1. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Solve compound inequalities in one variable involving absolute-value expressions. Objectives When an inequality contains an absolute-value expression, it can be written as a compound inequality. The inequality |x| –5 AND x < 5.

2. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities LESS THAN = LESS “AND”

3. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Additional Example 1A: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. |x|– 3 < –1 x > –2 AND x < 2 +3 |x| < 2 |x|– 3 < –1 –2 –1012 2 units

4. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities |x – 1| ≤ 2 Additional Example 1B: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions.

5. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Just as you do when solving absolute-value equations, you first isolate the absolute-value expression when solving absolute-value inequalities. Helpful Hint

6. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Check It Out! Example 1a Solve the inequality and graph the solutions. 2|x| ≤ 6

7. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Check It Out! Example 1b |x + 3|– 4.5 ≤ 7.5 Solve each inequality and graph the solutions.

8. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities 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.

9. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Solve the inequality and graph the solutions. Additional Example 2A: Solving Absolute-Value Inequalities Involving > |x| + 14 ≥ 19 |x| ≥ 5 x ≤ –5 OR x ≥ 5 –10 –8 –6–4 –2 0246810 5 units – 14 –14 |x| + 14 ≥ 19

10. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Solve the inequality and graph the solutions. 3 + |x + 2| > 5 Additional Example 2B: Solving Absolute-Value Inequalities Involving >

11. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Check It Out! Example 2a |x| + 10 ≥ 12 Solve each inequality and graph the solutions.

12. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Check It Out! Example 2b Solve the inequality and graph the solutions.

13. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities 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.

14. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Additional Example 4A: Special Cases of Absolute- Value Inequalities Solve the inequality. |x + 4|– 5 > – 8 + 5 |x + 4| > –3 Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers. All real numbers are solutions.

15. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Additional Example 4B: Special Cases of Absolute- Value Inequalities Solve the inequality. |x – 2| + 9 < 7 – 9 |x – 2| < –2 Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x. The inequality has no solutions.

16. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities An absolute value represents a distance, and distance cannot be less than 0. Remember!

17. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Check It Out! Example 4a Solve the inequality. |x| – 9 ≥ –11

18. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Check It Out! Example 4b Solve the inequality. 4|x – 3.5| ≤ –8

19. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. 3|x| > 15x 5 0 –5–10 5 10 2. |x + 3| + 1 < 3 –5 < x < –1 –2 0 –1–3–4–5–6

20. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Solve each inequality. 3. |3x| + 1 < 1 4. |x + 2| – 3 ≥ – 6 all real numbers no solutions Lesson Quiz: Part II