# Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental topic that learners should grasp owing to the fact that it becomes more critical as you advance to higher math.

If you see higher arithmetics, such as differential calculus and integral, in front of you, then knowing the interval notation can save you hours in understanding these ideas.

This article will talk about what interval notation is, what it’s used for, and how you can understand it.

## What Is Interval Notation?

The interval notation is merely a way to express a subset of all real numbers across the number line.

An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)

Fundamental difficulties you face mainly composed of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such straightforward applications.

However, intervals are usually employed to denote domains and ranges of functions in higher arithmetics. Expressing these intervals can progressively become complicated as the functions become further tricky.

Let’s take a simple compound inequality notation as an example.

x is higher than negative 4 but less than 2

Up till now we understand, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. However, it can also be expressed with interval notation (-4, 2), denoted by values a and b separated by a comma.

As we can see, interval notation is a way to write intervals elegantly and concisely, using predetermined rules that help writing and understanding intervals on the number line simpler.

In the following section we will discuss regarding the rules of expressing a subset in a set of all real numbers with interval notation.

## Types of Intervals

Various types of intervals place the base for denoting the interval notation. These kinds of interval are important to get to know because they underpin the complete notation process.

### Open

Open intervals are used when the expression do not include the endpoints of the interval. The prior notation is a great example of this.

The inequality notation {x | -4 < x < 2} describes x as being higher than -4 but less than 2, meaning that it does not include either of the two numbers mentioned. As such, this is an open interval expressed with parentheses or a round bracket, such as the following.

(-4, 2)

This means that in a given set of real numbers, such as the interval between negative four and two, those two values are not included.

On the number line, an unshaded circle denotes an open value.

### Closed

A closed interval is the opposite of the previous type of interval. Where the open interval does exclude the values mentioned, a closed interval does. In text form, a closed interval is expressed as any value “greater than or equal to” or “less than or equal to.”

For example, if the previous example was a closed interval, it would read, “x is greater than or equal to -4 and less than or equal to two.”

In an inequality notation, this can be written as {x | -4 < x < 2}.

In an interval notation, this is written with brackets, or [-4, 2]. This implies that the interval includes those two boundary values: -4 and 2.

On the number line, a shaded circle is utilized to represent an included open value.

### Half-Open

A half-open interval is a combination of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the last example as a guide, if the interval were half-open, it would read as “x is greater than or equal to negative four and less than two.” This implies that x could be the value -4 but couldn’t possibly be equal to the value 2.

In an inequality notation, this would be denoted as {x | -4 < x < 2}.

A half-open interval notation is denoted with both a bracket and a parenthesis, or [-4, 2).

On the number line, the shaded circle denotes the number included in the interval, and the unshaded circle denotes the value excluded from the subset.

## Symbols for Interval Notation and Types of Intervals

To summarize, there are different types of interval notations; open, closed, and half-open. An open interval doesn’t contain the endpoints on the real number line, while a closed interval does. A half-open interval consist of one value on the line but excludes the other value.

As seen in the examples above, there are various symbols for these types under the interval notation.

These symbols build the actual interval notation you develop when plotting points on a number line.

( ): The parentheses are employed when the interval is open, or when the two endpoints on the number line are excluded from the subset.

[ ]: The square brackets are used when the interval is closed, or when the two points on the number line are included in the subset of real numbers.

( ]: Both the parenthesis and the square bracket are used when the interval is half-open, or when only the left endpoint is not included in the set, and the right endpoint is not excluded. Also known as a left open interval.

[ ): This is also a half-open notation when there are both included and excluded values between the two. In this case, the left endpoint is included in the set, while the right endpoint is not included. This is also known as a right-open interval.

## Number Line Representations for the Various Interval Types

Apart from being denoted with symbols, the various interval types can also be described in the number line utilizing both shaded and open circles, relying on the interval type.

The table below will show all the different types of intervals as they are represented in the number line.

## Practice Examples for Interval Notation

Now that you know everything you are required to know about writing things in interval notations, you’re ready for a few practice problems and their accompanying solution set.

### Example 1

Convert the following inequality into an interval notation: {x | -6 < x < 9}

This sample problem is a straightforward conversion; simply use the equivalent symbols when denoting the inequality into an interval notation.

In this inequality, the a-value (-6) is an open interval, while the b value (9) is a closed one. Thus, it’s going to be expressed as (-6, 9].

### Example 2

For a school to take part in a debate competition, they require minimum of 3 teams. Express this equation in interval notation.

In this word question, let x stand for the minimum number of teams.

Since the number of teams needed is “three and above,” the number 3 is consisted in the set, which means that three is a closed value.

Additionally, since no upper limit was stated regarding the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.

Therefore, the interval notation should be denoted as [3, ∞).

These types of intervals, when one side of the interval that stretches to either positive or negative infinity, are also known as unbounded intervals.

### Example 3

A friend wants to participate in diet program limiting their regular calorie intake. For the diet to be a success, they should have at least 1800 calories every day, but maximum intake restricted to 2000. How do you express this range in interval notation?

In this word problem, the number 1800 is the lowest while the value 2000 is the highest value.

The question suggest that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.

Thus, the interval notation is written as [1800, 2000].

When the subset of real numbers is restricted to a range between two values, and doesn’t stretch to either positive or negative infinity, it is also known as a bounded interval.

## Interval Notation FAQs

### How To Graph an Interval Notation?

An interval notation is simply a way of representing inequalities on the number line.

There are rules of expressing an interval notation to the number line: a closed interval is written with a filled circle, and an open integral is denoted with an unshaded circle. This way, you can quickly check the number line if the point is included or excluded from the interval.

### How To Change Inequality to Interval Notation?

An interval notation is just a different way of expressing an inequality or a combination of real numbers.

If x is greater than or less a value (not equal to), then the number should be written with parentheses () in the notation.

If x is higher than or equal to, or less than or equal to, then the interval is expressed with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are employed.

### How To Exclude Numbers in Interval Notation?

Values excluded from the interval can be written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the value is ruled out from the combination.

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