# Absolute Value Inequalities -  Definition and Examples

Absolute value inequalities are inequalities that have an absolute value sign and a variable inside the absolute value. For example, |x| < 6 and |x - 1| > 2 are absolute value inequalities.

## Properties of absolute value inequalities

Let c represent a positive real number.

|x| ≥ c is equivalent to x ≥ c or x ≤ - c

|x| ≤ c is equivalent to -c ≤ x ≤ c

## Using the properties of absolute value inequalities to solve absolute value inequalities

1.

Solve |x - 1| > 2

|x - 1| > 2 is equivalent to x - 1 > 2 or x - 1 < -2

Solve x - 1 > 2

x - 1 > 2

x - 1 + 1 > 2 + 1

x > 3

Solve x - 1 < -2

x - 1 + 1 < -2 + 1

x < - 1

Choose any number bigger than 3 or any number smaller than -1

2.

Solve |2x + 1| < 7

|2x + 1| < 7 is equivalent to -7 < 2x + 1 < 7

-7 - 1 < 2x + 1 - 1 < 7 - 1

-8 < 2x < 6

-8 / 2 < 2x / 2 < 6 / 2

-4 < x < 3

Choose any number between -4 and 3.

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