Calculator inputs
Formula used
Linear inequalities
Rearrange ax + b ? c into ax ? c - b, then divide by a. Reverse the inequality sign whenever the divisor is negative.
Quadratic inequalities
Use the discriminant D = b² - 4ac to locate real roots. Then apply sign analysis across intervals determined by those roots.
Absolute-value inequalities
Convert |ax + b| < c or |ax + b| ≤ c into a double inequality. Convert |ax + b| > c or ≥ c into two outer rays.
Rational inequalities
Set numerator and denominator equal to zero. Build a sign chart on the resulting intervals. Exclude every denominator zero from the solution domain.
How to use this calculator
- Choose the inequality type that matches your expression.
- Select the sign: <, ≤, >, or ≥.
- Enter the needed coefficients for the selected model.
- Pick the output precision, then click Solve inequality.
- Read the interval notation, inspect the graph, and export the result as CSV or PDF.
Example data table
| Mode | Example inequality | Expected interval | Notes |
|---|---|---|---|
| Linear | 2x - 5 ≤ 9 | (-∞, 7] | Single boundary after isolating x. |
| Quadratic | x² - 3x + 2 ≤ 0 | [1, 2] | Roots at 1 and 2 form the middle interval. |
| Quadratic | -x² + 4 > 0 | (-2, 2) | Downward parabola is positive between roots. |
| Absolute | |2x - 1| < 5 | (-2, 3) | Convert to a double inequality. |
| Rational | (x - 2)/(x + 3) ≥ 0 | (-∞, -3) ∪ [2, ∞) | x = -3 is excluded because the denominator vanishes. |
| Rational | x/x > 0 | (-∞, 0) ∪ (0, ∞) | The shared factor still leaves x = 0 undefined. |
FAQs
1. What inequality types does this page solve?
It solves linear, quadratic, absolute-value, and rational inequalities with interval notation, critical-point reporting, graphing, and export support.
2. Why does the sign sometimes reverse?
A sign flips only when you divide or multiply both sides by a negative quantity. That rule preserves the true ordering relationship.
3. Why is a denominator root excluded?
A rational expression is undefined when its denominator equals zero. Even if the numerator is also zero there, that x-value cannot stay in the final domain.
4. How are quadratic intervals chosen?
The calculator finds real roots, checks the parabola direction, and assigns positive or negative regions through sign analysis across the root intervals.
5. Can it solve compound inequalities?
This version focuses on one inequality at a time. For compound statements, solve each part separately and then intersect or union the resulting intervals.
6. What does the shaded graph region mean?
The shaded x-range marks where the standard-form expression satisfies the selected inequality. The line itself shows how the expression behaves near boundaries.
7. Are the results exact or approximate?
The logic is analytical. Decimal formatting only affects how values are displayed, exported, and labeled on the graph.
8. Why can the solution be all real numbers or empty?
Some inequalities stay true everywhere or nowhere because the expression never changes sign, becomes constant, or compares a nonnegative absolute value against a negative bound.