Inequality to Interval Notation Calculator

About the Inequality to Interval Notation Calculator

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 — bounded intervals, left and right rays, strict and inclusive boundaries.

When This Page Helps

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.

How to Use the Inputs

1. Select the mode, method, or precision options that match your inequality to interval notation problem.
2. Read Interval Notation first, then use Inequality Notation to confirm your setup is correct.
3. Try a preset such as "x > 3" to test a known case quickly.
4. Compare the result with the formula and worked example so you can catch input, rounding, or setup mistakes.

Example Calculation

Tips & Best Practices

• Infinity (∞ or −∞) always gets an open parenthesis, never a bracket.
• The complement of an interval reverses open/closed boundaries at each endpoint.
• Union (∪) combines disjoint intervals; intersection (∩) finds overlap.
• Function domains are often expressed as intervals — practice switching between notations.
• When graphing, use a filled circle ● for included endpoints and open circle ○ for excluded ones.

How This Inequality to Interval Notation Calculator Works

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.

Interpreting Results

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.

Study Strategy

Frequently Asked Questions

• What is interval notation?

  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.

• Why is infinity always an open parenthesis?

  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.

• What is set-builder notation?

  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."

• How do you find the complement of an interval?

  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.

• Can an interval contain just one number?

  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.

• How do I convert interval notation back to an inequality?

  Replace brackets with ≤ and parentheses with <. For example, [−1, 4) becomes −1 ≤ x < 4. For rays like (3, ∞), write x > 3.