Solving Compound and Absolute Value Inequalities Absolute Value Inequalities Use the definition of absolute value to rewrite an absolute value inequality as a compound inequality. For all real numbers a and b, b > 0, the following statements are true. If |a| < then < < 1. b, a b. –b 2. If |a| > b, then a > b or a < –b. These statements are also true for and respectively. ≤ ≥, Example 1: Solve | + 2 | > 4. Graph Example 2: Solve | 2x 1 | < 5. x – the solution set on a number line. Graph the solution set on a number line. By statement 2 above, if |x + 2| > 4, By statement 1 above, if |2x 1| < 5, – then + 2 > 4 or + 2 < Subtracting 2 from both then < 2x 1 < 5. Adding 1 to all three parts of the x x –4. –5 – sides of each inequality gives > 2 or < inequality gives < 2x < 6. x x –6. –4 Dividing by 2 gives < < 3. x –2 Exercises Solve each inequality. Graph the solution set on a number line. 1. |3x + 4| < 8 2. |4k| + 1 > 27 3. |a + 9| 30 4. ≥ |2f 11| > 9 |5w + 2| < 28 5. 6. – |10 2k| < 2 + 2 > 10 7. 8. – 9. |4b 11| < 17 – 10. |100 3m| > 20 –
Solving Compound and Absolute Value Inequalities Absolute Value Inequalities Use the definition of absolute value to rewrite an absolute value inequality as a compound inequality. For all real numbers a and b, b > 0, the following statements are true. 1. If |a| < b, then –b < a < b. 2. If |a| > b, then a > b or a < –b. These statements are also true for ≤ and ≥, respectively. Example 1: Solve | x + 2 | > 4. Graph the solution set on a number line. By statement 2 above, if |x + 2| > 4, then x + 2 > 4 or x + 2 < –4. Subtracting 2 from both sides of each inequality gives x > 2 or x < –6. Example 2: Solve | 2x – 1 | < 5. Graph the solution set on a number line. By statement 1 above, if |2x – 1| < 5, then –5 < 2x – 1 < 5. Adding 1 to all three parts of the inequality gives –4 < 2x < 6. Dividing by 2 gives –2 < x < 3. Exercises Solve each inequality. Graph the solution set on a number line. 1. |3x + 4| < 8 2. |4k| + 1 > 27 3. 4. |a + 9| ≥ 30 5. |2f – 11| > 9 6. |5w + 2| < 28 7. |10 – 2k| < 2 8. + 2 > 10 9. |4b – 11| < 17 10. |100 – 3m| > 20