How to Convert Inequalities to Interval Notation
Interval notation is the standard mathematical format for expressing continuous subsets of real numbers $\mathbb{R}$. It replaces verbal inequalities like "greater than or equal to" with concise symbols that describe boundary points.
The Universal Bracket Rules
- 1Parentheses $( \text{ and } )$: Represent strict exclusion. Used for $<$ and $>$ where the endpoint is not part of the solution.
- 2Brackets $[ \text{ and } ]$: Represent inclusion. Used for $\le$ and $\ge$ where the endpoint is included.
- 3Infinities $\pm\infty$: Always paired with parentheses, e.g., $(-\infty, 5]$ or $(3, \infty)$, because infinity is a concept of unbounded growth rather than a reachable numerical value.
Step-by-Step Translation Blueprint
- Step 1: Identify if the inequality is single-sided (unbounded) or double-sided (compound bounded).
- Step 2: Write the lesser bound on the left and the greater bound on the right, separated by a comma: (left, right).
- Step 3: Check each endpoint: assign square brackets [ or ] if inclusive ($le, ge$), or round parentheses ( or ) if exclusive ($<, >$).
Comprehensive Notation Reference Table
| Inequality Relation | Interval Notation | Number Line Representation | Set-Builder Notation |
|---|---|---|---|
| $a < x < b$ | $(a, b)$ | Open circle at $a$ and $b$ | $\{x \in \mathbb{R} \mid a < x < b\}$ |
| $a \le x \le b$ | $[a, b]$ | Filled dot at $a$ and $b$ | $\{x \in \mathbb{R} \mid a \le x \le b\}$ |
| $a \le x < b$ | $[a, b)$ | Filled dot at $a$, open at $b$ | $\{x \in \mathbb{R} \mid a \le x < b\}$ |
| $x > a$ | $(a, \infty)$ | Open circle at $a$, arrow right | $\{x \in \mathbb{R} \mid x > a\}$ |
| $x \le b$ | $(-\infty, b]$ | Filled dot at $b$, arrow left | $\{x \in \mathbb{R} \mid x \le b\}$ |