# Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental topic that pupils are required grasp due to the fact that it becomes more essential as you grow to higher arithmetic.

If you see more complex math, such as integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these concepts.

This article will talk in-depth what interval notation is, what it’s used for, and how you can interpret it.

## What Is Interval Notation?

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

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

Fundamental difficulties you encounter mainly composed of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such straightforward utilization.

Though, intervals are generally employed to denote domains and ranges of functions in advanced arithmetics. Expressing these intervals can progressively become complicated as the functions become further complex.

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

x is greater than negative four but less than two

As we know, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), signified 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 about the principles of expressing a subset in a set of all real numbers with interval notation.

## Types of Intervals

Several types of intervals lay the foundation for writing the interval notation. These interval types are essential to get to know due to the fact they underpin the entire notation process.

### Open

Open intervals are used when the expression does not comprise the endpoints of the interval. The prior notation is a good example of this.

The inequality notation {x | -4 < x < 2} express x as being greater than -4 but less than 2, meaning that it does not include neither 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 implies that in a given set of real numbers, such as the interval between -4 and 2, those two values are excluded.

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 not include the values mentioned, a closed interval does. In text form, a closed interval is expressed as any value “higher than or equal to” or “less than or equal to.”

For example, if the last example was a closed interval, it would read, “x is greater than or equal to negative four and less than or equal to 2.”

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

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

On the number line, a shaded circle is used to describe 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 previous example for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than 2.” This means that x could be the value negative four but cannot possibly be equal to the value two.

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

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

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

## Symbols for Interval Notation and Types of Intervals

To recap, 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 prior example, there are different symbols for these types under the interval notation.

These symbols build the actual interval notation you create 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 employed when the interval is closed, or when the two points on the number line are not excluded in the subset of real numbers.

( ]: Both the parenthesis and the square bracket are employed when the interval is half-open, or when only the left endpoint is excluded in the set, and the right endpoint is included. Also called 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 excluded. This is also called 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 represented in the number line using both shaded and open circles, depending 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 question is a simple conversion; just utilize the equivalent symbols when writing 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 at least 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 value 3 is consisted in the set, which states that three is a closed value.

Furthermore, since no upper limit was referred to with concern to the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.

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

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

### Example 3

A friend wants to participate in diet program limiting their regular calorie intake. For the diet to be successful, they should have minimum of 1800 calories every day, but no more than 2000. How do you describe this range in interval notation?

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

The question implies that both 1800 and 2000 are included in the range, so the equation is a close interval, expressed with the inequality 1800 ≤ x ≤ 2000.

Therefore, the interval notation is described 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 called a bounded interval.

## Interval Notation FAQs

### How Do You Graph an Interval Notation?

An interval notation is basically a way of describing inequalities on the number line.

There are rules of expressing an interval notation to the number line: a closed interval is denoted with a shaded circle, and an open integral is written with an unfilled circle. This way, you can promptly see on a number line if the point is included or excluded from the interval.

### How Do You Transform Inequality to Interval Notation?

An interval notation is basically a diverse way of describing 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 written with closed brackets [ ] in the notation. See the examples of interval notation prior to check how these symbols are employed.

### How Do You Rule Out Numbers in Interval Notation?

Numbers ruled out from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which states that the number is excluded from the set.

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