Simplify to Interval Notation Calculator

Interval notation is a concise way to represent sets of real numbers. This calculator helps you convert complex inequalities to interval notation quickly and accurately.

What is Interval Notation?

Interval notation is a method of representing a set of real numbers that lie between two endpoints. It's commonly used in calculus, algebra, and other branches of mathematics to describe ranges of values.

There are four main types of intervals:

Closed interval: Includes both endpoints (e.g., [a, b])
Open interval: Excludes both endpoints (e.g., (a, b))
Half-open interval: Includes one endpoint but not the other (e.g., [a, b) or (a, b])
Infinite interval: Extends to infinity (e.g., [a, ∞) or (-∞, b])

Interval notation is particularly useful in describing domains of functions, solution sets of inequalities, and other mathematical concepts that involve ranges of numbers.

How to Convert to Interval Notation

Converting inequalities to interval notation involves a few simple steps:

Identify the inequality symbols (<, >, ≤, ≥)
Determine if the endpoints are included or excluded
Write the numbers in ascending order
Use the appropriate bracket notation

Conversion Rules:

Use square brackets [ ] for included endpoints
Use parentheses ( ) for excluded endpoints
Use ∞ for infinity when appropriate

For example, the inequality -3 ≤ x < 5 would be written as [-3, 5) in interval notation.

Examples

Here are some examples of inequalities and their interval notation equivalents:

Inequality	Interval Notation	Description
-2 < x < 4	(-2, 4)	Open interval excluding -2 and 4
-5 ≤ x ≤ 10	[-5, 10]	Closed interval including -5 and 10
x > -1	(-1, ∞)	All numbers greater than -1
x ≤ 7	(-∞, 7]	All numbers less than or equal to 7

These examples demonstrate how different types of inequalities translate to interval notation.

FAQ

What is the difference between [ ] and ( ) in interval notation?

Square brackets [ ] indicate that the endpoint is included in the interval, while parentheses ( ) indicate that the endpoint is excluded. For example, [2, 5] includes 2 and 5, while (2, 5) excludes both.

How do I represent all real numbers in interval notation?

All real numbers are represented as (-∞, ∞). This indicates that there are no restrictions on the values of x.

Can interval notation represent a single point?

Yes, a single point is represented by using the same number for both endpoints with square brackets, like [3, 3].