Solve equations of the form a|x + b| = c and see when the result has one solution, two solutions, no solution, or infinitely many solutions.
Last updated: June 2026 | By Patchworkr Team
An absolute value equation asks for values whose distance from a point matches a target value. That is why the same equation can produce two answers on opposite sides of the center point.
In this calculator, the equation has the form a|x + b| = c.
The symbols |x| mean the absolute value of x.
Absolute value measures distance from zero, so |x| is always zero or positive.
In an equation like a|x + b| = c, the expression inside the bars is the part you isolate before solving.
The first step is to isolate the absolute value expression:
If c / a is negative, there is no real solution, because an absolute value cannot be less than zero.
Suppose the equation is 2|x - 3| = 10.
The two answers come from the two directions that are the same distance from 3.
An absolute value equation can have one solution, two solutions, no solution, or infinitely many solutions.
If the absolute value part equals zero, the equation usually collapses to a single solution.
If the right side becomes negative after dividing, the equation has no solution.
If the coefficient a is zero, the equation either becomes true for every real number or impossible for every real number, depending on c.
Can there be one solution?
Yes. If the absolute value part equals zero, the positive and negative cases collapse to the same answer.
Can there be no solution?
Yes. If the isolated absolute value would need to equal a negative number, there is no real solution.
Does this accept decimals?
Yes. Coefficients and constants can be decimals or scientific notation.
What if a is zero?
If a is zero, the equation collapses to either all real numbers or no solution depending on c.
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