Absolute Value Equation Calculator
Solve equations of the form |mx + b| = c. Enter m, b and c — the calculator splits the equation into its two cases, solves each linear piece, and shows both solutions on a number line (or reports 'no solution' when c is negative).
Solve |m·x + b| = c. Enter m, b and c below.
What does absolute value mean?
The absolute value of a number is its distance from 0 on the number line, ignoring direction. Because distance is never negative, |x| is always 0 or positive: |3| = 3 and |−3| = 3. You can think of the bars as stripping off the sign.
How the equation is solved, piece by piece
Each card explains one thing this tool does with the equation you enter — how the bars split it into two ordinary linear equations, and how it detects an equation that has no real solution.
Formula — split into two cases
Because |X| equals either X or −X, the equation |X| = a (with a ≥ 0) breaks into two ordinary linear equations that the tool solves separately.
Answer — no solution (RHS < 0)
Since |X| is a distance, it is never negative. If the right side is negative the tool reports ∅ immediately — there is no real value of x that can satisfy the equation.
Features of this calculator
- Solves any equation of the form |mx + b| = c with integer, decimal or negative coefficients
- Shows both cases side by side — mx + b = c and mx + b = −c
- Reports 'no solution' when c is negative, and a single repeated solution when c = 0
- Handles the degenerate m = 0 case (|b| = c) and reports 'all real numbers' or 'no solution' correctly
- Plots both solutions on an interactive number-line visual for quick sanity checks
- Full step-by-step working, so it doubles as a study aid — not just an answer key
Frequently asked questions
+How do I solve an absolute value inequality?
Use the same case-split idea. For c > 0, |X| < c becomes −c < X < c (an interval), and |X| > c becomes X < −c OR X > c (a union). Solve each linear piece with the Inequality Solver.
+Why can there be two solutions?
Because the number line has two points at any given distance from 0 — one on each side. So an equation like |x − 4| = 3 is satisfied by both x = 7 (three to the right of 4) and x = 1 (three to the left).
+What if my equation has an absolute value on both sides?
|A| = |B| leads to two cases as well: A = B or A = −B. Solve both and check each candidate in the original equation — sometimes one is extraneous. This calculator handles the more common single-bar form.
+How do I check my answer?
Substitute each candidate back into the original equation and confirm the left side (after taking the absolute value) really equals the right side. If it doesn't, the value is extraneous and should be discarded.