# Absolute Value Inequality (One Variable)

How to solve an absolute value inequality (one variable): 2 formulas, 2 examples, and their solutions.

## Formula: |x| < a

### Formula

|x| < a can be solved as

-a < x < a.

By the same way, |x| ≤ a can be solved as

-a ≤ x ≤ a.

## Example 1

### Example

### Solution

|x - 2| < 5

So, by using the previous formula, -5 < x - 2 < 5.

-5 < x - 2 < 5 means

[x - 2] is less than -5 and greater than 5.

To remove -2 in the middle side, write +2 to each side.

The left side is -5 + 2.

The middle side is, remove -2, x.

And the right side is 5 + 2.

Linear Inequality (One Variable)

-5 + 2 = -3

5 + 2 = 7

So -3 < x < 7.

-3 < x < 7

This is the answer.

## Formula: |x| > a

### Formula

|x| > a can be solved as

x < -a or x > a.

By the same way, |x| ≥ a can be solved as

x ≤ -a or x ≥ a.

## Example 2

### Example

### Solution

|2x + 1| ≥ 9

So, by using the previous formula, think of two cases:

Case 1) 2x + 1 ≤ -9

Case 2) 2x + 1 ≥ 9

Case 1) 2x + 1 ≤ -9

Solve this inequality.

Linear Inequality (One Variable)

Then x ≤ -5.

Case 2) 2x + 1 ≥ 9

Solve this inequality.

Then x ≥ 4.

From case 1, x ≤ -5.

From case 2, x ≥ 4.

So the answer is [x ≤ -5 or x ≥ 4].