1. Warm-Up • |x |=3 • |x |= -7 • |2x |=10 • |x+3 |=6

2. 2.2 (M2) Solve Absolute Value Equations & Inequalities

3. Vocabulary • An extraneous solution is an apparent solution that must be rejected because it does not satisfy the original equation.

4. Example 1 – Solve an absolute value equation Solve | 2x-5 | = 9 Because this is an absolute value equation, the expression within the abs. value can equal 9 or -9. Set the equation equal to 9 & to -9 and solve.

5. 2x – 5 = 9 2x – 5 = -9 2x = 14 2x = -4 x = 7 x = -2 The solutions are -2, 7.

6. Try these . . . • | x + 3 | = 7 • | x – 2 | = 6 • | 2x + 1 | = 9

7. Example 2 – Check for extraneous solutions Solve | x + 2 | = 3x. Just like example 1, we will set the expression within the abs. value to 3x & to -3x. Make sure to check for extraneous solutions.

8. x + 2 = 3x x + 2 = -3x 2 = 2x 2 = -4x x = 1 x = -½ Substitute into original equation. | 1 + 2 | = 3(1) | -½ + 2 | = 3(-½) | 3 | = 3 | 1½ | = -1½ 3 = 3 1½ ≠ -1½ Therefore, x = 1 is the only solution.

9. Example 3 – Solve an inequality of the form |ax + b | > c • This absolute value inequality is equivalent to ax + b > cOR ax + b < -c. • same sign, same inequality • opposite sign, opposite inequality • Write the two inequalities, solve and graph the solutions on a number line.

10. Solve | 2x – 1 | > 5. Write as 2 inequalities. 2x – 1 > 5 2x – 1 < -5 2x > 6 2x < -4 x > 3 OR x < -2 Show solution on a number line.

11. Example 4 – Solve an inequality of the form | ax + b | ≤ c • This absolute value inequality can be rewritten as a compound inequality. • Solve the compound inequality and graph solution on a number line.

12. Solve | 2x – 3 | ≤ 5. Write as compound inequality. -5 ≤ 2x – 3 ≤ 5 Add 3 to each expression. -2 ≤ 2x ≤ 8 Divide each expression by 2. -1 ≤ x ≤ 4 Graph solution on number line.

13. Try these . . . 1. | x + 7 | > 2 2. | 2x + 1 | ≥ 5 3. | x – 6 | ≤ 4 4. | x + 7 | < 2