < 1

7. Write as a compound inequality.
Example 1C Continued
x > –9 AND x < 1
Write as a compound inequality.
–10 –8 –6 –4 –2 2 4 6 8 10
–9 < x < 1

8. Think, “The distance from x to 0 is less than 3 units.”
Check It Out! Example 1a
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| + 12 < 15
Since 12 is added to |x|, subtract 12 from both sides to undo the addition.
|x| + 12 < 15
– 12 –12
|x| < 3 –5 –4 –3 –2 –1 1 2 3 4 5
3 units
Think, “The distance from x to 0 is less than 3 units.”
x is between –3 and 3.
x > –3 AND x < 3
Write as a compound inequality.
–3 < x < 3

9. Think, “The distance from x to 0 is less than 1 unit.”
Check It Out! Example 1b
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| – 6 < –5
Since 6 is subtracted from |x|, add 6 to both sides to undo the subtraction.
|x| – 6 < –5
|x| < 1
1 unit
Think, “The distance from x to 0 is less than 1 unit.” –2 –1 1 2
x is between –1 and 1.
x > –1 AND x < 1
Write as a compound inequality.
–1 < x < 1

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 written as the compound inequality x < –5 OR x > 5.

12. Example 2A: Solving Absolute-Value Inequalities Involving >
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| + 2 > 7
|x| + 2 > 7
Since 2 is added to |x|, subtract 2 from both sides to undo the addition.
– 2 –2
|x| > 5
5 units
–10 –8 –6 –4 –2 2 4 6 8 10
x < –5 OR x > 5
Write as a compound inequality.

13. Example 2B: Solving Absolute-Value Inequalities Involving >
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| – 12 ≥ –8
Since 12 is subtracted from |x|, add 12 to both sides to undo the subtraction.
|x| – 12 ≥ –8
|x| ≥ 4
4 units
–10 –8 –6 –4 –2 2 4 6 8 10
x ≤ –4 OR x ≥ 4
Write as a compound inequality.

14. Example 2C: Solving Absolute-Value Inequalities Involving >
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x + 3| – 5 > 9
|x + 3| – 5 > 9
Since 5 is subtracted from |x + 3|, add 5 to both sides to undo the subtraction.
|x + 3| > 14
14 units
–16
–12 –8 –4 4 8 12 16
x + 3 < –14 OR x + 3 > 14

15. Solve the two inequalities. – 3 –3 –3 –3
Example 2C Continued
x + 3 < –14 OR x + 3 > 14
Solve the two inequalities.
– – –3 –3
x < –17 OR x > 11
–24
–20
–16
–12 –8 –4 4 8 12 16
–17 11
Graph.

16. Write as a compound inequality.
Check It Out! Example 2a
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| + 10 ≥ 12
|x| + 10 ≥ 12
Since 10 is added to |x|, subtract 10 from both sides to undo the addition.
– 10 –10
|x| ≥ 2
2 units –5 –4 –3 –2 –1 1 2 3 4 5
x ≤ –2 OR x ≥ 2
Write as a compound inequality.

17. Write as a compound inequality.
Check It Out! Example 2b
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
|x| – 7 > –1
|x| – 7 > –1
Since 7 is subtracted from |x|, add 7 to both sides to undo the subtraction.
|x| > 6
6 units
–10 –8 –6 –4 –2 2 4 6 8 10
x < –6 OR x > 6
Write as a compound inequality.

18. Check It Out! Example 2c
Solve the inequality and graph the solutions. Then write the solutions as a compound inequality.
Since is added to |x |, subtract from both sides to undo the addition.
3.5 units –5 –4 –3 –2 –1 1 2 3 4 5 OR

19. Check It Out! Example 2c ContinuedOR
Solve the two inequalities. OR
–10 –8 –6 –4 –2 2 4 6 8 10 1
Graph.