Analyze numerator, denominator, signs, exclusions, and interval endpoints. View tested ranges, notation, and decimal approximations. Download polished reports for study, teaching, or review sessions.
Enter a rational expression of the form ((ax² + bx + c) / (dx² + ex + f)) compared with a constant value.
((ax² + bx + c) / (dx² + ex + f)) ☐ k
((ax² + bx + c) - k(dx² + ex + f)) / (dx² + ex + f) ☐ 0
The calculator transforms the inequality so the right side becomes zero. It then finds all real zeros of the transformed numerator and denominator.
Those critical values split the number line into intervals. A test point from each interval reveals the sign of the rational expression. Matching intervals form the final solution set.
| Numerator | Denominator | Inequality | Right Side | Expected Solution |
|---|---|---|---|---|
| x² - 3x - 4 | x² - x - 6 | ≥ | 0 | (-∞, -2) ∪ [-1, 3) ∪ [4, ∞) |
| x² + x - 6 | x² - 4 | < | 0 | (-3, -2) ∪ (2, 3) |
| 2x² - 5x + 2 | x² - 1 | ≤ | 1 | Use calculator to test shifted intervals. |
It solves rational inequalities where both numerator and denominator are quadratic, linear, or constant expressions, compared with a numeric constant using greater than, less than, or inclusive forms.
Any denominator root makes the expression undefined. Those x-values are removed from the domain, even if surrounding intervals satisfy the inequality.
Rewriting everything over one denominator makes sign analysis easier. Once the comparison is against zero, interval testing becomes direct and reliable.
Yes. Repeated roots may touch the axis without changing sign. The calculator still tests each interval so the correct sign behavior is preserved.
It lists each interval, a sample test point, the transformed numerator sign, denominator sign, resulting expression sign, and whether that interval satisfies your chosen inequality.
Yes. Decimal and negative coefficients are supported. The output roots and interval endpoints are displayed using your selected precision setting.
Yes. It captures the solved result section and saves a clean PDF summary, which is useful for homework, notes, or classroom review.
Then the expression equals zero wherever it is defined. Non-strict inequalities may hold across the full domain, while strict ones return no real solution.
The solver handles shifted inequalities, such as comparing a rational expression with 1, -2.5, or any other constant.
Use the exported CSV file to keep the interval sign chart. Use the PDF button for a printable summary.
For best results, check whether denominator coefficients produce domain breaks before interpreting included endpoints.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.