How do I write “x < −1 or x ≥ 2”?

As a union of two intervals: (−∞, −1) ∪ [2, ∞). Each piece follows the same bracket rules. This page shows such unions as the complement of a single interval.

Interval Notation Calculator (interval ↔ inequality ↔ set-builder)

Q: Why does infinity always get a round bracket?

Because ∞ is not a number the interval could reach or include — it only means “without end”. Writing [3, ∞] is a common mistake; the correct form is [3, ∞).

Q: How do I write “x < −1 or x ≥ 2”?

As a union of two intervals: (−∞, −1) ∪ [2, ∞). Each piece follows the same bracket rules. This page shows such unions as the complement of a single interval.

Understanding the interval notation calculator — concept and principle

This calculator converts between interval notation, inequalities and set-builder notation. Enter the two ends and whether each is included, and it gives all three forms, the kind of interval, its length and midpoint, its complement and the smallest whole number in it. Use it for algebra and precalculus homework on inequalities and domains.

A square bracket and filled dot include the end; a round bracket and open dot leave it out.

How it is calculated

A square bracket [ or ] means the end is included (≤ or ≥); a round bracket ( or ) means it is excluded (< or >).
−∞ and ∞ are not numbers, so they always take round brackets: x ≥ 2 is [2, ∞).
Set-builder notation states the condition: {x | −3 ≤ x < 5}.
Closed: both ends included; open: neither; half-open: one.
Length = b − a and midpoint = (a + b) ÷ 2 when both ends are finite; the complement is every other real number.

The principle

An interval is a single unbroken stretch of the number line. Because it has no gaps, its two ends and whether each end belongs to it describe it completely. The three notations say the same thing for different purposes: inequalities are natural when solving, interval notation is compact for answers and domains, and set-builder notation is the most general and also works for sets that are not intervals.

Worked example

Default inputs: lower end −3 included, upper end 5 excluded.

Interval notation: [−3, 5).
Inequality: −3 ≤ x < 5; set-builder: {x | −3 ≤ x < 5}.
Kind: half-open and bounded.
Length = 5 − (−3) = 8; midpoint = (−3 + 5) ÷ 2 = 1.
Complement: (−∞, −3) ∪ [5, ∞); smallest whole number in it: −3.

How to read the result

The bracket types flip in the complement: −3 is in the interval, so the complement stops just before it with a round bracket, and 5 is not in the interval, so the complement starts at 5 with a square bracket. Whole numbers in [−3, 5) are −3 through 4, eight of them.

Common mistakes

Using a square bracket next to ∞.
Writing the larger number first, such as (5, −3].
Mixing up open and closed ends when reading ≤ and <.
Forgetting the union symbol when an answer has two pieces.

Keep in mind: Interval notation describes real numbers. If only whole numbers are meant, list them or say so explicitly.

Frequently asked questions

What does [−3, 5) mean?

All real numbers from −3, included, up to 5, not included.

Why does infinity always get a round bracket?

Infinity is not a number that can be reached or included.

How do I write x > 2 in interval notation?

(2, ∞).

Sources: OpenStax, College Algebra 2e (2021), §2.7 Linear Inequalities and Absolute Value Inequalities

Checked 9 Nov 2026. OpenStax, College Algebra 2e (2021), §2.7 Linear Inequalities and Absolute Value Inequalities — “Using Interval Notation”, Table 1 (inequality, set-builder and interval notation; parentheses with ±∞; union symbol ∪).

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