Solve absolute value equations of the form a·|b·x + c| + d = e·x + f by entering the six coefficients below. Choose between a one absolute value or two absolute value equation type, fill in the fields, and get back all solutions for x — including verification of each case.
Solutions for x
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Solution x₁
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Solution x₂
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Number of Solutions
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Equation Solved
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Absolute value represents the distance of a number from zero on the number line, always returning a non-negative result. For example, |−5| = 5 and |5| = 5. It is written using two vertical bars surrounding the expression, such as |x|.
An absolute value equation is any equation that contains an absolute value expression. For example, |2x + 3| = 7 is an absolute value equation. Because the expression inside the absolute value can be positive or negative, these equations typically produce two separate cases to solve.
First, isolate the absolute value expression on one side. Then split into two cases: one where the inner expression is non-negative (Case 1: inner = right side) and one where it is negative (Case 2: inner = −right side). Solve each linear equation separately, then check that each solution satisfies its corresponding condition.
Yes. If the right-hand side of an isolated absolute value expression is negative — for example |x + 1| = −3 — there is no real solution, since absolute value is always non-negative. The calculator will indicate when no valid solutions exist.
Yes, this happens when both cases yield the same value of x, or when one case produces a solution that violates its condition and is discarded. For instance, |x| = 0 yields only x = 0.
A one absolute value equation has a single absolute value expression, such as a·|bx + c| + d = ex + f. A two absolute value equation has an absolute value on both sides, like a·|bx + c| + d = e·|fx + g| + h, which creates up to four possible cases to consider.
The most frequent errors are: forgetting to check that solutions satisfy their case conditions, not isolating the absolute value before splitting into cases, and assuming there is always exactly two solutions. Always verify each answer by substituting back into the original equation.
Absolute value equations appear in physics (distance and displacement), engineering (tolerance and error margins), finance (measuring deviation from a target), and computer science (algorithms comparing distances or differences). Any scenario measuring how far a value strays from a reference point can involve absolute value.