1. Algebra 1
Chapter 3 Section 7

2. 3-7: Solving Absolute-Value Inequalities
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.

4. Example 1: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions.
|x| < 2
|x|– 3 < –1
Since 3 is subtracted from |x|, add 3 to both sides to undo the subtraction.
{x: –2 < x < 2}.
Write as a compound inequality.
The solution set is
x > –2 AND x < 2 –2 –1 1 2
2 units

5. Example 2: Solving Absolute-Value Inequalities Involving <
Solve the inequality |x – 1| ≤ 2 and graph the solutions.
x – 1 ≥ –2 AND x – 1 ≤ 2
Write as a compound inequality.
+1 +1
Solve each inequality.
x ≥ –1
x ≤ 3
AND
{x: –1 ≤ x ≤ 3}.
Write as a compound inequality.
The solution set is –2 –1 1 2 3 –3

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

7. 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. Example 3: Solving Absolute-Value Inequalities Involving >
Solve the inequality |x| + 14 ≥ 19 and graph the solutions.
– 14 –14
|x| + 14 ≥ 19
Since 14 is added to |x|, subtract 14 from both sides to undo the addition.
|x| ≥ 5
x ≤ –5 OR x ≥ 5
Write as a compound inequality. The solution set is {x: x ≤ –5 OR x ≥ 5}.
5 units
–10 –8 –6 –4 –2 2 4 6 8 10

10. Example 4: Solving Absolute-Value Inequalities Involving >
Solve the inequality 3 + |x + 2| > 5 and graph the solutions.
Since 3 is added to |x + 2|, subtract 3 from both sides to undo the addition.
|x + 2| > 2
– – 3
3 + |x + 2| > 5
Write as a compound inequality. Solve each inequality.
x + 2 < –2 OR x + 2 > 2
–2 –2
–2 –2
x < –4 OR x > 0
Write as a compound inequality.
The solution set is
{x: x < –4 or x > 0}.
–10 –8 –6 –4 –2 2 4 6 8 10

11. 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. Its solution set is ø.

12. Example 5: Special Cases of Absolute-Value Inequalities
Solve the inequality |x + 4|– 5 > – 8 .
|x + 4|– 5 > – 8
|x + 4| > –3
Add 5 to both sides.
Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
The solution set is all real numbers.

13. Example 6: Special Cases of Absolute-Value Inequalities
Solve the inequality |x – 2| + 9 < 7.
|x – 2| + 9 < 7
– 9 – 9
|x – 2| < –2
Subtract 9 from both sides.
Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x.
The inequality has no solutions. The solution set is ø.

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