Equation
|u| = c ⇔ u = −c or u = c
Solve both resulting linear equations. When c = 0, solve u = 0 only once.
x = −1 or x = 4
Interval or set notation{−1, 4}
Absolute Value Equation and Inequality Solver
Get exact solutions, interval or set notation, boundary checks, step-by-step transformations, and an accessible number-line graph.
Enter an absolute value statement
Solution and number line
Solution
Equivalent statement2x − 3 = −5 or 2x − 3 = 5
Boundary values−1 and 4 (included)
Check|2(−1) − 3| = 5 and |2(4) − 3| = 5 ✓
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Step-by-step solution
The algebra steps will appear here.
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Absolute value equation and inequality rules
Absolute value is distance from zero, so it is always nonnegative. For c > 0:
Inside inequality
|u| < c ⇔ −c < u < c
“Less than” means values inside the two boundaries. Use closed endpoints for ≤.
Outside inequality
|u| > c ⇔ u < −c or u > c
“Greater than” means two outside rays. Use closed endpoints for ≥.
Nonnegative check
If c < 0, equality and “less than” cases have no solution. “Greater than” cases are true for every real input.
Worked absolute value examples
Equation with two solutions
|2x−3|=5 becomes 2x−3=−5 or 2x−3=5. Therefore x=−1 or x=4.
Inside interval
|3x+1|<7 becomes −7<3x+1<7, then −8/3<x<2. Interval: (−8/3,2).
Outside rays
|−2x+4|≥6 has boundaries x=−1 and x=5. The solution is (−∞,−1] ∪ [5,∞).
Zero threshold
|x|≥0 is true for every real number. In contrast, |x|<0 has no solution.
Calculation method, privacy, and limits
Integers, fractions, and finite decimals are converted to reduced fractions and processed with BigInt arithmetic. Solution boundaries and comparisons are exact.
Each numeric part is limited to 30 digits to keep browser calculations responsive. Scientific notation and repeating-decimal notation are not accepted; enter an exact fraction instead. All solving happens in your browser, and the calculator does not transmit or store equation or inequality inputs. Last reviewed: 4 August 2026.
Absolute value solver FAQs
How do you solve an absolute value equation?
For positive c, split |u|=c into u=−c or u=c, then solve both linear equations. A negative right side gives no real solution.
How do you solve an absolute value inequality?
For positive c, a less-than statement becomes a compound inequality between −c and c. A greater-than statement becomes two outside inequalities joined by “or.”
Why can an absolute value problem have no solution?
An absolute value cannot be negative. A constant expression can also make the comparison false for every value of x.
Can the solution be all real numbers?
Yes. For example, |x|≥0 is always true. If a=0, the expression is constant and the original comparison is either true for every real x or false for all of them.
Can I enter fractions and decimals?
Yes. Use integers, finite decimals, or fractions such as -3/4. Exact fractions are reduced automatically.
Is this absolute value solver private?
Yes. It solves locally in your browser and does not send or store the values you enter.