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

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

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

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

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

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 2 2 |x| ≤ 3 x ≥ –3 AND x ≤ 3 –2 –1 1 2 3 units –3 3

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 + 4.5 +4.5 |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

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

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 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

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

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

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

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.

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

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

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.

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

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