Interval Notation Calculator

What this calculator does

So, you want to move between interval notation and inequalities, in either direction. This tool converts both ways. It can turn an interval into the inequality it stands for, and it can turn an inequality into interval notation, including compound inequalities joined by and or or.

A dropdown chooses the direction, and further menus set the exact type of interval or inequality you are working with.

How to use it

1. Choose the conversion direction: interval to inequality, or inequality to interval.
2. Pick the type and enter the endpoint values.
3. Press Calculate.

What interval notation is

Interval notation is a compact way of writing a range of numbers on the real line. Instead of saying x is greater than 2 and less than 5, you write an interval with those two numbers as its ends. It is the standard shorthand for describing solution sets, the collection of all numbers that satisfy a condition. An interval is simply a connected stretch of the number line, and the notation captures two things about it: where it starts and stops, and whether those endpoints are themselves included.

Round brackets and square brackets

The whole system rests on two kinds of bracket. A round bracket, a parenthesis, means the endpoint is not included: the interval gets arbitrarily close to it but stops just short. This matches a strict inequality, less than or greater than. A square bracket means the endpoint is included, matching an inequality with an "or equal to" part. So the interval from 2 to 5 with round brackets means 2 is less than x is less than 5, while square brackets would mean 2 is less than or equal to x is less than or equal to 5. You can even mix them, a square bracket on one end and a round on the other, for a range that includes one endpoint but not the other.

Infinity is always open

When a range has no bound on one side, it runs off to infinity, and the notation uses the infinity symbol for that end. An inequality like x is greater than 2 has no upper limit, so its interval starts at 2 and stretches to infinity. Infinity always takes a round bracket, never a square one, because infinity is not a number you can ever reach or include; it just indicates the range goes on without end. The tool covers all these one-sided cases, both the ones open to the right and the ones open to the left.

Compound inequalities, and or or

Things get more interesting when two conditions are combined. A compound inequality joins two simpler ones with and or with or, and the two words mean very different things. And means a number must satisfy both conditions at once, which is the overlap, or intersection, of the two intervals. Or means a number need satisfy only one of them, which is the combination, or union, of the two intervals. The tool handles both, writing the result with a union or intersection symbol and simplifying where it can, including the cases where two conditions overlap in a single interval, split into two separate pieces, or contradict each other so that nothing satisfies them at all.

A worked example

Converting the interval from 2 to 5 with a square bracket on the left and a round bracket on the right gives the inequality 2 is less than or equal to x is less than 5. Going the other way, the inequality x is greater than 3 becomes the interval from 3 to infinity with round brackets. And a compound condition like x is less than 1 or x is greater than or equal to 4 becomes the union of two intervals: from negative infinity to 1 with a round bracket, joined to 4 through infinity with a square bracket on the 4.

Questions people ask

What is interval notation?

A compact way to write a range of numbers, giving its endpoints and whether each is included, used to describe solution sets.

What is the difference between round and square brackets?

A round bracket excludes the endpoint, matching a strict inequality. A square bracket includes it, matching an "or equal to" inequality.

Why does infinity get a round bracket?

Because infinity is not a number you can reach or include. It only signals that the range continues without end, so it is always open.

What is the difference between and and or?

And is the overlap of two intervals, their intersection. Or is the combination of two intervals, their union.

Can a compound inequality have no solution?

Yes. If two "and" conditions contradict each other, no number satisfies both, and the tool reports that no interval works.

References

On interval notation. An interval is a connected part of the real line, written with brackets that show whether each endpoint is included.

1. Eric W. Weisstein, "Interval," from MathWorld, a Wolfram resource, on open, closed, and half-open intervals and rays to infinity.
2. "Linear Inequalities and Absolute Value Inequalities," Algebra and Trigonometry, OpenStax, on interval notation for describing solution sets of inequalities.