Calculator Inputs
Example data table
| Type | Example | Expected solution set |
|---|---|---|
| Linear | 2x - 6 ≥ 0 | [3, ∞) |
| Quadratic | x² - 5x + 6 ≤ 0 | [2, 3] |
| Rational | (x - 2)/(x - 4) < 0 | (2, 4) |
| Absolute | |2x - 3| ≤ 5 | [-1, 4] |
Formula used
Linear: Solve ax + b ? 0 by finding the boundary x = -b/a. If a is negative, the inequality direction flips when dividing.
Quadratic: Find roots using x = (-b ± √(b² - 4ac)) / 2a. Use the roots to split the number line, then test each interval sign.
Rational: Solve numerator zeros from ax + b = 0 and denominator restrictions from cx + d = 0. Test each interval, but always exclude undefined denominator points.
Absolute: Rewrite |ax + b| ? k as |ax + b| - k ? 0. Boundaries occur where ax + b = k or ax + b = -k.
How to use this calculator
- Select the inequality type you want to solve.
- Choose the comparison operator: >, >=, <, or <=.
- Enter coefficients for the chosen form.
- Set the number of decimal places for displayed results.
- Press Solve Inequality to generate the solution set.
- Review interval tests, boundary checks, and notes.
- Use the CSV or PDF buttons to export the result.
FAQs
1. What types of inequalities can this tool solve?
It solves linear, quadratic, rational, and absolute-value inequalities. Each mode identifies critical points, tests intervals, checks boundaries, and shows the final interval notation.
2. Why does the solution use interval notation?
Interval notation is compact and standard in mathematics. It shows ranges, endpoints, and unions clearly, making the final answer easier to read and verify.
3. Why are some boundary points excluded?
A strict operator excludes equality points. Rational expressions also exclude denominator zeros because division by zero is undefined, even if the numerator is also zero there.
4. What happens when a quadratic has no real roots?
The quadratic never crosses the x-axis. Its sign stays constant, so the inequality is either true for all real numbers or false for all real numbers.
5. Can the calculator handle repeated roots?
Yes. A repeated root is still listed as a boundary. The sign usually stays the same on both sides, and inclusion depends on whether equality is allowed.
6. How does the absolute-value mode work?
It converts the question into an expression of the form |ax+b| - k ? 0. Then it locates boundary values and tests the resulting intervals.
7. What does the interval testing table show?
It lists each interval between critical points, the sample point used, the expression value there, and whether that interval belongs in the solution set.
8. Can I use the export buttons for reports?
Yes. CSV exports the computed tables, while PDF saves a printable version of the result section. Both are useful for study notes and homework records.