Absolute value equation solver

This absolute value equation solver takes the three coefficients a, b and c of |ax + b| = c and returns every real solution, distinguishing the four cases that matter: two solutions, exactly one, none at all when c is negative, and infinitely many when a is zero and |b| equals c. It runs deterministically, with no approximation methods, so the answer you test in the browser is the same answer the API returns.

Why one equation becomes two

The absolute value of a number is its distance from zero, so saying |ax + b| = c really says the quantity ax + b sits exactly c units away from zero. When c is positive, that happens in two places: ax + b = c or ax + b = −c. Each branch is an ordinary linear equation, solved by isolating x, which gives x = (−b + c)/a on one side and x = (−b − c)/a on the other. The absolute value equation solver performs exactly this split for you, then sorts the two roots ascending so the output is stable and easy to compare against a textbook answer. Take |2x − 4| = 10: the branches are 2x − 4 = 10 and 2x − 4 = −10, giving x = 7 and x = −3. Both are returned, and you can verify each by substituting it back into the original expression. This is the same procedure taught in algebra courses, just executed without sign slips: the most common student mistake is forgetting the negative branch entirely, which loses half the solution set. Automating the split removes that whole class of error.

The edge cases: one solution, none, or infinitely many

Not every absolute value equation has two answers, and a solver that always returns two roots is wrong often enough to be dangerous. When c equals zero, the two branches collapse into the single equation ax + b = 0, so there is exactly one solution at x = −b/a; |x + 3| = 0 is satisfied only by x = −3. When c is negative, there is nothing to find at all: an absolute value is a distance and a distance cannot be negative, so |3x + 1| = −2 has no real solutions, and the solver says so with an empty list rather than inventing numbers. The trickiest case is a = 0, where x disappears from the equation entirely. If |b| happens to equal c, every real number is a solution and the solver reports infinitely many; otherwise the equation is a contradiction like |5| = 3 and the call is rejected as invalid input, because a coefficient set that cannot involve x is a modelling mistake, not a solvable problem. Covering these four cases explicitly is what separates a correct solver from a quadratic-formula-style guess.

Using the API and interpreting the output

You send three numbers — a, b and c — and receive the reconstructed equation, the coefficients echoed back, and the solution set. The solutions field is a list of real numbers rounded to ten decimal places so that results are byte-identical across machines and safe to snapshot in tests; solution_count tells you how many there are without counting elements. When the solution set is empty you still get a successful response with an empty list, because 'no real solutions' is a valid mathematical answer, not an error. When a is zero and |b| equals c, the response carries infinite_solutions instead of a list. Only genuinely malformed requests are rejected: a missing or non-numeric coefficient, a non-finite value like Infinity, or the contradictory a = 0 case described above. The capability runs on our edge with nothing stored, and the identical code powers the free widget on this page, so you can prototype by hand and only pay — $0.002 per call — once you wire it into a pipeline. Typical integrations are tutoring backends checking student answers, symbolic pre-processors, and validation of generated exercise sets.