Calculator Inputs
This calculator solves inequalities of the form ax² + bx + c relation 0. Set a = 0 for a linear inequality.
Example Data Table
| Inequality | Key Roots | Interval Notation Answer | Notes |
|---|---|---|---|
| x² - 5x + 6 > 0 | 2, 3 | (-∞, 2) ∪ (3, ∞) | Positive outside both roots. |
| x² - 9 ≤ 0 | -3, 3 | [-3, 3] | Negative or zero between the roots. |
| 2x - 8 ≥ 0 | 4 | [4, ∞) | Linear case with one boundary point. |
| -x² + 4x - 3 < 0 | 1, 3 | (-∞, 1) ∪ (3, ∞) | Downward parabola is negative outside roots. |
Formula Used
Standard form: ax² + bx + c relation 0
Discriminant: Δ = b² - 4ac
Quadratic roots: x = (-b ± √Δ) / (2a)
Linear root: x = -c / b when a = 0
After finding real roots, split the number line at each boundary point. Then test the sign of the expression on each interval. Keep the intervals where the expression matches the selected inequality sign. Use closed brackets for equality and open brackets for strict inequalities.
How to Use This Calculator
- Enter coefficients a, b, and c for the expression ax² + bx + c.
- Choose the inequality sign you want to solve.
- Set the graph viewing range and decimal precision.
- Click Solve Inequality to generate interval notation.
- Review the steps, roots, discriminant, and sign rule.
- Use the graph to confirm where the expression sits above or below the x-axis.
- Download the result as CSV or PDF for records or teaching material.
FAQs
1) What does interval notation mean?
Interval notation writes solution sets using parentheses, brackets, and unions. Parentheses exclude endpoints. Brackets include endpoints. It is a compact way to describe all valid x-values.
2) When do I use brackets instead of parentheses?
Use brackets when the endpoint is included, usually with ≥ or ≤. Use parentheses when the endpoint is excluded, usually with > or <.
3) What happens if a = 0?
The quadratic becomes a linear inequality. The calculator automatically switches to a one-boundary-point method and still returns interval notation correctly.
4) Why is the discriminant important?
The discriminant tells how many real roots exist. Real roots create boundary points. Those points divide the number line into intervals for sign testing.
5) Can the solution be all real numbers?
Yes. That happens when the expression always stays above or below zero in a way that matches your selected inequality. The answer appears as (-∞, ∞).
6) Can the solution be empty?
Yes. If no real x-value makes the inequality true, the calculator returns ∅. This often happens when the expression never reaches the required sign.
7) Why does the sign flip in some linear cases?
When you divide or multiply an inequality by a negative number, the inequality direction reverses. The calculator applies that rule automatically.
8) How does the graph help?
The graph shows where the expression is above, below, or touching the x-axis. That visual check makes the interval answer easier to trust and explain.