Absolute Value Equation Solver
Solve an absolute value equation of the form |ax + b| = c for x. Splits into the two linear cases, handles the no-solution and infinite-solutions edge cases, and shows a step-by-step derivation.
Input
Coefficient of x inside the absolute value.
Constant added to ax, inside the absolute value.
Value the absolute value equals.
Output
one_solution, two_solutions, no_solution, or infinite_solutions.
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/absolute-value-equation-solver \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"a": "2",
"b": "3",
"c": "7"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `absolute-value-equation-solver` tool (Absolute Value Equation Solver) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/absolute-value-equation-solver/"
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title="Absolute Value Equation Solver — iotools.cloud"
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allow="clipboard-write"
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<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
Also available with
Guides
The Absolute Value Equation Solver finds x in any equation of the form |ax + b| = c. Enter the three coefficients and it returns the solution (or solutions), a status flag, and a step-by-step derivation showing how each answer was reached.
It's built for algebra students checking homework, teachers preparing worked examples, and anyone who needs a quick, exact answer without splitting the equation into cases by hand.
How to use it
- Enter a — the coefficient of x inside the absolute value bars.
- Enter b — the constant added to ax, also inside the bars.
- Enter c — the value the absolute value equals.
- Read the Solution or the Step-by-Step Derivation for the full working.
Results update automatically as you type. For example, entering a = 2, b = 3, c = 7 returns x = 2 or x = -5 — solved from |2x + 3| = 7.
How is an absolute value equation solved?
An absolute value strips the sign off whatever's inside it, so |2x + 3| = 7 is really asking: what value of x makes 2x + 3 equal to either 7 or -7? That gives two ordinary linear equations to solve separately:
- Case 1: ax + b = c → x = (c − b) / a
- Case 2: ax + b = −c → x = (−c − b) / a
Both cases are always checked, since an equation like |x| = 5 is satisfied by both x = 5 and x = -5 — either one, substituted back in, makes the original equation true.
Why does it sometimes say "no solution"?
An absolute value measures a distance, and distances are never negative. If c is negative — for example |x| = -1 — no real number can ever satisfy the equation, so the calculator reports no solution rather than a blank or invalid result.
There's a second no-solution case: when a = 0, the x term disappears entirely and the equation collapses to a plain statement about b and c (|b| = c). If that statement is false, no x can rescue it — the equation has no solution regardless of what x is.
What if a is 0 and the equation is true?
When a = 0 and |b| = c actually holds — for example |0x + 5| = 5 — the x term contributes nothing to either side, so the equation is true no matter what x is. That's infinitely many solutions, not zero and not a specific number.
What if c is 0?
When c = 0, both cases (ax + b = 0 and ax + b = 0) are identical, so there's only one solution rather than two — the calculator collapses the duplicate automatically instead of listing the same value twice.
A number-line way to think about it
|ax + b| = c can also be read as "the expression ax + b is a distance of c from zero." On a number line, that means ax + b lands at either +c or −c — the same two cases the calculator solves. This framing is useful for word problems phrased as a distance ("how far from zero," "within c units of a target") rather than as a bare equation.
For a plain linear equation without the absolute value bars, see the Linear Equation Solver. For a curved (quadratic) equation instead of a straight line, see the Quadratic Formula Solver.
Privacy
This calculator runs entirely in your browser. Your coefficients are never uploaded, logged, or stored — the computation happens locally on your device.