# Absolute Value Inequality (One Variable)

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

## Formula|x| < a

|x| < a

→ -a < x < a.

By the same way,

|x| ≤ a

→ -a ≤ x ≤ a.

Absolute Value

## Example|x - 2| < 5

|x - 2| < 5

The left side is an absolute value.

And it is [less than] the right side.

Then, by the above formula,

-5 < x - 2 < 5.

To remove -2 in the middle side,

+2 to each side.

Linear Inequality (One Variable)

-5 + 2 = -3

5 + 2 = 7

So

-3 < x < 7

This is the answer.

## Formula|x| > a

|x| < a

→ x < -a or x > a.

By the same way,

|x| ≤ a

→ x ≤ -a or x ≥ a.

## Example|2x + 1| ≥ 9

|2x + 1| ≥ 9

The left side is an absolute value.

And it is [greater than] (or equal to) the right side.

Then, by the above formula,

think of two cases.

Case 1: 2x + 1 ≤ -9

Case 2: 2x + 1 ≥ 9

First write case 1:

2x + 1 ≤ -9

Move +1 to the right side.

Then 2x ≤ -10.

Divide both sides by 2.

Then x ≤ -5.

This is the answer for case 1.

Case 2: 2x + 1 ≥ 9

Move +1 to the right side.

Then 2x ≥ 8.

Divide both sides by 2.

Then x ≥ 4.

This is the answer for case 2.

Case 1: x ≤ -5

Case 2: x ≥ 4

Then x ≤ -5 or x ≥ 4.

So

x ≤ -5 or x ≥ 4

is the answer.