An absolute value equation measures distance from zero. If c is nonnegative, split into ax+b=c and ax+b=-c. If c is negative, there is no real solution because distance cannot be negative.
⚠Note: This solver handles real-number equations in the form |ax+b|=c. If a=0, the expression is constant and must be checked separately.
How Absolute Value Equations Work
The core idea is distance. The expression inside the absolute value can be positive or negative, but the output is always nonnegative. Splitting into two cases captures both possible signs.
Positive Case
Set ax+b equal to c and solve.
Negative Case
Set ax+b equal to -c and solve.
No Solution
A negative right side is impossible for absolute value.
Verification
Substitute each solution back into the original equation.
💡 Example: |2x-3|=5 gives 2x-3=5 or 2x-3=-5, so x=4 or x=-1.
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