Solve real equations of the form a|x + b| = c and identify whether there are two solutions, one solution, no real solution, or all real numbers.
Last updated: August 2026 | By Summa Calculator
This calculator solves equations with one absolute-value expression in the specific form a|x + b| = c.
The expression |x + b| is the distance from x to -b. After accounting for the coefficient a, the equation asks which points lie a fixed distance from that center.
It does not solve arbitrary typed equations, equations with absolute-value expressions on both sides, nonlinear expressions inside the bars, or systems of equations.
When a ≠ 0, divide by a:
Two solutions: x = -b ± c/a.
One solution: x = -b.
No real solution, because absolute value cannot be negative.
A negative coefficient a is valid. What matters is the sign of the isolated value c/a, not the sign of c by itself.
Check: 2|8 - 3| = 10 and 2|-2 - 3| = 10, so both values satisfy the original equation.
If a = 0, the left side is zero for every real x, regardless of b. The equation becomes 0 = c.
Therefore, a = 0, c = 0 gives all real numbers, while a = 0, c ≠ 0 gives no real solution.
Why are there usually two answers?
A positive distance from the center -b can be reached on either side, giving x = -b + c/a and x = -b - c/a after the absolute value is isolated.
Can a negative coefficient be used?
Yes. Divide by a first. If c/a is non-negative, real solutions may exist; if c/a is negative, there is no real solution.
Does the solver accept decimals?
Yes. It accepts finite decimals, zero, negative values, and scientific notation for a, b, and c.
Should I check the answers?
Yes. Substituting each reported x into the original equation is a direct way to confirm the equality.
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