Absolute Value Equation Solver
This absolute value equation solver finds the values of x that satisfy |ax + b| = c. Enter the coefficient a, the constant b, and the right-hand side c, and it splits the equation into its two cases - ax + b = c and ax + b = −c - then solves each for x. It only returns solutions when c ≥ 0, because an absolute value can never equal a negative number, and it verifies each answer for you.
Formula
|ax + b| = c has solutions when c ≥ 0: x = (c - b) / a or x = (-c - b) / a
- An absolute value equation |ax + b| = c is solved by removing the bars in two ways: ax + b = c and ax + b = −c.
- Solving each case gives x = (c − b) / a and x = (−c − b) / a.
- There is no solution when c < 0, because |ax + b| can never be negative.
- When c = 0 the two cases coincide, so there is exactly one solution: x = −b / a.
- The coefficient a must not be 0; otherwise there is no x term to solve for.
Example Calculation
Inputs
- Coefficient (a): 2
- Constant (b): 3
- Result (c): 5
For |2x + 3| = 5, the two cases give 2x + 3 = 5 → x = 1, and 2x + 3 = −5 → x = −4. Both check out: |2(1) + 3| = 5 and |2(−4) + 3| = 5.
Frequently asked questions
How do you solve an absolute value equation?
Rewrite |ax + b| = c as two separate equations, ax + b = c and ax + b = −c, then solve each for x. That gives x = (c − b)/a and x = (−c − b)/a.
When does |ax + b| = c have no solution?
When c is negative. An absolute value is always zero or positive, so it can never equal a negative number.
Why does it sometimes give only one solution?
When c = 0, the two cases ax + b = 0 and ax + b = −0 are identical, so they share the single solution x = −b/a.
What if the coefficient a is zero?
Then there is no x term to solve and the calculator reports an error. The coefficient a must be non-zero.
How can I check a solution?
Substitute it back into |ax + b| and confirm the result equals c. The calculator does this verification automatically.