# Absolute Value Equation (One Variable)

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

## Example|x - 1| = 2

The number inside the absolute value sign,

x - 1,

is either (+) or (-).

So think of two cases:

Case 1: x - 1 ≥ 0

Case 2: x - 1 < 0

Case 1: x - 1 ≥ 0

(x - 1) is not (-).

So the sign of (x - 1) doesn't change.

So |x - 1| = x - 1.

So x - 1 = 2.

Move -1 to the right side.

Then x = 3.

This is the answer for case 1.

Linear Equation (One Variable)

Case 2: x - 1 < 0

(x - 1) is (-).

Then the sign of (x - 1) changes.

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

So -(x - 1) = 2.

Multiply the minus sign to both x and -1.

Then -(x - 1) = -x + 1.

Multiply a Monomial and a Polynomial

Move +1 to the right side.

Then -x = 1.

Divide both sides by -1.

Then x = -1.

This is the answer for case 2.

Case 1: x = 3

Case 2: x = -1

So x = -1, 3.

(This means x = -1 or x = 3.)

So

x = -1, 3

is the answer.

## Example|2x - 3| + 1 = 0

Move +1 to the right side.

Then |2x - 3| = -1.

Recall that

the absolute value sign changes (-) to (+).

So the right side,

the result of an absolute value,

cannot be (-).

But the right side is -1: (-).

So there's no solution.

So

no solution

is the answer.