# Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a essential concept that learners should understand because it becomes more important as you advance to more difficult math.

If you see higher arithmetics, such as integral and differential calculus, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these theories.

This article will discuss what interval notation is, what it’s used for, and how you can interpret it.

## What Is Interval Notation?

The interval notation is simply a method 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 ∞ denotes infinity.)

Basic problems you face essentially composed of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such straightforward utilization.

Despite that, intervals are generally used to denote domains and ranges of functions in advanced math. 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 higher than negative 4 but less than 2

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

So far we know, interval notation is a way to write intervals concisely and elegantly, using fixed principles that make writing and understanding intervals on the number line simpler.

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

## Types of Intervals

Several types of intervals place the base for denoting the interval notation. These interval types are necessary to get to know due to the fact they underpin the entire notation process.

### Open

Open intervals are used when the expression do not comprise the endpoints of the interval. The previous 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 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 negative four and two, 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 last 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 stated with brackets, or [-4, 2]. This implies that the interval contains those two boundary values: -4 and 2.

On the number line, a shaded circle is utilized 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 last example for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to negative four and less than two.” This states that x could be the value -4 but couldn’t 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 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 excludes 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 different symbols for these types subjected to interval notation.

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

( ): The parentheses are employed 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 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 not excluded 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 written with symbols, the various interval types can also be described in the number line employing both shaded and open circles, relying on the interval type.

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

## Practice Examples for Interval Notation

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

### Example 1

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

This sample question is a straightforward conversion; just use 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 participate in a debate competition, they should have a at least three teams. Express this equation in interval notation.

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

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

Furthermore, since no upper limit was mentioned regarding 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 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 minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you describe this range in interval notation?

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

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

Therefore, the interval notation is denoted as [1800, 2000].

When the subset of real numbers is confined 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 Frequently Asked Questions

### How To Graph an Interval Notation?

An interval notation is simply a technique of describing inequalities on the number line.

There are rules to writing an interval notation to the number line: a closed interval is expressed 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 Do You Transform Inequality to Interval Notation?

An interval notation is just a diverse way of describing an inequality or a set of real numbers.

If x is higher than or lower than a value (not equal to), then the value 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 denoted with closed brackets [ ] in the notation. See the examples of interval notation above to check how these symbols are used.

### 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 means that you’re expressing an open interval, which states that the number is excluded from the set.

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