# Absolute Value Equation (One Variable)

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

## Example 1

### Example

### Solution

The number inside the absolute value sign, x - 1, can either be plus or minus.

So think of two cases:

Case 1) If x - 1 ≥ 0

Case 2) If x - 1 < 0

Case 1) If x - 1 ≥ 0

So |x - 1| = x - 1.

So |x - 1| = 2 becomes x - 1 = 2.

Solve x - 1 = 2.

Linear Equation (One Variable)

Move -1 to the right side.

Then x = 3.

This is the answer for case 1.

Case 2) If x - 1 < 0

So |x - 1| = -(x - 1).

Recall that the absolute value sign changes minus to plus.

So the minus number (x - 1) is changed to a plus number -(x - 1).

So |x - 1| = 2 becomes -(x - 1) = 2.

Solve the equation -(x - 1) = 2.

-(x - 1) = -x + 1

So -x + 1 = 2.

Move +1 to the right side.

Then -x = 1.

Divide both sides by -1.

Then x = -1.

This is the answer for case 2.

From case 1, x = 3.

From case 2, x = -1.

So x = 3, -1 is the answer.

## Example 2

### Example

### Solution

Move +1 to the right side.

Then |2x - 3| = -1.

The absolute value sign changes the sign of a number to 0 or plus.

So the left side, |2x - 3| can be either 0 or plus.

But the right side, -1, is minus.

So there's no x that makes this equation true.

So there's no solution.

So [no solution] is the answer.