Calculator Inputs
Choose a solving mode, enter coefficients, and the result will appear above this form after submission.
Formula Used
Linear inequality: Solve ax + b ≥ 0 by locating the boundary x = -b/a. If you divide by a negative number, the inequality direction reverses.
Quadratic inequality: Compute the discriminant D = b² - 4ac. Real roots come from x = (-b ± √D)/(2a). Those roots divide the real line into intervals for sign testing.
Product inequality: Set each factor to zero, then multiply factor signs on each interval. Zeros of the factors become candidate boundaries.
Rational inequality: Find numerator zeros and denominator zeros. Numerator zeros may be included for non-strict operators. Denominator zeros are always excluded from the domain.
How to Use This Calculator
- Select the expression type that matches your inequality.
- Choose the operator: greater than, less than, or their non-strict versions.
- Enter the needed coefficients in the visible input boxes.
- Pick the display precision for roots, test points, and interval values.
- Press Solve Inequality to generate the solution above the form.
- Review the summary, interval chart, boundary table, and solution notation.
- Use the CSV or PDF buttons to export the computed report.
Example Data Table
| Case | Sample Inequality | Method | Solution Set |
|---|---|---|---|
| Linear | 2x - 6 ≥ 0 | Isolate x at the boundary 3 | [3, ∞) |
| Quadratic | x² - 5x + 6 ≤ 0 | Roots at 2 and 3, then sign test | [2, 3] |
| Product | (x - 4)(x + 1) < 0 | Factor sign multiplication | (-1, 4) |
| Rational | (x - 1)/(x + 2) ≤ 0 | Include numerator root, exclude denominator root | (-2, 1] |
FAQs
1. What kinds of inequalities can this calculator solve?
It solves linear, quadratic, product-form, and rational inequalities. Each mode builds the real-line intervals, checks signs, and returns interval notation with any included or excluded boundary points.
2. Why do some endpoints use parentheses and others use brackets?
Parentheses mark values not included in the solution. Brackets mark values included in the solution. Strict operators use open endpoints. Non-strict operators may include root values if the expression is defined there.
3. Why are denominator zeros always excluded?
A rational expression is undefined where its denominator equals zero. Those points break the domain, so they can never belong to the solution set, even when the simplified expression looks harmless nearby.
4. How does the sign chart help?
The sign chart shows one test point inside each interval. If the evaluated value satisfies the chosen operator, that whole interval belongs to the solution set until the next boundary changes the sign behavior.
5. What happens when a quadratic has no real roots?
Its sign never changes on the real line. The leading coefficient tells whether the expression stays positive or negative everywhere, so the solution becomes either all real numbers or no real solution.
6. Can this calculator handle repeated roots?
Yes. Repeated roots appear as single boundary points. The sign may stay the same on both sides, but non-strict operators can still include the repeated root when the expression equals zero there.
7. What does the CSV export contain?
The CSV export includes the solution summary table shown in the result card. It is useful for saving final answers, sharing reports, or importing solved results into worksheets and notes.
8. What if every real number satisfies the inequality?
The solver returns (-∞, ∞). This means every real x makes the inequality true, with the usual exception that undefined rational-expression points are still excluded from the domain.