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].