Enter a rational inequality
This version compares two fractions with linear numerators and denominators, then solves the resulting rational inequality using sign analysis.
Example data table
| Example inequality | Restrictions | Solution set |
|---|---|---|
| x / (x - 2) > 1 / (x + 1) | x ≠ 2, -1 | (-∞, -1) ∪ (2, ∞) |
| (x + 1) / (x - 3) ≥ 0 | x ≠ 3 | (-∞, -1] ∪ (3, ∞) |
| (2x - 5) / (x + 4) < 0 | x ≠ -4 | (-4, 2.5) |
Formula used
For a general comparison, solve
ax + bcx + d □ ex + fgx + h
Move everything to one side without assuming denominator signs:
(ax + b)(gx + h) - (ex + f)(cx + d)(cx + d)(gx + h) □ 0
Then follow four steps. First, find the zeros of the new numerator. Second, mark denominator zeros as restrictions. Third, test one point inside each interval. Fourth, keep intervals whose sign matches the chosen inequality.
How to use this calculator
- Enter the left fraction using its numerator and denominator coefficients.
- Select the required inequality symbol: greater than, less than, or inclusive versions.
- Enter the right fraction coefficients. Use 0 and 1 to compare against zero.
- Press Solve inequality to show the result above the form.
- Read the standard form, restrictions, critical points, interval checks, and highlighted graph.
- Download the summary as CSV or PDF for revision or documentation.
Frequently asked questions
1. What kind of inequalities can this page solve?
It solves inequalities formed by two fractions whose numerators and denominators are linear in x. After simplification, the comparison becomes a rational quadratic-over-quadratic sign problem.
2. Why are some x-values excluded?
Any x-value that makes an original denominator equal zero is excluded. Those values create undefined expressions, so they can never be part of the final solution set.
3. Why not multiply both sides by denominators directly?
A denominator may be positive on one interval and negative on another. Direct multiplication can accidentally reverse the inequality. Sign analysis avoids that mistake.
4. What does the interval table show?
Each row tests one sample x-value from an interval. The calculator evaluates N(x), D(x), their ratio, and whether that interval satisfies the selected inequality.
5. When are endpoints included?
Endpoints are included only for ≥ or ≤ when the numerator becomes zero there and the expression remains defined. Restricted points are always excluded.
6. Can I compare a fraction with zero?
Yes. Set the right fraction to 0/1. That turns the tool into a standard rational inequality solver against zero while keeping the same workflow.
7. What does the graph highlight?
The curve represents the transformed rational expression. Blue shaded x-regions show where the inequality is true, and dashed vertical lines show forbidden x-values.
8. Are decimal coefficients allowed?
Yes. The inputs accept integers and decimals, so you can test classroom examples, textbook exercises, or applied algebra models with non-integer coefficients.