Convert inequalities to interval notation, set-builder notation, and back. Visualize intervals on a number line with open/closed endpoints and test membership.
Interval notation is a concise mathematical language for describing sets of real numbers. Instead of writing "all x such that −3 is less than x and x is less than or equal to 5," you write (−3, 5]. Every algebra student encounters this notation when solving inequalities, defining function domains and ranges, and expressing solution sets.
There are four fundamental interval types. An open interval (a, b) excludes both endpoints. A closed interval [a, b] includes both. Half-open (or half-closed) intervals [a, b) or (a, b] include one endpoint and exclude the other. Unbounded intervals extend to infinity in one or both directions and always use an open parenthesis next to the infinity symbol, because infinity is not a number that can be reached.
The set-builder notation equivalent writes the same idea more formally: {x ∈ ℝ | −3 < x ≤ 5}. This is common in higher mathematics and formal proofs. Inequality notation (−3 < x ≤ 5) is the form most students see first.
This calculator converts freely among all three notations. Enter the bounds and boundary types, and it produces interval notation, inequality notation, set-builder notation, the complement, interval length, midpoint, and a number-line visualization with filled or hollow circles for closed or open endpoints. A membership test table lets you verify which sample points fall inside or outside the interval. Use the eight presets to explore common types instantly — bounded intervals, left and right rays, strict and inclusive boundaries.
Inequality to Interval Notation Calculator helps you solve inequality to interval notation problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter your inputs once and immediately inspect Interval Notation, Inequality Notation, Set-Builder Notation to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
Open interval: (a, b) ↔ a < x < b. Closed: [a, b] ↔ a ≤ x ≤ b. Half-open: [a, b) ↔ a ≤ x < b. Ray: (a, ∞) ↔ x > a. Always parenthesis with ±∞.
Result: Interval Notation shown by the calculator
Using the preset "x > 3", the calculator evaluates the inequality to interval notation setup, applies the selected algebra rules, and reports Interval Notation with supporting checks so you can verify each transformation.
This calculator takes the problem inputs and applies the relevant inequality to interval notation relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Interval Notation, Inequality Notation, Set-Builder Notation, Interval Length to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
Interval notation is a shorthand for describing a set of numbers between two endpoints. Parentheses ( ) mean the endpoint is excluded; brackets [ ] mean it is included. For example, [2, 7) means all numbers from 2 (included) up to but not including 7.
Infinity is not a real number — it is a concept representing unboundedness. You can never "reach" infinity, so it cannot be included in the set, and the parenthesis reflects this exclusion.
Set-builder notation defines a set by stating a property its members must satisfy. For example, {x ∈ ℝ | x > 3} reads "the set of all real x such that x is greater than 3."
The complement is all real numbers NOT in the interval. For (a, b), the complement is (−∞, a] ∪ [b, ∞). For [a, b], it is (−∞, a) ∪ (b, ∞). Open and closed boundaries swap at each endpoint.
Yes. The degenerate interval [a, a] = {a} is a single-point set. Open intervals (a, a) are empty because no number is strictly between a and a.
Replace brackets with ≤ and parentheses with <. For example, [−1, 4) becomes −1 ≤ x < 4. For rays like (3, ∞), write x > 3.