THE NUMORIX GUIDE
How to use the Absolute Value Equation Calculator
Last reviewed September 14, 2026
What this calculator does
Solve |a*x+b|=c by requiring a*x+b=c or a*x+b=-c.
Formula and method
Solve |a*x+b|=c by requiring a*x+b=c or a*x+b=-c. Rearranging gives x1=(c-b)/a and x2=(-c-b)/a when c is positive; c=0 gives one repeated solution.
Variables and inputs
The defaults are a=2, b=-4, c=6. The UI displays the equation |a*x+b|=c and has no units; a and c determine whether real solutions exist.
Worked example
For |2x-4|=6: case 1, 2x-4=6 -> 2x=10 -> x=5; case 2, 2x-4=-6 -> 2x=-2 -> x=-1. Both check to an absolute value of 6.
How to interpret the result
The equation asks for points whose transformed expression is at distance c from zero, so a positive c commonly produces two solutions.
Common mistakes to avoid
Do not solve only a*x+b=c; include the negative case. If c is negative, stop because an absolute value cannot equal it over the reals.
Assumptions and limitations
The engine assumes a is nonzero and does not explicitly handle a=0. It returns NaN for negative c, and outputs are not rounded by the engine.
Practical use and checks
This calculator solves equations in the form |a*x + b| = c by considering both possible signs of the expression. Enter a = 2, b = -4, and c = 6 as a check. The two cases are 2x - 4 = 6 and 2x - 4 = -6, giving x = 5 and x = -1. Substitute both values into the original absolute-value equation; each should produce 6. This two-case check is essential because keeping only the positive case loses a valid solution. If c is negative, there is no real solution because an absolute value cannot be negative. If c is zero, there is one solution when a is nonzero. When a is zero, the equation is a constant statement: it has all real numbers as solutions when |b| equals c, and no solutions otherwise. The page assumes real coefficients and does not enforce domain restrictions that an applied problem may add. A rounded result should be checked by substitution, especially when coefficients are fractional or near zero. Use the solution set, not just the first displayed field, when deciding which values are admissible.