# Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a essential concept that learners need to learn owing to the fact that it becomes more critical as you grow to higher mathematics.

If you see advances mathematics, something like differential calculus and integral, on your horizon, then knowing the interval notation can save you hours in understanding these theories.

This article will discuss what interval notation is, what are its uses, and how you can decipher it.

## What Is Interval Notation?

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

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

Fundamental difficulties you face essentially consists 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 typically used to denote domains and ranges of functions in higher math. Expressing these intervals can progressively become difficult as the functions become more tricky.

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

x is greater than negative four but less than two

As we know, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be written with interval notation (-4, 2), denoted by values a and b separated by a comma.

So far we understand, interval notation is a method of writing intervals elegantly and concisely, using predetermined principles that help writing and comprehending intervals on the number line less difficult.

The following sections will tell us more regarding the rules of expressing a subset in a set of all real numbers with interval notation.

## Types of Intervals

Many types of intervals place the base for denoting the interval notation. These kinds of interval are necessary to get to know because they underpin the entire notation process.

### Open

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

The inequality notation {x | -4 < x < 2} describes x as being higher than -4 but less than 2, which means that it excludes 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 -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 contrary of the last type of interval. Where the open interval does not contain the values mentioned, a closed interval does. In text form, a closed interval is written as any value “higher 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 negative four and less than or equal to 2.”

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

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

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

### Half-Open

A half-open interval is a blend 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 two.” This means that x could be the value -4 but couldn’t possibly be equal to the value 2.

In an inequality notation, this would be written 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 which are not included from the subset.

## Symbols for Interval Notation and Types of Intervals

In brief, there are different types of interval notations; open, closed, and half-open. An open interval doesn’t include the endpoints on the real number line, while a closed interval does. A half-open interval includes one value on the line but does not include the other value.

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

These symbols build the actual interval notation you create when expressing points on a number line.

( ): The parentheses are used when the interval is open, or when the two endpoints on the number line are not included in the subset.

[ ]: The square brackets are used 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 not included in the set, and the right endpoint is included. Also known as a left open interval.

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

## Number Line Representations for the Various Interval Types

Apart from being denoted with symbols, the different interval types can also be represented in the number line utilizing 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 prepared 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 stating 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 join in a debate competition, they should have a at least 3 teams. Express this equation in interval notation.

In this word problem, let x be the minimum number of teams.

Because the number of teams required is “three and above,” the number 3 is included on the set, which implies that three is a closed value.

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

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

These types of intervals, where there is 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 undertake a diet program constraining their daily calorie intake. For the diet to be a success, they must have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you describe 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 implies that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, written 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 Frequently Asked Questions

### How To Graph an Interval Notation?

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

There are laws of expressing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is denoted with an unfilled circle. This way, you can promptly 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 describing an inequality or a combination of real numbers.

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

If x is higher than or equal to, or lower 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 ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the number is excluded from the set.

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