Solve equations with absolute value. Solve inequalities with absolute value.

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3.5 Solving Equations and Inequalities with Absolute Value

Equations with Absolute Value

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For a > 0 and an algebraic expression x:

| x | = a

is equivalent to

x = a or x = a.

Example

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Solve: x 5.

Example

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Solve: x 3 1 4.

More About Absolute Value Equations

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When a = 0, | x | = a is equivalent to x = 0.

Note that for a < 0, | x | = a has no solution, because the absolute value of an expression is never negative.

The solution is the empty set, denoted .

Inequalities with Absolute Value

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Inequalities sometimes contain absolute-value notation. The following properties are used to solve them.

For a > 0 and an algebraic expression x:

| x | < a is equivalent to a < x < a.

| x | > a is equivalent to x < a or x > a.

Similar statements hold for | x | a and | x | a.

Inequalities with Absolute Value

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For example,

| x | < 3 is equivalent to 3 < x < 3

| y | ≥ 1 is equivalent to y ≤ 1 or y ≥ 1

| 2x + 3 | ≤ 4 is equivalent to 4 < 2x + 3 < 4

Example

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Solve: Solve and graph the solution set:

3x 2 5.

Example

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Solve: Solve and graph the solution set:

5 2x 1.

Absolute Value InequalitiesAbsolute Value Inequalities Absolute Value EquationsAbsolute Value Equations

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Practice