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Absolute-Value Inequalities — result sheet
Solve |a*x+b| in relation to a nonnegative threshold and report the interval or solution meaning explicitly.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: Let m=-b/a and r=c/|a|. Then |a*x+b| <, <=, >, or >= c is solved from the boundaries m-r and m+r.
The absolute value is centered at x=-b/a and the threshold creates a half-width c/|a|. Less-than relations keep the inside interval, less-than-or-equal relations close its endpoints, and greater-than relations keep the outside rays with the appropriate endpoint openness.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Coefficient a — coefficient units; The nonzero coefficient multiplying x inside the absolute value.; minimum -1000000; maximum 1000000
- Constant b — coefficient units; The constant term inside |a*x+b|.; minimum -1000000; maximum 1000000
- Threshold c — absolute-value units; A nonnegative comparison threshold.; minimum 0; maximum 1000000
- Relation — comparison; Choose the relation between |a*x+b| and c.; 4 choices
Worked example
| Input | Value |
|---|---|
| Coefficient a | 2 |
| Constant b | -4 |
| Threshold c | 6 |
| Relation | lessEqual |
|2x-4| <= 6 gives center m=2 and half-width r=3, so the solution is -1 <= x <= 5, a closed interval.
Assumptions and limits
- a is a nonzero real coefficient and c is a nonnegative real threshold.
- The entered relation applies to the linear expression |a*x+b|, not to a parsed equation or a more complicated expression.
- The answer uses the real-number line and the displayed finite bounds; interval endpoints are numeric boundaries, while infinity is rendered as text.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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