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Absolute Value Equation Solver

Math

|2x − 3| = 7

Inside the bars
Inside the bars
Right-hand side

Try an example

Two Solutions

Two distinct values of x satisfy the equation.

Solution x1

x = -2

Solution x2

x = 5

Verification

x = -2

LHS = 7

RHS = 7

Valid

x = 5

LHS = 7

RHS = 7

Valid

Branch Equations

Case 1 (ax + b = c)

2x − 3 = 0x + 7

x = 5

Case 2 (ax + b = −c)

2x − 3 = 0x − 7

x = -2

Graph

y = |ax + b|

Right-hand side

Solution(s)

Step-by-Step Derivation

Start with the equation |2x − 3| = 7.

Case 1: the inside expression is non-negative, so ax + b = c.

Step 1: Start with the equation 2x − 3 = 0x + 7

Step 2: Move the x-terms to the left side by subtracting 0x from both sides: 2x − 0x − 3 = 7

Step 3: Combine like terms: (2 − 0)x − 3 = 7 → 2x − 3 = 7

Step 4: Move the constant to the right side by subtracting -3 from both sides: 2x = 7 − (-3) = 10

Step 5: Divide both sides by 2: x = 10 / 2 = 5

Case 2: the inside expression is negative, so ax + b = −c.

Step 1: Start with the equation 2x − 3 = 0x − 7

Step 2: Move the x-terms to the left side by subtracting 0x from both sides: 2x − 0x − 3 = -7

Step 3: Combine like terms: (2 − 0)x − 3 = -7 → 2x − 3 = -7

Step 4: Move the constant to the right side by subtracting -3 from both sides: 2x = -7 − (-3) = -4

Step 5: Divide both sides by 2: x = -4 / 2 = -2

Final solution(s): x = -2, x = 5.

About This Tool

Absolute Value Equation Solver – Branches, Extraneous Roots & Graphs

An absolute value equation contains an expression wrapped in| | bars, such as |ax + b| = c. Because|expression| strips away the sign of whatever is inside it, every absolute value equation actually represents two possible cases: the expression inside the bars could be non-negative, or it could be negative. Solving the equation means splitting it into these two branches, solving each branch as an ordinary linear equation, and then checking every candidate to make sure it truly satisfies the original equation.

The Three Supported Forms

This tool covers the three forms of absolute value equations that appear throughout algebra courses:

|ax + b| = c

A constant on the right. Two, one, or zero solutions depending on the sign of c.

|ax + b| = cx + d

A linear expression on the right. Requires checking every candidate for extraneous roots.

|ax + b| = |cx + d|

Two absolute values. Every branch root is automatically valid — no extraneous check needed.

Why the Right-Hand Side Sign Matters

For the |ax + b| = c form, the sign of c immediately tells you what to expect. Since an absolute value can never be negative, c < 0 guarantees no solution. When c = 0, the two branches collapse to the same equation, giving exactly one solution. When c > 0, the two branches produce two distinct values, unless they coincide.

Distance interpretation
|x − k| = r asks: which points are exactly a distance r fromk on the number line? The two solutions are k − r and k + r — the branch split is really just this distance idea in algebraic form.

Extraneous Roots in the Linear-RHS Form

When the right-hand side contains x — the |ax + b| = cx + d form — splitting into branches can introduce extraneous solutions: candidate values that solve a branch equation but fail the original equation because they would requirecx + d to be negative. This tool substitutes every branch candidate back into|ax + b| = cx + d and discards any candidate whose left- and right-hand sides do not match, showing the exact numeric check that caused the rejection.

Double Absolute Value Equations

For |ax + b| = |cx + d|, both branch equations — ax + b = cx + d and ax + b = −(cx + d) — are exact algebraic splits of the original equation, so every root found this way is automatically valid. The only cleanup needed is deduplicating roots when both branches happen to produce the same value.

Degenerate Cases (a = 0)

If the coefficient a is zero, the expression inside the bars becomes a plain constant |b|, and the equation no longer depends on x at all. Each branch is still solved through the shared linear-equation logic, which correctly reports an identity (infinitely many solutions) when the constant equation is true for every x, or a contradiction (no solution) when it is false.

Reading the Graph

The graph plots y = |ax + b| — a V-shaped curve with its corner (the vertex) at x = −b/a — against the right-hand side, drawn as a horizontal line, a sloped line, or a second V-curve depending on the selected form. Every accepted solution is marked where the two curves cross; these intersection points are the graphical meaning of "solving the equation."

Exact Fractions and Precision

Solutions are displayed both as decimals, rounded to your chosen precision (0 to 10 places), and as exact simplified fractions when the result is not a whole number. Toggle Show fractions to hide the fraction line if you only need decimal answers.

Practical Applications

  • Tolerance and quality control — a part is acceptable when its measurement differs from a target by at most a fixed amount.
  • Temperature ranges — finding values a fixed number of degrees away from a set point.
  • Error bounds — expressing how far a measured or estimated value can deviate from the true value.
  • Distance problems — any "how far apart" question on a number line reduces to an absolute value equation.

Tips for Accurate Results

  • Use the Try an example buttons to quickly see two-solution, one-solution, no-solution, extraneous-root, and double-absolute-value cases.
  • Toggle Show step-by-step to see the full branch derivation and every verification check.
  • Remember: for the |ax + b| = c form, a negative c always means no solution — no calculation is needed to know that in advance.

Frequently Asked Questions

Is the Absolute Value Equation Solver free?

Yes, Absolute Value Equation Solver is totally free :)

Can I use the Absolute Value Equation Solver offline?

Yes, you can install the webapp as PWA.

Is it safe to use Absolute Value Equation Solver?

Yes, any data related to Absolute Value Equation Solver only stored in your browser (if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

How does the Absolute Value Equation Solver work?

Enter coefficients for one of three forms — |ax + b| = c, |ax + b| = cx + d, or |ax + b| = |cx + d| — and the tool splits the equation into its two branch cases, solves each branch as a linear equation, substitutes every candidate back into the original equation to reject extraneous roots, and reports the final valid solutions along with a full step-by-step derivation.

What is an extraneous solution and why does it happen?

An extraneous solution is a candidate produced by a branch equation that does not actually satisfy the original absolute value equation. It happens in the |ax + b| = cx + d form because replacing |ax + b| with −(ax + b) can create a root where the right-hand side is negative, which is impossible for an absolute value. The tool checks every candidate and discards any that fail this test.

Why does |ax + b| = c have no solution sometimes?

An absolute value is always greater than or equal to zero, so it can never equal a negative number. Whenever the right-hand side c is negative, the equation |ax + b| = c has no real solution — the tool detects this automatically and explains why.

What does it mean when the two branches give the same root?

When the right-hand side of |ax + b| = c is exactly zero, both branch equations (ax + b = c and ax + b = −c) reduce to the same equation, so they produce the same root. The tool automatically deduplicates matching roots and reports a single solution in that case.

Can this tool solve equations with two absolute values?

Yes. For |ax + b| = |cx + d|, the tool splits the equation into ax + b = cx + d and ax + b = −(cx + d). Both cases are exact algebraic splits of the original equation, so no extraneous-root check is needed — every root found is automatically valid.

How accurate are the results, and can I see exact fractions?

Calculations use 64-bit floating point arithmetic and are accurate to the decimal precision you select (0–10 places). Solutions are also converted to exact simplified fractions and mixed numbers whenever a small-denominator fraction closely matches the decimal result.