Absolute Value Equation Solver

Solve absolute value equations |ax + b| = c with step-by-step solutions.

Solve |ax + b| = c

What Is the Absolute Value Equation Solver?

The Absolute Value Equation Solver finds all values of x satisfying |ax + b| = c. Because |y| = c means y = c or y = −c, absolute value equations typically produce two solutions. The tool handles all three cases: no solution (c < 0), one solution (c = 0), and two solutions (c > 0).

Formula

Equation: |ax + b| = c Case 1: c < 0 → No solution (absolute value ≥ 0 always) Case 2: c = 0 → ax + b = 0 → x = −b/a (one solution) Case 3: c > 0 → Two equations: ax + b = c → x₁ = (c − b)/a ax + b = −c → x₂ = (−c − b)/a

How to Use

Enter the coefficients a and b for the expression inside the absolute value bars, and the constant c on the right side. Click Solve to see the case analysis and all solutions with verification.

Example Calculation

Solve |2x − 3| = 7 a=2, b=−3, c=7 Case 1: 2x − 3 = 7 → 2x = 10 → x = 5 Case 2: 2x − 3 = −7 → 2x = −4 → x = −2 Verify: |2(5)−3| = |7| = 7 ✓ |2(−2)−3| = |−7| = 7 ✓

Understanding Absolute Value Equation

Absolute value equations model many real-world situations involving distance and tolerance. In manufacturing, a tolerance specification like "the part must be 50mm ± 2mm" is an absolute value inequality. In signal processing, absolute value measures the magnitude of a signal.

Geometrically, |ax + b| = c defines two points on the number line equidistant from −b/a. Extending to two dimensions, |x − a| + |y − b| = r defines a diamond (rotated square) centered at (a, b).

Absolute value appears throughout advanced mathematics: in the definition of limits (|f(x) − L| < ε), in metric spaces, in the complex modulus |z| = √(a² + b²), and as the L¹ norm in optimization and machine learning.

Frequently Asked Questions

What is absolute value?

The absolute value |x| of a number is its distance from zero on the number line — always non-negative. |3| = 3, |−3| = 3, |0| = 0.

Why are there two solutions when c > 0?

Because |ax+b| = c means the expression ax+b is distance c from zero. There are two numbers at distance c from zero: +c and −c, leading to two equations.

When is there no solution?

When c < 0, since |ax+b| ≥ 0 for all x, it can never equal a negative number. For example, |x+1| = −5 has no solution.

What if a = 0?

If a = 0, the equation becomes |b| = c. This has the solution set of all real numbers if |b| = c, or no solution otherwise.

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