Absolute Value Equation Solver

Solve equations with absolute values by entering your equation in the form a|bx + c| + d = e. The Absolute Value Equation Solver breaks your equation into two separate cases and returns both solution values of x, along with a step-by-step breakdown showing the positive and negative cases. Enter coefficients for the absolute value expression and the constant on the right-hand side to get your solutions instantly.

The number multiplying the absolute value expression. Enter 1 if there is no coefficient.

The coefficient of x inside the absolute value bars.

The constant added to bx inside the absolute value bars.

The constant added outside the absolute value expression on the left side.

The value on the right side of the equation.

Results

Solution x₁ (Positive Case)

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Solution x₂ (Negative Case)

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Isolated Absolute Value

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Number of Solutions

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Solutions on the Number Line

Results Table

Frequently Asked Questions

What is absolute value?

Absolute value represents the distance of a number from zero on the number line, always resulting in a non-negative value. For example, |−5| = 5 and |5| = 5. It is denoted by two vertical bars surrounding the expression, such as |x|.

What is an absolute value equation?

An absolute value equation is an equation that contains an absolute value expression, such as |2x + 1| = 7. Because absolute value can represent two opposite values, these equations typically produce two solutions — one from the positive case and one from the negative case.

How do you solve an absolute value equation step by step?

First, isolate the absolute value expression on one side of the equation. Then split into two cases: set the inner expression equal to the positive right-hand side, and set it equal to the negative right-hand side. Solve each linear equation separately to find both solutions, and verify them in the original equation.

Can an absolute value equation have no solution?

Yes. If, after isolating the absolute value expression, the right-hand side is negative (e.g. |2x + 1| = −3), the equation has no solution. Absolute value is always non-negative, so it can never equal a negative number.

Can an absolute value equation have only one solution?

Yes. When the right-hand side equals zero after isolation (e.g. |2x + 1| = 0), both cases collapse into a single equation, giving exactly one solution: x = −1/2 in this example.

What are common mistakes to avoid when solving absolute value equations?

A frequent mistake is forgetting to isolate the absolute value before splitting into two cases. Another is neglecting the negative case entirely, which means missing one of the two solutions. Always check your answers by substituting back into the original equation, as extraneous solutions can sometimes appear.

Where are absolute value equations used in real life?

Absolute value equations appear in quality control (acceptable tolerance ranges), physics (distance and displacement), finance (profit/loss thresholds), and engineering (measuring deviation from a target value). Any scenario requiring a measure of magnitude without regard to direction uses absolute value.

What does this solver calculate, and what format should I use?

This solver handles equations in the form a|bx + c| + d = e. Enter the outer coefficient (a), the coefficient of x inside the bars (b), the constant inside the bars (c), the outer constant (d), and the right-hand side value (e). The solver returns both solutions, the isolated absolute value, and a full step-by-step table.

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