What is interval notation in math?
Interval notation is a compact way to describe a set of real numbers between two endpoints. Square brackets [ ] indicate that an endpoint is included (closed), while parentheses ( ) indicate it is excluded (open). Infinity is always paired with a parenthesis because it is not a real number and can never be included. See also our Y-Intercept (b) — Y-Intercept.
How do you convert an inequality to interval notation?
Replace the variable's bounds with the endpoint values and decide on brackets vs. parentheses based on the inequality sign. Use ( or ) for strict inequalities (< or >) and [ or ] for non-strict inequalities (≤ or ≥). For example, x ≥ 3 becomes [3, ∞) and x < 5 becomes (-∞, 5).
What are the main types of intervals?
There are four main types: open intervals (a, b) where neither endpoint is included; closed intervals [a, b] where both endpoints are included; half-open (or half-closed) intervals like [a, b) or (a, b]; and unbounded intervals that extend to positive or negative infinity, such as [a, ∞) or (-∞, b).
How do you handle compound inequalities in interval notation?
Compound inequalities use either 'and' (intersection) or 'or' (union). An 'and' compound like 2 < x < 8 is written as a single interval (2, 8). An 'or' compound like x < 2 or x > 8 is written as a union: (-∞, 2) ∪ (8, ∞). The union symbol ∪ separates the two separate solution sets. You might also find our use the Rational Zeros Calculator useful.
What is the interval notation for -1 ≤ x ≤ 1?
The interval notation is [-1, 1]. Since both endpoints are included (≤ on both sides), square brackets are used on both ends. This is called a closed interval.
What is set-builder notation and how does it differ from interval notation?
Set-builder notation describes a set by specifying a property its elements must satisfy, written as {x | condition} — for example, {x | x > 3}. Interval notation conveys the same idea more compactly as (3, ∞). Both are equivalent ways to describe the same set of numbers; interval notation is often preferred for its brevity.
Can infinity be written with a bracket in interval notation?
No. Infinity (∞) and negative infinity (-∞) are concepts, not real numbers, so they can never be reached or included in a set. They are always written with a parenthesis: for example (5, ∞) or (-∞, 3]. Using a bracket next to infinity is considered a mathematical error.
What does an open circle vs. a closed circle mean on a number line?
An open circle on a number line means that endpoint is not included in the solution — corresponding to a strict inequality (< or >) and a parenthesis in interval notation. A closed (filled) circle means the endpoint is included — corresponding to a non-strict inequality (≤ or ≥) and a square bracket in interval notation.