Overview
The absolute value |E| is the distance of E from zero: it equals E when E is nonnegative and −E when E is negative. An absolute value equation therefore often has two solutions, since c and −c share the same absolute value.
Formula
|ax + b| = c with c > 0 means ax + b = c or ax + b = −c. If c < 0 there is no solution.
How it works
Check the sign of c
If c is negative, there is no real solution.
Write both cases
Set the expression inside the absolute value equal to c and to −c.
Solve each case
Solve the two resulting linear equations.
Verify
Substitute each solution into the original equation.
Worked examples
Example 1: |2x − 4| = 6
2x − 4 = 6 gives x = 5, and 2x − 4 = −6 gives x = −1.
Example 2: |x + 1| = −2
No solution, because an absolute value cannot be negative.
Frequently asked questions
When does the equation have one solution?
When c = 0, because the expression inside the absolute value must equal zero.
Why solve two cases?
Because c and −c have the same absolute value when c > 0.
Are there complex solutions?
Over the real numbers there are none when c < 0; over the complex numbers the absolute value is defined differently.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.