Math
Interval Notation Calculator
Type an inequality such as -3 < x <= 5, a compound statement such as x > 2 or x < -1, or interval notation such as [-3, 5) and the interval notation calculator rewrites it in every form: inequality, interval notation with ∪ for unions, set-builder notation, and a description of the number-line graph with open and closed dots.
Use <, <=, >, >=, or ≤ ≥. Combine with 'and' / 'or' (or ∪). Write infinity as inf or ∞. Any single letter works as the variable.
Try an example
Result
Interval notation
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- Inequality
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- Set-builder notation
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- Number line
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- Type of set
- —
- Endpoints
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- Bounded?
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More details (2 more)
- Total length
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- Integers in the set
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the interval notation calculator solves
The same set of numbers can be written as an inequality, as an interval, as a set-builder statement, or drawn on a number line. Round parentheses ( ) mean the endpoint is not included; square brackets [ ] mean it is. Infinity always gets a parenthesis because it is not a number you can reach.
Notation cheat sheet
| Inequality | Interval | Set-builder | Number line |
|---|---|---|---|
| -3 < x ≤ 5 | (-3, 5] | {x | -3 < x ≤ 5} | open dot at -3, closed dot at 5, shade between |
| x ≥ 4 | [4, ∞) | {x | x ≥ 4} | closed dot at 4, shade right |
| x < -1 or x > 2 | (-∞, -1) ∪ (2, ∞) | {x | x < -1 or x > 2} | open dots at -1 and 2, shade outward |
| x ≠ 3 | (-∞, 3) ∪ (3, ∞) | {x | x ≠ 3} | open dot at 3, shade everything else |
| all real numbers | (-∞, ∞) | ℝ | shade the whole line |
How to use it
- Type the inequality as written in your problem, such as -3 < x <= 5, or the interval such as [-3, 5).
- Join conditions with 'and' or 'or'. Unions can also be typed with U or ∪, and set-builder input like {x | x >= 4} is accepted.
- Read the Writing a range with brackets and parentheses, where brackets include the endpoint. at the top and the other forms below it. The steps show how each endpoint was read.
How to read the answer
A parenthesis means the endpoint is excluded (open dot); a bracket means it is included (closed dot). ∪ joins separate pieces. The type row tells you whether the set is open, closed, half-open, unbounded, empty, or all real numbers.
Common mistakes and edge cases
- Using a bracket with infinity: it is always (-∞, 3], never [-∞, 3].
- Writing the larger number first: intervals go from left to right on the number line, so (5, -3) is wrong.
- Mixing up and/or: 'x > 2 or x < -1' is two rays, not an empty set.
- Writing a union as a single interval: (-∞, -1) ∪ (2, ∞) cannot be simplified further.
Worked examples
Double inequality
-3 < x <= 5 → (-3, 5]
Interval notation
(-3, 5]
Compound 'or'
x > 2 or x < -1 → (-∞, -1) ∪ (2, ∞)
Interval notation
(-∞, -1) ∪ (2, ∞)
Interval to inequality
[-3, 5) → -3 ≤ x < 5
Interval notation
[-3, 5)
Compound 'and'
x >= 1 and x < 4 → [1, 4)
Interval notation
[1, 4)
Not equal
x ≠ 3 → (-∞, 3) ∪ (3, ∞)
Interval notation
(-∞, 3) ∪ (3, ∞)
Overlapping union simplifies
x < 2 or x <= 5 → (-∞, 5]
Interval notation
(-∞, 5]
Decimal endpoints
0.5 <= t < 2.75 with variable t
Interval notation
[0.5, 2.75)
Empty set
x > 5 and x < 2 has no solution
Interval notation
∅ (empty set)
Bracket on infinity
[-inf, 2] is not valid
Interval notation
Error
Frequently asked questions
What is the difference between ( and [ in interval notation?+
A parenthesis means the endpoint is not part of the set (strict inequality, open dot). A bracket means it is included (≤ or ≥, closed dot).
How do I write x ≥ 4 in interval notation?+
[4, ∞). The 4 is included so it gets a bracket; infinity always gets a parenthesis.
How do I write 'or' inequalities in interval notation?+
Write each piece as its own interval and join them with the union symbol ∪. For x < -1 or x > 2 the answer is (-∞, -1) ∪ (2, ∞).
What is set-builder notation?+
A way of describing a set by a condition: {x | x > 2} reads 'the set of all x such that x is greater than 2'. Some books use a colon instead of the bar.
Why can't I use a bracket with infinity?+
Infinity is not a number in the set; it only shows the interval keeps going. Since it can never be reached, it can never be included, so it always takes a parenthesis.
How do I show x ≠ 3 in interval notation?+
Take the whole line and remove the point: (-∞, 3) ∪ (3, ∞).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026