# Absolute Value Equations (One Variable)

How to solve absolute value equations problems: meaning, examples, and their solutions.

## Meaning

The absolute value of a number means

the distance between 0 and the number.

(regardless of its direction, sign)

So the absolute value sign

makes the number's sign to (+):

|(+)| = (+)

|(-)| = (+)

## Example 1

First, see if |*x* - 1| ≥ 0.

|*x* - 1| ≥ 0

because the result of the absolute value sign

cannot be (-).

(-) distance doesn't make sense.

So if it's (-), then the equation has no solution.

In this case, |*x* - 1| = 2 ≥ 0.

So move on to the next step.

|*x* - 1| = 2

So *x* - 1 can be 2.

(|2| = 2)

So *x* = 2 + 1 = 3.

|*x* - 1| = 2

So *x* - 1 can be -2.

(|-2| = 2)

So *x* = -2 + 1 = -1.

So *x* = -1, 3.

## Example 2

First, see if |2*x* + 8| ≥ 0.

But |2*x* - 1| = -1 < 0.

So this equation has no solution.