Inequality Symbols
| Symbol | Meaning | Interval Notation |
|---|---|---|
| < | Less than (strict) | (a, b) — open bracket |
| ≤ | Less than or equal to | [a, b] — closed bracket |
| > | Greater than (strict) | (a, ∞) — open bracket |
| ≥ | Greater than or equal to | [a, ∞) — closed bracket |
Key Rule
Example: −2x < 6 → x > −3 (sign flipped when dividing by −2)
Example Solutions
| Inequality | Solution | Interval |
|---|---|---|
| 2x + 3 < 11 | x < 4 | (−∞, 4) |
| −3x ≥ 9 | x ≤ −3 | (−∞, −3] |
| 5x − 2 > 13 | x > 3 | (3, ∞) |
| 4x + 1 ≤ 2x + 9 | x ≤ 4 | (−∞, 4] |
Frequently Asked Questions
What is interval notation?
Interval notation is a concise way to express a range of values. Square brackets [ ] indicate the endpoint IS included (≤ or ≥). Parentheses ( ) indicate the endpoint is NOT included (< or >). Infinity (∞) always uses a parenthesis since it is not a real number. Example: [3, ∞) means x ≥ 3.
Why does the inequality sign flip when dividing by a negative?
Consider 4 > 2. If we multiply both sides by −1, we get −4 and −2. On the number line, −4 is to the LEFT of −2, so −4 < −2 — the relationship has flipped. This is not an arbitrary rule; it reflects the geometry of the number line for negative multiplications.
What is a compound inequality?
A compound inequality combines two inequalities: "and" (both must hold, e.g. 2 < x ≤ 5) or "or" (either must hold, e.g. x < 1 or x > 4). This calculator solves single linear inequalities; compound inequalities require splitting and solving each part.