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An absolute value equation typically has two solutions, not one, because the expression inside the absolute value bars could be either positive or negative and still produce the same result.
This solver finds both solutions (when they exist) for an equation of the form \(|ax+b|=c\), with a full step-by-step solution.
How it works
Enter the coefficients a, b, and c for the equation \(|ax+b|=c\). The solver splits the equation into its two possible cases (\(ax+b=c\) and \(ax+b=-c\)) and solves each for x.
- Enter a (coefficient of x).
- Enter b (constant inside).
- Enter c (right-hand side).
- Click Calculate to see your results.
Examples
Solving |2x + 3| = 7
For the equation |2x+3|=7, splitting into the two cases 2x+3=7 and 2x+3=-7 gives the two solutions x=2 and x=-5.
Who should use it
- Solving absolute value equations for algebra coursework.
- Finding both boundary values in a symmetric range or tolerance problem.
Industry applications
- Mathematics education
- Engineering tolerance and error-margin calculations
Advantages
- Finds both solutions (when they exist) in one step.
- Includes a complete, formula-based step-by-step solution.
Limitations
- Limited to the single-absolute-value form |ax+b|=c — equations with absolute values on both sides need a different approach.
Common mistakes to avoid
- Solving only one case and missing the second valid solution.
- Forgetting that a negative right-hand side (c<0) means there's no real solution at all.
Best practices
- Always check whether c is negative before solving — it immediately tells you whether a real solution exists.
Tips
- Absolute value equations often model "how far from a target value" problems — the two solutions represent the two directions (above and below) that reach the same distance.