Absolute Value Equation Calculator

An absolute-value equation splits into two ordinary equations, giving up to two solutions.

How the Math Works

The Absolute Value Equation Calculator solves equations of the form |ax + b| = c by leveraging the definition of absolute value. Since |x| = c means x = c or x = -c, the equation |ax + b| = c splits into two separate linear equations: ax + b = c and ax + b = -c. Solving each gives x = (c - b)/a and x = (-c - b)/a, which are the two solutions. This approach works for any real numbers a, b, and c (with a ≠ 0 and c ≥ 0 for real solutions).

Practical Applications

To use this calculator, simply enter the coefficients a, b, and c from your absolute value equation |ax + b| = c into the respective fields. The calculator will instantly compute both possible solutions using the formula x = (±c - b)/a. This is particularly useful when solving problems in algebra homework, checking work for systems of equations, or verifying solutions in larger mathematical models where absolute value constraints appear.

Day-to-Day Use

Understanding absolute value equations helps in everyday situations involving tolerances and margins of error. For example, when manufacturing products, quality control often requires items to be within a certain distance from a target measurement (like a part needing to be 5cm ± 0.1cm). This calculator can quickly determine the acceptable range of variations. It's also helpful for financial planning when setting budget ranges, or in construction when measuring distances that must meet minimum and maximum requirements.

Worked example

|x − 3| = 5 → x − 3 = 5 (x = 8) or x − 3 = −5 (x = −2).

FAQ