Absolute Value

Absolute Value

Rules and Properties

Absolute-Value Equations

If a > 0 and |x| = a, then x = a or x = –a.

1.8 Solving Absolute-Value Equations and Inequalities

Absolute-Value Inequalities

If a > 0 and |x| < a, then x > –a and x < a.

If a > 0 and |x| > a, then x < –a or x > a.

Similar statements are true for |x| a and |x| a.

Absolute Value Rules

If you start with an “=“ sign then you will have an “or” statement.

If you start with a “<“ or a “≤” then you will have an “and” statement.

If you start with a “>” or a “≥” then you will have an “or” statement.

Example 1

Solve and graph absolute-value equations.

|x + 1| = 7

1.8 Solving Absolute-Value Equations and Inequalities

Example 1

Solve and graph absolute-value equations.

|x + 1| = 7

1.8 Solving Absolute-Value Equations and Inequalities

x + 1 = 7 or x + 1 = –7

x = 6 or x = –8

Example 2

Solve and graph absolute-value equations.

4|x| = 8

1.8 Solving Absolute-Value Equations and Inequalities

Example 2

Solve and graph absolute-value equations.

4|x| = 8

1.8 Solving Absolute-Value Equations and Inequalities

4x = 8 or 4x = –8

x = 2 or x = –2

Example 3

|x - 4| = x + 1

Example 3|x - 4| = x + 1

x = x + 5 or x – 4 = -x - 1

0 = 5 or 2x – 4 = -1

x - 4 = x + 1 x - 4 = -(x + 1)

No Solution or 2x = 3No Solution or x = 3/2

3/2

Example 4

Solve and graph absolute-value inequalities.

|x + 52| 76

1.8 Solving Absolute-Value Equations and Inequalities

Example 4

Solve and graph absolute-value inequalities.

|x + 52| 76

x + 52 76 or x + 52 -76

x 24 or x –128

1.8 Solving Absolute-Value Equations and Inequalities

Example 5

Solve and graph absolute-value inequalities.

|x + 52| 76

1.8 Solving Absolute-Value Equations and Inequalities

Example 5

Solve and graph absolute-value inequalities.

|x + 52| 76

x + 52 76 and x + 52 -76

x 24 and x –128

1.8 Solving Absolute-Value Equations and Inequalities