How this calculator works
Absolute value measures distance from zero, so it cannot be negative. For a non-negative right side, solve the two linear equations ax + b = c and ax + b = -c. Remove a duplicate when c is zero. If a is zero, compare the two constants instead of dividing.
Formula
|ax + b| = c gives x = (-b + c)/a or x = (-b - c)/a when a is non-zero and c is non-negative.
Worked example
For |2x - 3| = 5, the positive branch gives 2x = 8, hence x = 4. The negative branch gives 2x = -2, hence x = -1. Both values make the absolute value equal 5.
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Frequently asked questions
Can the answer be negative?
Yes. The unknown x can be negative even though an absolute value cannot.
What happens when c is zero?
Only ax + b = 0 remains. The two branches coincide, so there is one root when a is non-zero.
Why can every real number work?
If a = 0 and |b| = c, the equation is true without depending on x.