Absolute Value Inequalities (One Variable)

Absolute Value Inequalities (One Variable)

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

Formulas

|x| < a: -a < x < a.

|x| < a means
the distance between 0 and x is lesser than a.

So it's -a < x < a.

|x| > a: x < -a or 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

Solve the inequality. |x - 2| < 5

|x - 2| < 5

Then, -5 < x - 2 < 5.

To remove -2 in the middle term,
add +2 on each side.

Example 2

Solve the inequality. |2x + 1| >= 9

|2x + 1| ≥ 9

Then case 1:
2x + 1 ≤ -9

|2x + 1| ≥ 9

Then case 2:
2x + 1 ≥ 9

Example 3

Solve the inequality. |-x + 7| <= 0

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.