This inequality to interval notation calculator is the first place you should visit when you need to convert between these two very popular types of mathematical notation.
Importantly, our inequality to interval notation converter can work both ways! That is, it can teach you how to write an inequality in interval notation, and also how to convert interval notation to inequality notation. It can even deal with compound inequalities! Once you're done here, take one more step in your mathematical journey and discover how to graph inequalities on a number line.
What is the interval notation in math?
An interval is a subset of real numbers that consists of all numbers contained between two given numbers called the endpoints of the interval.
Intervals are directly linked to inequalities: the numbers contained in an interval are exactly those that satisfy certain inequalities related to the endpoints of our interval. For example, the set of numbers x satisfying the inequality 0 < x < 7 is the set that contains all numbers that are simultaneously greater than 0 and less than 7, so the interval has the endpoints 0 and 7.
💡 Intervals may look innocent and very simple, but they are vital in various branches of science — one of their most important applications is in statistics, where they appear as confidence intervals.
What are the types of intervals?
There are different types of intervals. They relate to whether or not each of the endpoints belongs to the interval. In general, there are three types of intervals:
- Open intervals — do not include the endpoints;
- Closed intervals — do include the endpoints; and
- Half-open intervals — include only one of the endpoints.
To determine whether or not a given endpoint is included, just take a look at the symbols used to denote the interval — parentheses (round brackets) mean the endpoint is not included, while square brackets mean the endpoint is included:
| Type | Interval notation | Meaning |
|---|---|---|
| Open interval | (a, b) | a < x < b |
| Closed interval | [a, b] | a ≤ x ≤ b |
| Half-open interval | (a, b] or [a, b) | a < x ≤ b or a ≤ x < b |
How to use this inequality to interval notation calculator?
Here we explain how our inequality to interval notation calculator works:
- Start by choosing the calculator mode, that is, the conversion direction:
- From inequality to interval notation; or
- From interval notation to inequality.
- For the inequality → interval notation mode, first choose the inequality type:
- One-sided (e.g.
x > 3); - Two-sided (e.g.
-2 ≤ x < 5); or - Compound (two inequalities joined with and / or).
- One-sided (e.g.
- For the interval → inequality mode, pick the bracket type on each side — a parenthesis
( ), a square bracket[ ], or infinity — and enter the endpoints. - The result appears at the bottom: the interval notation, the equivalent inequality, the set-builder notation, and a number-line picture. If possible, the calculator will simplify the solution and display the subset of real numbers in the most compact way possible!
How to convert inequality notation to interval notation?
You can very quickly write an inequality in interval notation with the help of the following table, which you can think of as an interval-inequality dictionary 😉
| Inequality | Interval notation |
|---|---|
| x < a | (-∞, a) |
| x ≤ a | (-∞, a] |
| x > a | (a, ∞) |
| x ≥ a | [a, ∞) |
| a < x < b | (a, b) |
| a ≤ x ≤ b | [a, b] |
| a < x ≤ b | (a, b] |
| a ≤ x < b | [a, b) |
Notice that infinity (∞) and negative infinity (-∞) always take a parenthesis, never a square bracket — infinity is not a number, so it can never be "included" as an endpoint.
How to solve compound inequalities in interval notation?
A compound inequality combines two simple inequalities joined by the word and or the word or. The joining word decides how we combine the two solution sets.
Compound inequalities with and
An and compound is the intersection of the two solution sets — only the numbers that satisfy both inequalities survive. For example:
x > 2andx ≤ 5→(2, 5]
Only the numbers greater than 2 and at most 5 belong to the answer. If the two half-lines don't overlap (for instance x < 1 and x > 4), the intersection is empty — there is no solution.
Compound inequalities with or
An or compound is the union of the two solution sets — a number qualifies if it satisfies at least one of the inequalities. For example:
x < 1orx ≥ 4→(-∞, 1) ∪ [4, ∞)
The symbol ∪ is the union operator: it glues two separate pieces of the number line together. When the two pieces overlap, our calculator automatically merges them into one interval.
Frequently Asked Questions
How do I write x ≥ 4 in interval notation?
The inequality x ≥ 4 means every number from 4 upward, with 4 included. In interval notation this is [4, ∞) — a square bracket next to 4 (because 4 is included) and a parenthesis next to infinity (because infinity is never included).
What is the difference between a parenthesis and a square bracket?
A parenthesis ( ) means the endpoint is excluded from the interval (a strict inequality, < or >). A square bracket [ ] means the endpoint is included (a non-strict inequality, ≤ or ≥).
Why does infinity always get a parenthesis?
Infinity is not an actual real number — it is a concept describing an unbounded direction. Because you can never "reach" or include infinity, it is always written with a parenthesis: (a, ∞) or (-∞, b).
What does an empty set look like?
If an inequality has no solution — for example an and compound whose two conditions can never be true together — the answer is the empty set, written ∅ or { }.
Can this tool convert interval notation back to an inequality?
Yes! Switch the conversion direction to "interval → inequality", choose the bracket on each side, and type the endpoints. The calculator returns the matching inequality, set-builder notation, and a number line.