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An inequality expresses a relationship where two expressions are not (necessarily) equal, using symbols <, >, ≤, or ≥. Solving an inequality finds all values of the variable that make the statement true — the solution set — rather than a single answer.
This calculator solves linear inequalities of the form ax + b < c (or >, ≤, ≥), compound inequalities connected by "and" or "or", and quadratic inequalities of the form ax² + bx + c < 0. The solution is displayed in inequality notation, set notation, and interval notation.
The most important rule when solving inequalities is the sign-flip rule: when you multiply or divide both sides by a negative number, the inequality symbol must reverse direction. For example, −2x > 6 divides by −2 to give x < −3 (not x > −3). Forgetting to flip is the most common algebra error with inequalities.
Interval notation is the standard way to express solution sets compactly. Parentheses ( ) denote strict inequalities (< or >), while brackets [ ] denote non-strict inequalities (≤ or ≥). For example, x > 3 is written (3, ∞), x ≤ 5 is written (−∞, 5], and −1 < x ≤ 4 is written (−1, 4]. The union symbol ∪ combines two disjoint solution intervals.
Linear: ax + b < c → x < (c−b)/a (flip sign if a < 0) | Quadratic: find roots, test intervals
ax + b [symbol] c
| Inequality | Interval Notation | Graph Description | Sign Flip? |
|---|---|---|---|
| x < a | (−∞, a) | Open dot at a, arrow left | Only if dividing by negative |
| x ≤ a | (−∞, a] | Closed dot at a, arrow left | Only if dividing by negative |
| x > a | (a, ∞) | Open dot at a, arrow right | Only if dividing by negative |
| x ≥ a | [a, ∞) | Closed dot at a, arrow right | Only if dividing by negative |
| a < x < b | (a, b) | Open dots at both ends | Apply to each part |
| a ≤ x ≤ b | [a, b] | Closed dots at both ends | Apply to each part |
| x < a or x > b | (−∞,a) ∪ (b,∞) | Two rays, open dots | For |...| > c type |
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.