Absolute Value Equation Solver
Solve any absolute value equation of the form |ax+b| = c and see its solutions as the points where the V-shaped graph crosses the horizontal line y = c.
The equation has two solutions.
- 1
Positive branch: ax + b = c
8 ÷ 1 = 8 - 2
Negative branch: ax + b = −c
-2 ÷ 1 = -2 - 3
First solution (smaller root)
-2Roots are sorted in ascending order.
How does this calculator work?
To solve |ax+b| = c, check c: if c is negative there is no solution; if c is zero the only answer is x = −b/a; if c is positive there are two answers, x = (c−b)/a and x = (−c−b)/a, found where the V-shaped graph y = |ax+b| crosses the line y = c.
Formula
How this is calculated
Enter the three numbers a, b and c that appear in |ax + b| = c. The coefficient a multiplies x inside the bars, b is the constant added inside, and c is the value on the right-hand side. The coefficient a must be non-zero, otherwise there is no x to solve for.
An absolute value measures distance from zero, so it is never negative. If c < 0 the equation |ax + b| = c has no solution. If c = 0 the expression inside the bars must equal zero, giving the single solution x = −b/a. When c > 0 the expression inside can equal either +c or −c, producing two linear equations whose solutions are x = (c − b)/a and x = (−c − b)/a.
Graphically, y = |ax + b| is a V-shaped curve and y = c is a horizontal line. The solutions are exactly the x-values where the two graphs meet: two crossings when c > 0, a single touch at the vertex when c = 0, and no crossing when c < 0. The plot marks these intersection points in red.
Frequently asked questions
The quantity inside the bars can be either positive or negative and still have the same absolute value, so |ax+b| = c splits into ax+b = c and ax+b = −c, giving up to two solutions.
When c is negative. Absolute value is a distance and can never be negative, so |ax+b| = c is impossible for any c < 0.
There is exactly one solution. The inside expression must be zero, so ax + b = 0, giving x = −b/a, the vertex of the V touching the x-axis.
Also known as
TG we-Calculate Editorial Team. (2026). Absolute Value Equation Solver [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/absolute-value-equation-solver
TG we-Calculate Editorial Team. "Absolute Value Equation Solver." TG we-Calculate. 2026. https://we-calculate.com/calculator/absolute-value-equation-solver.
TG we-Calculate Editorial Team, "Absolute Value Equation Solver," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/absolute-value-equation-solver
@misc{wecalculate_absolute_value_equation_solver, title = {Absolute Value Equation Solver}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/absolute-value-equation-solver}}, year = {2026}, note = {TG we-Calculate} }
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