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Interval Notation Calculator

Enter your inequality to see the correct interval notations with step-by-step explanation.

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Interval Notation Calculator

This interval notation calculator converts the inequalities into interval notations. Whether you are a student, teacher, or mathematician, it helps you express the solution sets and number ranges quickly and accurately. Our calculator is designed to clarify what every parentheses, bracket, and infinity symbol truly means. It saves time, eliminates errors, and enhances understanding of interval notation.

What is Interval Notation?

The interval notation is the way to write all the numbers between two endpoints on a number line. Instead of writing the inequalities in a complex form, it lets you write them in a simple interval notation form.

For instance, suppose you have the inequality 2 ≤ x < 5, you can express it in the interval notation form as [2, 5). Here, brackets are used to define whether the endpoints are included or excluded

[ ] This bracket indicates that the endpoint is included (closed interval)
( ) If this parenthesis is used, then it means the endpoint is excluded (open interval)
∞ (infinity) is always written with parentheses because it’s not a real number

The interval notation makes it easier to describe domains, ranges, or solution sets in mathematics. It is widely used to simplify expressions containing the set of real numbers in calculus and algebra.

Why Use Interval Notation?

The interval notation allows you to write the inequalities in a simple way to show continuous ranges of values. It's very useful in:

Defining the domains and ranges of functions
Showing the solution sets of inequalities
Presenting the intervals on the number line

By grasping the concept of interval notations, the students can easily simplify complex mathematical problems and can understand the graphs more effectively.

Using the interval notation calculator ensures accuracy and saves time, especially when dealing with inequalities, complex ranges, infinity, or compound intervals.

How to Find the Interval Notation?

See the following example to understand how to find interval notation:

Example:

Let’s convert the inequality 7 - x/6 > 8 into interval notation step by step:

Step #1: Move constants to one side

7 - x 6 > 8

Subtract 7 from both sides:

-7 +7 x 6 > 1

Step #2: Eliminate the negative sign

Multiply both sides by -1 and reverse the inequality (because multiplying or dividing by a negative number flips the sign):

x 6 < -1

Step #3: Remove the denominator

Multiply both sides by 6:

6 * x 6 < -1 * 6

Step #4: Write the answer in interval notation

Since x is less than -6, the interval is written as:

(-∞, -6)

Result Interpretation:

The parentheses ( ) are used in the result, because -6 is not included in the solution ( inequality).
Remember, the infinity symbol ∞ is always written with parentheses because it’s not a real number

Types of Intervals

Intervals are categorized into three types based on the numbers that are present in a set. Some sets include the endpoints, some may not, or some may contain only partially. The three types of intervals are: