Last updated: July 26, 2026
Absolute Value Equation Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
a|bx + c| + d = e
Where:
- a,b,c,d,e=Equation coefficients/constants
- x=Unknown variable
Worked Examples
Two-solution case: |x - 3| = 5
Most common absolute value equation pattern.
- 1|x - 3| = 5
- 2Case 1: x - 3 = 5 → x = 8
- 3Case 2: x - 3 = -5 → x = -2
Single-solution case: |2x - 4| = 0
Absolute value equals zero gives one root.
- 1|2x - 4| = 0
- 2Inside must be zero: 2x - 4 = 0
- 3x = 2
No-solution case: |x| = -3
Absolute value cannot be negative.
- 1|x| + 3 = 0 → |x| = -3
- 2Impossible because |x| ≥ 0
Introduction
This calculator solves linear absolute-value equations in the form a|bx + c| + d = e. It returns the full solution set and correctly handles all result types: two roots, one root, no solution, or all real numbers.
Core Solving Idea
Isolate the absolute value term first, then split into two linear equations.
Rearrange to |bx+c| = (e-d)/a
If right side < 0 → no solution
If right side = 0 → one solution from bx+c=0
If right side > 0 → two linear cases
Two-Case Method
When |u| = k and k > 0, solve two equations.
Case 1: u = k
Case 2: u = -k
Here u = bx+c, k = (e-d)/a
Solve both linear equations and combine unique roots
Possible Result Types
Absolute-value equations can produce four outcome classes.
Two roots (most common)
One root (typically when target is zero)
No solution (target negative)
All real numbers (identity case)
How to Use This Calculator
Enter coefficients a, b, c, d, e from your equation a|bx+c|+d=e.
Input all five numeric coefficients
Read Solution Set for final answer
Read solution count and equation type for interpretation
Use normalized target to verify algebra manually
Common Mistakes
Most mistakes happen when the split-case step is skipped or sign handling is wrong.
Forgetting the negative case (u = -k)
Trying to solve with negative target as if valid
Sign errors while moving constants
Not deduplicating repeated roots when target = 0
Where This Appears
Absolute-value equations model distance constraints and tolerance bands.
Distance from a point on a line
Quality-control tolerance intervals
Piecewise linear modeling
Intro algebra and precalculus problems
FAQs
Why can absolute value equations have two solutions?
Because |u| = k means u can be +k or -k, creating two linear cases.
When is there no solution?
If the isolated right side is negative, because absolute value is never negative.
When is there only one solution?
Usually when the isolated target equals zero, forcing bx+c=0.
Can there be infinitely many solutions?
Yes, in identity cases such as 0|bx+c|+d=e when d=e.
Do I always need to divide by a first?
Yes, if a is nonzero. Isolate |bx+c| before splitting cases.
What if b=0?
Then |bx+c| is constant (|c|), so the equation is either always true or never true depending on constants.