Calculator inputs
Example data table
| Inequality | Solution set |
|---|---|
| (x² − 3x − 4) ≥ 0 | (-∞, -1] ∪ [4, ∞) |
| (2x − 10) < 0 | (-∞, 5) |
| (−x² + 4) > 0 | (-2, 2) |
| (x² + 1) ≤ 0 | ∅ |
| (x² − 2x + 1) < 0 | ∅ |
Formula used
The calculator solves inequalities in the form ax² + bx + c (op) 0. It finds real roots (when they exist) and applies sign analysis on the intervals split by those roots.
- Δ = b² − 4ac decides how many real roots exist.
- Roots: x = (−b ± √Δ) / (2a) when a ≠ 0 and Δ ≥ 0.
- For a > 0, the quadratic is positive outside roots and negative between.
- For a < 0, the quadratic is positive between roots and negative outside.
- Linear case (a=0): solve bx + c (op) 0 using the root x = −c/b.
- Compound mode: AND uses interval intersection; OR uses interval union.
How to use this calculator
- Enter a, b, and c for ax² + bx + c.
- Choose an operator such as ≥ or <.
- Optionally set a domain minimum and maximum to restrict the solution.
- Enable the second inequality if you want a compound condition.
- Press Submit to see the solution set above the form.
- Use Download CSV or PDF to save your latest result.
FAQs
1) What types of inequalities does this solve?
It solves linear and quadratic inequalities written as ax² + bx + c compared to zero. You can also combine two inequalities using AND or OR, then optionally clip results to a chosen domain range.
2) What does “solution set” mean?
The solution set is the collection of all x values that make the inequality true. The calculator presents it using interval notation, such as (−∞, 5] or [2, 7).
3) Why are there brackets and parentheses?
Brackets mean the endpoint is included, usually for ≤ or ≥. Parentheses mean the endpoint is excluded, usually for < or >. Infinity endpoints are always written with parentheses.
4) What happens if the discriminant is negative?
If Δ is negative, the quadratic has no real roots. The expression stays always positive when a > 0, and always negative when a < 0. The result becomes either all real numbers or an empty set, depending on the operator.
5) How does the AND option work?
AND means both inequalities must be true at the same x value. The calculator finds each solution interval set, then computes their intersection, keeping only the overlap.
6) How does the OR option work?
OR means either inequality may be true. The calculator unions the two solution interval sets and merges overlaps, producing one combined solution set in interval notation.
7) Can I solve bx + c compared to zero?
Yes. Set a = 0 and enter b and c. The calculator computes the root x = −c/b and chooses the correct side of that root based on the operator and whether b is positive or negative.
8) Do exports include my steps?
The exports include the inequality text, the final interval-notation solution, and a small set of sample checks. If you need a full handwritten solution, use the displayed formula section as a guide.