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.