Warm-Up  |x |=3  |x |= -7  |2x |=10  |x+3 |=6

2.2 (M2) Solve Absolute Value Equations & Inequalities

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

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.

2x – 5 = 92x – 5 = -9 2x = 142x = -4 x = 7x = -2 The solutions are -2, 7.

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

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.

x + 2 = 3xx + 2 = -3x 2 = 2x2 = -4x x = 1x = -½ Substitute into original equation. | | = 3(1) | -½ + 2 | = 3(-½) | 3 | = 3 | 1½ | = -1½ 3 = 3 1½ ≠ -1½ Therefore, x = 1 is the only solution.

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

Solve | 2x – 1 | > 5. Write as 2 inequalities. 2x – 1 > 52x – 1 < -5 2x > 6 2x < -4 x > 3 ORx < -2 Show solution on a number line.

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.

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

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