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 = 9 2x – 5 = -9

2x = 14 2x = -4

x = 7 x = -2

The solutions are -2, 7.

Try these . . .

1. | 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 = 3x x + 2 = -3x2 = 2x 2 = -4xx = 1 x = -½Substitute 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.

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 > 5 2x – 1 < -5

2x > 6 2x < -4

x > 3 OR x < -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 2.

-1 ≤ x ≤ 4

Graph solution on number line.

Try these . . .

1. | x + 7 | > 2

2. | 2x + 1 | ≥ 5

3. | x – 6 | ≤ 4

4. | x + 7 | < 2