The formula this page uses
|x| = x when x ≥ 0, |x| = −x when x < 0
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Absolute Value Calculator
Solve absolute value problems with clear steps, notation, and a final check.
|x|
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Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
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Algebra
Absolute Value Calculator explained
The short version
- Absolute value is distance from zero, so the answer is never negative.
- |2x − 5| = 9 has two answers, because the inside can be 9 or −9 and both are 9 units from zero.
- Simplify everything inside the bars first, then let the bars strip the sign.
What each part means
SymbolWhat it means
x — The number or expression insideWhatever sits between the bars. Finish all its arithmetic before applying the bars.
|x| — The absolute valueHow far x is from zero on the number line. Always zero or positive, same unit as x.
|a − b| — Distance between two numbersThe gap between a and b, which is why the order of subtraction does not matter.
Show your work: a full example
- The equation|2x − 5| = 9
- The inside is 9 units from zero, so it is 9 or −92x − 5 = 9 or 2x − 5 = −9
- First branch, add 52x = 14
- Divide by 2x = 7
- Second branch, add 52x = −4
- Divide by 2x = −2
- Check both|2(7) − 5| = |9| = 9 and |2(−2) − 5| = |−9| = 9
A second, different case
- A different case: an inequality|3x + 4| < 10
- Less-than traps the inside between −10 and 10−10 < 3x + 4 < 10
- Subtract 4 from all three parts−14 < 3x < 6
- Divide all three parts by 3−4.667 < x < 2
- Test a value insidex = 0 gives |4| = 4, and 4 < 10, so 0 belongs
- Test a value outsidex = 3 gives |13| = 13, and 13 is not less than 10, so 3 is correctly excluded
- Compare with greater-than|3x + 4| > 10 would instead split into two separate rays, x < −4.667 or x > 2
Copy-ready example
|2x − 5| = 9
The equation
Questions about the Absolute Value Calculator
Can an absolute value ever come out negative?
No, because it measures distance. That is why |x − 3| = −5 has no solution at all: no number can sit −5 units away from 3. If your working produces a negative on the right-hand side, stop and write 'no solution'.
Why do absolute value equations usually have two answers?
Because two different numbers share every distance except zero. Anything 9 units from zero is either 9 or −9, so each equation splits into two ordinary equations. The single exception is |something| = 0, which has one answer.
What is the difference between |x| < 5 and |x| > 5?
|x| < 5 means 'closer to zero than 5', which is one connected stretch: −5 < x < 5. |x| > 5 means 'further from zero than 5', which is two disconnected pieces: x < −5 or x > 5.
Is |a + b| the same as |a| + |b|?
Only when a and b share a sign. |3 + 5| = 8 and |3| + |5| = 8 agree, but |3 + (−5)| = 2 while |3| + |−5| = 8. The left side is never bigger, which is known as the triangle inequality.