Convert between interval notation, inequality notation, set-builder notation, and number line graph. Supports open, closed, half-open, and unbounded intervals.
Interval notation is a concise way to describe a range of numbers on the real number line. Instead of writing "all x such that 2 < x ≤ 5," mathematicians write (2, 5]. Parentheses indicate the endpoint is excluded (open), while brackets indicate it is included (closed). Mastering these conversions is essential for algebra, calculus, and beyond.
This calculator instantly converts between four representations: interval notation (e.g., [−3, 7)), inequality notation (−3 ≤ x < 7), set-builder notation ({x ∈ ℝ | −3 ≤ x < 7}), and a visual number-line graph. It handles bounded, unbounded (half-line), and entire-real-line intervals, and computes useful properties like interval length, midpoint, and classification.
Whether you are learning algebra, checking homework, or need a quick conversion reference, this tool provides all four representations simultaneously with an interactive number-line visualization. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case.
Interval notation is used throughout mathematics, from algebra through real analysis. It is the standard way to express domains of functions, ranges of solutions, confidence intervals in statistics, and constraint sets in optimization. Being fluent in converting between interval, inequality, set-builder, and graphical forms saves time and prevents errors.
This converter also helps students visualize abstract concepts by providing an instant number-line graph alongside the algebraic representations.
Interval → Inequality: (a, b) means a < x < b [a, b] means a ≤ x ≤ b (a, b] means a < x ≤ b [a, b) means a ≤ x < b ∞ always uses parentheses: (−∞, b) or [a, ∞)
Result: 2 < x ≤ 5
The interval (2, 5] means all real numbers greater than 2 (not including 2) and less than or equal to 5 (including 5). In set-builder: {x ∈ ℝ | 2 < x ≤ 5}. On a number line, open dot at 2 and closed dot at 5.
Intervals are classified by their boundedness and bracket types. Bounded intervals have two finite endpoints and come in four flavors: open (a, b), closed [a, b], and two half-open variants [a, b) and (a, b]. Unbounded intervals extend to infinity in one or both directions: (−∞, b), [a, ∞), or (−∞, ∞). The last represents the entire real number line. Empty intervals (∅) arise when the endpoints are contradictory.
In calculus, interval notation is used extensively. The domain of f(x) = √x is [0, ∞). The range of sin(x) is [−1, 1]. Continuity, differentiability, and integrability are all defined on intervals. Open intervals are particularly important because many theorems (like the Mean Value Theorem) require the function to be continuous on a closed interval [a, b] and differentiable on the open interval (a, b).
To convert from any representation to another, identify three pieces of information: (1) the left endpoint and whether it is included, (2) the right endpoint and whether it is included, and (3) whether either endpoint is infinite. From these three facts, you can write the interval, inequality, set-builder, and number-line forms. Practice all four conversions until they become automatic — this fluency pays dividends throughout mathematics.
Parentheses ( ) mean the endpoint is excluded (open). Brackets [ ] mean the endpoint is included (closed). For example, (3, 7] excludes 3 but includes 7.
Infinity is not a real number and cannot be "reached" or included. Therefore, endpoints at ±∞ always use parentheses: (−∞, 5] or [2, ∞).
Set-builder notation describes a set by stating the property its members satisfy. For example, {x ∈ ℝ | 2 < x ≤ 5} reads "the set of all real x such that x is greater than 2 and at most 5."
A half-open (or half-closed) interval includes one endpoint but not the other. Examples: [a, b) includes a but excludes b; (a, b] excludes a but includes b.
Yes. If the left endpoint exceeds the right, or if the endpoints are equal but at least one bracket is open, the interval is empty (∅).
Identify the endpoints and whether each is included (≤ or ≥ → bracket) or excluded (< or > → parenthesis). For example, −1 ≤ x < 4 becomes [−1, 4).