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Home › Math & Conversions › Core Math & Algebra › Interval Notation Interval Notation Calculator Convert inequalities, endpoints, and interval notation in one place. See the result on a number line and get the corresponding set-builder notation explained step by step. Inequality → Interval Interval → Inequality Endpoints → Interval Enter an inequality in x Supported patterns include: x > a , x ≥ a , x < b , x ≤ b , and double inequalities like a < x ≤ b . Convert to interval notation Enter an interval This tool parses a single interval such as (a, b) , [a, b] , (−∞, b] , or [a, ∞) . Unions like (-∞, 0) ∪ (1, ∞) are not yet supported. Convert to inequalities Left endpoint ( a ) Right endpoint ( b ) Left side ( open (not included) [ closed (included) Right side ) open (not included) ] closed (included) Leave an endpoint blank if it is not bounded on that side and choose the appropriate parenthesis for (−∞ or ∞) . Build interval Result Interval notation — Inequality form — Set-builder notation — Plain-language description Enter an inequality, interval, or endpoints above to see a detailed explanation here. Number line visualization The shaded segment shows the interval on a simplified real number line. Open circles represent excluded endpoints; filled circles represent included endpoints. What is interval notation? Interval notation is a compact way to describe sets of real numbers. Instead of writing “all real numbers x such that -1 < x ≤ 3 ”, we can write the interval (-1, 3] . Basic types of intervals (a, b) : open interval, excludes both endpoints a and b . [a, b] : closed interval, includes both endpoints. [a, b) or (a, b] : half-open (half-closed) intervals. (−∞, b) , (−∞, b] : intervals unbounded on the left. (a, ∞) , [a, ∞) : intervals unbounded on the right. How to convert inequalities to interval notation 1. Single-ended inequalities Consider inequalities involving a single bound: x > a → (a, ∞) x ≥ a → [a, ∞) x < b → (−∞, b) x ≤ b → (−∞, b] 2. Double inequalities For a double inequality like a < x ≤ b , the middle term tells you the variable ( x ), and the inequalities tell you about inclusion or exclusion: If the inequality is strict (< or >), use parentheses: ( ) . If the inequality is inclusive (≤ or ≥), use brackets: [ ] . Example: -1 < x ≤ 3 becomes the interval (-1, 3] . From interval notation back to inequalities To reverse the process, read the interval from left to right: (a, b) means a < x < b . [a, b] means a ≤ x ≤ b . (−∞, b] means x ≤ b . [a, ∞) means x ≥ a . Interval notation, sets, and domains In calculus and algebra, interval notation is often used to describe the domain and range of functions. For example, the function f(x) = 1/x has domain (-∞, 0) ∪ (0, ∞) : all real numbers except 0 . In set-builder notation, you might see { x ∈ ℝ | x > 2 } , which is equivalent to the interval (2, ∞) . Set-builder and interval notation { x ∈ ℝ | a < x < b } ↔ (a, b) { x ∈ ℝ | a ≤ x ≤ b } ↔ [a, b] { x ∈ ℝ | x ≥ a } ↔ [a, ∞) { x ∈ ℝ | x < b } ↔ (−∞, b) Interval notation FAQ What do the different brackets mean? Parentheses ( ) always mean “open” or “not included”, while brackets [ ] mean “closed” or “included”. Infinity symbols ∞ and −∞ are always paired with parentheses, because infinity itself is not a real number. Can an interval be empty? Yes. For example, the condition x < 1 and x ≥ 1 at the same time describes the empty set. In practice, inconsistent inequalities lead to no interval; this calculator reports such cases as invalid input. What about unions like (-∞, 0) ∪ (1, ∞)? Many real-world domains are a union of disjoint intervals. This calculator currently focuses on a single interval at a time. You can handle unions by converting each piece separately and then combining them with the union symbol ∪ in your work. Frequently Asked Questions How is interval notation different from inequalities? Interval notation compresses an entire inequality (or system of inequalities in one variable) into a short expression using brackets and parentheses. Both notations describe the same set; interval notation is just more compact and easier to scan. Do I always use ℝ (real numbers) in set-builder notation? For most introductory algebra and calculus problems, the underlying set is the real numbers ℝ. In more advanced settings you might restrict to integers ℤ or other sets, but interval notation itself is designed for real intervals. How do I write the domain of a function in interval notation? Start by identifying values that must be excluded (for example, denominators equal to zero or even roots of negative numbers). Then express the remaining real numbers as one or more intervals, using parentheses or brackets according to whether the endpoints are included. Core Math & Algebra tools Set theory Set operations Venn diagram generator Base converter Modular arithmetic Modulo Equation solver Number theory Graphing Probability Prime number Interval notation (current) Study tips for interval notation Always sketch a quick number line to visualize the set. Match strict inequalities with parentheses and inclusive ones with brackets. Remember that infinity symbols are never included in the set. Practice converting domains and ranges between graphs, inequalities, and intervals.
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