Absolute Value Equation and Inequality Calculator
Solve |ax+b|=c or an absolute-value inequality. See exact fractional boundaries, the inside or outside solution and a number line with endpoint rules.
YOUR CALCULATION
Result
—
Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
For a positive right side, |ax+b|=c means ax+b=c or ax+b=−c. A smaller absolute value selects between the boundaries; a larger value selects outside them. Handle zero or negative right sides and a=0 before dividing.
|t|=c ⇒ t=±c for c≥0; |t|<c ⇔ −c<t<c for c>0
Worked example
|2x−3|=5 gives 2x−3=5 or −5, so x=4 or −1. The inequality |2x−3|≤5 includes every x from −1 to 4, including both endpoints. The strict inequality excludes the endpoints.
Assumptions and input details
Coefficients and c accept whole numbers, decimals with up to six places and fractions. Magnitudes are limited to 1e6 and reduced input denominators to 1e6. Absolute value is never negative, so a negative right side can make the relation always true or impossible. A zero coefficient makes the expression constant. Exact fractional boundaries determine the answer; number-line positions are rounded illustrations. Filled endpoints are included and hollow endpoints are excluded.
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
The solution boundaries and relation classification use exact fraction arithmetic. Number-line positions are approximate.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
Solve a linear or quadratic inequality against zero. See the solution intervals, open or closed endpoints, a sign chart and a number line.
Solve ax+b=cx+d with whole numbers, decimals or fractions. Follow the same operation on both sides and check the answer by substitution.
Add, subtract, multiply or divide fractions and mixed numbers. Follow exact fraction steps, simplification and a number-line diagram.
Reserved advertising space.