How do I write interval notation?
Type an inequality, such as -2 <= x < 6 or x > 4, or an interval in interval notation, such as [-2, 6) or (-inf, 1]. The interval notation calculator writes it in the other forms, says what kind of interval it is, how to draw each end on a number line, and gives its length and midpoint.
- Interval notation
- [−2, 6)
In interval notation it is [−2, 6).
- Inequality
- −2 ≤ x < 6
- Set-builder notation
- {x | −2 ≤ x < 6}
- Type of interval
- half-open and bounded
- On a number line
- closed (filled) dot at −2; open dot at 6
- Length
- 8
- Midpoint
- 2
Interval notation: [−2, 6). In interval notation it is [−2, 6).
How to calculate
Writes an inequality such as −2 ≤ x < 6 in interval notation, [−2, 6), and interval notation back as an inequality and in set-builder notation, with the length, midpoint and number-line endpoints.
Example with the default inputs (Inequality or interval -2 <= x < 6): In interval notation it is [−2, 6).
Method: A bracket [ ] includes an endpoint (≤ or ≥), a parenthesis ( ) excludes it (< or >), and ±∞ always takes a parenthesis. Length = right − left; midpoint = (left + right) ÷ 2.
- One interval at a time; unions such as (−∞, 2) ∪ (3, ∞) are not read.
- In interval notation the comma separates the ends, so numbers there have no thousands separators.
- The variable is any single letter; the answer uses the letter you typed (x for interval notation).
Worked examples
Each example is checked against the calculator on every build.
- Inequality or interval -2 <= x < 6 gives Interval notation [−2, 6), Set-builder notation {x | −2 ≤ x < 6}, Type of interval half-open and bounded, Length 8, Midpoint 2.Source: OpenStax, Algebra and Trigonometry 2e, §2.7 Linear Inequalities and Absolute Value Inequalities (a bracket includes an endpoint, a parenthesis excludes it, infinity always takes a parenthesis; {x | −2 ≤ x < 6} = [−2, 6), {x | x ≤ 1} = (−∞, 1]), https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-7-linear-inequalities-and-absolute-value-inequalities (retrieved 2026-10-02)
- Inequality or interval (-inf, 1] gives Inequality x ≤ 1, Set-builder notation {x | x ≤ 1}, On a number line arrow toward −∞; closed (filled) dot at 1.Source: OpenStax, Algebra and Trigonometry 2e, §2.7 Linear Inequalities and Absolute Value Inequalities (a bracket includes an endpoint, a parenthesis excludes it, infinity always takes a parenthesis; {x | −2 ≤ x < 6} = [−2, 6), {x | x ≤ 1} = (−∞, 1]), https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-7-linear-inequalities-and-absolute-value-inequalities (retrieved 2026-10-02)
- Inequality or interval -1 < x < 0 gives Interval notation (−1, 0), Type of interval open and bounded, Length 1, Midpoint -0.5.Source: OpenStax, Algebra and Trigonometry 2e, §2.7 Linear Inequalities and Absolute Value Inequalities (a bracket includes an endpoint, a parenthesis excludes it, infinity always takes a parenthesis; {x | −2 ≤ x < 6} = [−2, 6), {x | x ≤ 1} = (−∞, 1]), https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-7-linear-inequalities-and-absolute-value-inequalities (retrieved 2026-10-02)
- Inequality or interval 4 < t gives Interval notation (4, ∞), Inequality t > 4, Type of interval unbounded (a ray), open.Source: OpenStax, Algebra and Trigonometry 2e, §2.7 Linear Inequalities and Absolute Value Inequalities (a bracket includes an endpoint, a parenthesis excludes it, infinity always takes a parenthesis; {x | −2 ≤ x < 6} = [−2, 6), {x | x ≤ 1} = (−∞, 1]), https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-7-linear-inequalities-and-absolute-value-inequalities (retrieved 2026-10-02)
- Inequality or interval 1,000 >= x >= -2.5 gives Interval notation [−2.5, 1000], Length 1,002.5, Midpoint 498.75.Source: OpenStax, Algebra and Trigonometry 2e, §2.7 Linear Inequalities and Absolute Value Inequalities (a bracket includes an endpoint, a parenthesis excludes it, infinity always takes a parenthesis; {x | −2 ≤ x < 6} = [−2, 6), {x | x ≤ 1} = (−∞, 1]), https://openstax.org/books/algebra-and-trigonometry-2e/pages/2-7-linear-inequalities-and-absolute-value-inequalities (retrieved 2026-10-02)
How it works
An interval is a set of numbers between a left end a and a right end b. Each end is either included or not:
| Inequality | Interval notation | Set-builder notation |
|---|---|---|
| a ≤ x ≤ b | [a, b] | {x | a ≤ x ≤ b} |
| a < x < b | (a, b) | {x | a < x < b} |
| a ≤ x < b | [a, b) | {x | a ≤ x < b} |
| a < x ≤ b | (a, b] | {x | a < x ≤ b} |
| x ≥ a | [a, ∞) | {x | x ≥ a} |
| x > a | (a, ∞) | {x | x > a} |
| x ≤ b | (−∞, b] | {x | x ≤ b} |
| x < b | (−∞, b) | {x | x < b} |
- a bracket means the end is included (≤, ≥, a closed dot), a parenthesis that it is not (< or >, an open dot); ±∞ always takes a parenthesis and is drawn as an arrow
- an inequality written the other way round is read the same: 4 < t is t > 4, and 6 > x ≥ −2 is −2 ≤ x < 6
- length = b − a and midpoint = a ÷ 2 + b ÷ 2, for a bounded interval only
- type: "closed and bounded" (both ends included), "open and bounded" (neither), "half-open and bounded" (one), "a single point (closed)" ([a, a]), "unbounded (a ray), closed" or "open" (by its finite end), or "all real numbers (unbounded)"
Rules
- Accepted forms: interval notation
[a, b); a double inequalitya <= x < b(both signs the same way);x > a,x <= b,a < x; andall real numbers(alsoall,Rorℝ, in any case). Signs may be typed as<=,≤,=<,>=,≥or=>. For infinity type ∞, inf or infinity, with − for minus infinity. - The variable is one letter; interval notation uses x. Numbers are decimals with an optional minus sign (- or −) and an optional exponent (2.5e3). In an inequality a comma may appear only as a thousands separator (1,000); in interval notation the comma separates the ends, so numbers there have none.
- There is no answer when the left end is greater than the right, when a = b and an end is excluded, when ∞ has a bracket, when −∞ is on the right or ∞ on the left, or when a number is beyond the double-precision range.
Output format. Ends are written with up to 15 significant digits, no thousands separators, and the minus sign "−". Length and midpoint to 12 significant figures. The number line reads "closed (filled) dot at a" or "open dot at a", or "arrow toward −∞" / "arrow toward ∞", left end first, joined with "; ".
Worked examples by hand
−2 ≤ x < 6. −2 is included, 6 is not: [−2, 6), {x | −2 ≤ x < 6}, half-open; length 6 − (−2) = 8, midpoint 2.
(−∞, 1]. x ≤ 1: an arrow toward −∞ and a closed dot at 1.
−1 < x < 0. (−1, 0), open; length 1, midpoint −0.5.
4 < t. Read as t > 4: (4, ∞), an open ray.
1,000 ≥ x ≥ −2.5. Read as −2.5 ≤ x ≤ 1,000: [−2.5, 1000], length 1,002.5, midpoint 498.75.
Other questions people ask
How do I write an inequality in interval notation?
Write the left end, a comma, and the right end. Use a bracket [ or ] where the end is included (≤ or ≥) and a parenthesis ( or ) where it is not (< or >). −2 ≤ x < 6 becomes [−2, 6).
Does infinity get a bracket or a parenthesis?
Always a parenthesis, because infinity is not a number the interval can include. x ≤ 1 is (−∞, 1], and x > 4 is (4, ∞).
What is set-builder notation?
It describes a set by the rule its members follow: {x | −2 ≤ x < 6} reads "the set of x such that x is at least −2 and less than 6". It is the same set as the interval [−2, 6).
What do open and closed dots mean on a number line?
A closed (filled) dot means the end is included, matching a bracket. An open dot means it is not, matching a parenthesis. An end at infinity is drawn as an arrow.
What is a half-open interval?
One end included and the other not, such as [−2, 6) or (3, 7]. A closed interval includes both ends, [a, b]; an open interval includes neither, (a, b).
Can I type x ≠ 3 or a union?
Not yet: the page reads one interval at a time. x ≠ 3 is the union of two intervals, (−∞, 3) ∪ (3, ∞); type each part on its own.