Inequality to Interval Notation Calculator

Enter the lower and/or upper bound of an inequality and choose whether each end is open, closed, or absent, to get the interval notation, the rewritten inequality, and set-builder notation.

Quick Facts

Open interval
( )
Used for strict inequalities (< or >) and always next to ±∞, since infinity can never be "included".
Closed interval
[ ]
Used for inclusive inequalities (≤ or ≥); the endpoint itself is part of the solution set.
Union
∪
Joins two disjoint intervals, e.g. "x < 2 or x > 5" is (-∞, 2) ∪ (5, ∞).

Your Results

Calculated
Interval Notation
-
Standard bracket/parenthesis form
Inequality
-
Rewritten in a ≤/< x ≤/< b form
Set-Builder Notation
-
{x | condition}
Interval Type
-
Bounded, unbounded, empty, or all reals

Ready

Set your bounds and press Calculate.

Formula and Method for Inequality to Interval Notation

Interval notation is a compact way to write the solution set of an inequality using parentheses and brackets instead of inequality symbols. A bounded compound inequality such as a < x ≤ b is written (a, b]: the parenthesis on the left says a itself is not included (strict <), and the bracket on the right says b is included (≤). A one-sided inequality like x ≥ a becomes [a, ∞) — infinity always gets a parenthesis, since it is not a number the set can actually "reach" and include.

How the calculation works

Enter a lower bound and an upper bound, and for each one choose whether it is open (strict < or >, endpoint excluded), closed (≤ or ≥, endpoint included), or absent (no bound on that side, meaning the interval extends to -∞ or +∞). The calculator combines these choices using the standard convention: parentheses for open endpoints and for ±∞, brackets for closed endpoints. It also rewrites the result as an inequality (a < x ≤ b) and as set-builder notation ({x | a < x ≤ b}). If the two bounds are equal with at least one side strict, the solution set is empty (∅). If the lower bound exceeds the upper bound, the calculator treats this as invalid input and asks you to correct the values rather than returning an empty set. If neither bound is set, the interval covers all real numbers, (-∞, ∞).

Common mistakes

  • Bracket next to infinity: ∞ and -∞ are never enclosed by a bracket — always use a parenthesis, e.g. (-∞, 5], not (-∞, 5].
  • Reversed order: the smaller number always goes first inside the interval; (7, 2) is not valid notation even if the inequality was written "backwards" as 7 > x > 2.
  • Mixing up ≤ and <: phrases like "at least" or "no less than" mean ≥ (closed bracket), while "more than" or "greater than" mean strict > (open parenthesis).

Real-world applications

  • Describing the domain or range of a function, such as the domain of √x being [0, ∞).
  • Writing the solution set of an algebra or calculus inequality after solving it step by step.
  • Specifying valid input ranges for forms, sensors, or programming validation logic (e.g., a percentage input restricted to [0, 100]).
  • Expressing confidence intervals and acceptable tolerance ranges in statistics and engineering.

Frequently Asked Questions

What does a parenthesis versus a bracket mean in interval notation?
Parentheses ( ) mean the endpoint is excluded, used for strict inequalities (< or >) and always next to infinity, since infinity is not a number that can be included. Brackets [ ] mean the endpoint is included, used for inclusive inequalities (≤ or ≥). For example, 2 < x ≤ 7 becomes (2, 7].
How do I write x ≥ 3 in interval notation?
A one-sided inequality with a closed endpoint uses a bracket on the finite side and a parenthesis at infinity: x ≥ 3 becomes [3, ∞). If it were the strict inequality x > 3, it would be (3, ∞).
What if the inequality has no solution or includes all real numbers?
If the bounds are equal with at least one side strict (e.g., 5 < x < 5), the solution set is empty, written ∅. If the lower bound is greater than the upper bound, that's an invalid input rather than a valid empty-set case; the calculator flags it as an error so you can correct the values. If there is no lower or upper restriction at all, the interval is (-∞, ∞), meaning x can be any real number.
How do I convert a compound inequality with "or" into interval notation?
Inequalities joined by "or" (such as x < 2 or x > 5) describe two separate, non-overlapping rays, written as a union: (-∞, 2) ∪ (5, ∞). This calculator evaluates one continuous interval at a time; for an "or" compound, compute each piece separately and join the results with ∪.

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