# Absolute Value Inequalities (One Variable)

How to solve absolute value inequalities problems: formulas, examples, and their solutions.

## Formulas

|*x*| < *a* means

the distance between 0 and *x* is lesser than *a*.

So it's -*a* < *x* < *a*.

|*x*| > *a* means

the distance between 0 and *x* is greater than *a*.

So it's *x* < -*a* or *x* > *a*.

## Example 1

|*x* - 2| < 5

Then, -5 < *x* - 2 < 5.

To remove -2 in the middle term,

add +2 on each side.

## Example 2

|2*x* + 1| ≥ 9

Then case 1:

2*x* + 1 ≤ -9

|2*x* + 1| ≥ 9

Then case 2:

2*x* + 1 ≥ 9

## Example 3

Recall that

the absolute value of a number cannot be (-).

Absolute value equations (one variable)

So |-*x* + 7| ≤ 0 becomes |-*x* + 7| = 0.

Then -*x* + 7 = 0.