Calculator
Example Data Table
| Example | Numerator | Denominator | Operator | Solution Summary |
|---|---|---|---|---|
| 1 | x² - 3x - 4 | x² - x - 6 | ≥ 0 | Use zeros and undefined points to split intervals. |
| 2 | x² - 1 | x - 3 | < 0 | Check signs on each interval and exclude x = 3. |
| 3 | 2x - 8 | x² - 9 | > 0 | Critical points occur at x = 4, -3, and 3. |
Formula Used
General form:
R(x) = (ax² + bx + c) / (dx² + ex + f)
Quadratic roots:
x = (-b ± √(b² - 4ac)) / (2a)
To solve a rational inequality, find numerator zeros and denominator zeros first. Denominator zeros are excluded from the domain.
These values create critical points. The real line is split into intervals around them. A test point from each interval reveals the sign of the rational expression.
Then match interval signs to the chosen operator. Include numerator zeros only for ≥ or ≤, and only when the denominator is nonzero.
How to Use This Calculator
- Enter the numerator coefficients for x², x, and the constant.
- Enter the denominator coefficients for x², x, and the constant.
- Select the required inequality sign: >, ≥, <, or ≤.
- Choose graph minimum, graph maximum, and sample steps.
- Press Solve Inequality to display the result section.
- Review the solution set, critical points, interval table, and graph.
- Use the export buttons to save the result table as CSV or PDF.
FAQs
1. What does this solver actually compute?
It solves rational inequalities of the form polynomial over polynomial compared with zero. It identifies zeros, undefined points, interval signs, and the final real-number solution set.
2. Why are denominator roots excluded?
A denominator root makes the rational expression undefined. Even if the numerator is also zero there, that x-value is not allowed in the domain unless the expression is simplified before modeling.
3. When are endpoints included?
Endpoints are included only for ≥ or ≤ when the rational expression equals zero there. That happens at numerator roots where the denominator stays nonzero.
4. Can it solve linear cases too?
Yes. Set the x² coefficient to zero for any linear numerator or denominator. The solver automatically treats the expression as linear where appropriate.
5. What does the graph show?
The graph shows the rational function across your chosen x-range. Breaks appear near denominator zeros, helping you see vertical asymptotes and sign changes visually.
6. Why use interval testing?
Critical points are the only places where the sign can change. Testing one value inside each interval is enough to determine the sign of the whole interval.
7. Does this handle all rational expressions?
This page is designed for quadratic or linear numerator and denominator inputs. Higher-degree expressions need more advanced factorization and root-finding methods.
8. What export options are included?
You can download the interval sign table as CSV and create a compact PDF summary containing the inequality, solution set, and interval results.