Inequality Symbols & Interval Notation Glossary
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Full bio →Inequality notation packs a lot of meaning into a few symbols. Here’s what each one means for the inequality calculator.
The Four Inequality Symbols
| Symbol | Meaning | Strict? |
|---|---|---|
< | Less than | Strict — boundary excluded |
> | Greater than | Strict — boundary excluded |
≤ | Less than or equal to | Non-strict — boundary included |
≥ | Greater than or equal to | Non-strict — boundary included |
Strict vs. Non-Strict Inequality
A strict inequality (< or >) excludes the boundary value itself from the solution — x < 5 means every number less than 5, but not 5 itself. A non-strict inequality (≤ or ≥) includes the boundary — x ≤ 5 means every number less than 5, plus 5 itself. This distinction determines whether interval notation uses a parenthesis or a bracket — see below.
Interval Notation
A compact way to write an inequality’s solution set as a range rather than a symbol-based statement. Parentheses ( ) mark excluded (strict) boundaries; square brackets [ ] mark included (non-strict) boundaries. Infinity (∞ or -∞) always gets a parenthesis, since infinity isn’t a specific number the solution actually reaches. See the inequality calculator for how this notation gets generated automatically from a solved inequality.
Sign-Flip Rule
The rule that multiplying or dividing both sides of an inequality by a negative number reverses its direction (< becomes >, ≤ becomes ≥, and vice versa). Addition and subtraction never trigger this flip, regardless of sign. See common mistakes solving inequalities for how often this rule gets missed.
Compound Inequality
An inequality with three parts chained together, like a < mx + b < c, meaning the middle expression must satisfy both bounds simultaneously. Solved by applying the same operation to all three parts at once — see graphing inequality solutions on a number line for what a compound inequality’s solution looks like visually.
”And” vs. “Or” in Compound Statements
A compound inequality written as a chain (a < x < c) is implicitly an “and” statement — x must satisfy both conditions at once. Some inequality problems instead use an explicit “or” between two separate inequalities (like x < 2 or x > 8), describing two disconnected solution regions rather than one continuous range — these can’t be written as a single chained compound inequality, since “or” solutions don’t overlap the way “and” solutions do.
Boundary / Endpoint
The specific value where an inequality’s solution set begins or ends — for x ≥ -3, the boundary is -3. Whether the boundary itself is part of the solution depends entirely on whether the inequality is strict or non-strict at that point.
Open Interval vs. Closed Interval
An open interval (both ends use parentheses, like (-3, 5)) excludes both boundaries. A closed interval (both ends use brackets, like [-3, 5]) includes both boundaries. A half-open interval (one of each, like [-3, 5)) includes one boundary and excludes the other — common in compound inequality solutions where the two bounds use different operators.
Ready to solve a real inequality? Use the inequality calculator, or see inequality calculator examples for the math worked out step by step.