# Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental concept that pupils should understand owing to the fact that it becomes more critical as you progress to higher mathematics.

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

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

## What Is Interval Notation?

The interval notation is merely a method to express a subset of all real numbers along 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 encounter essentially composed of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless utilization.

However, intervals are generally used to denote domains and ranges of functions in more complex math. Expressing these intervals can increasingly become difficult as the functions become more tricky.

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

x is greater than negative 4 but less than 2

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

So far we understand, interval notation is a way to write intervals elegantly and concisely, using set rules that help writing and understanding intervals on the number line easier.

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

Many 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 does not comprise the endpoints of the interval. The previous notation is a fine example of this.

The inequality notation {x | -4 < x < 2} describes x as being greater than -4 but less than 2, which means that it excludes either of the two numbers referred to. As such, this is an open interval denoted 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 2 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 not contain the values mentioned, a closed interval does. In word 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 -4 and less than or equal to two.”

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

In an interval notation, this is expressed 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 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 last example as a guide, 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 couldn’t possibly be equal to the value 2.

In an inequality notation, this would be expressed 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

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 excludes the other value.

As seen in the prior example, there are numerous symbols for these types subjected to interval notation.

These symbols build the actual interval notation you develop 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 excluded from the subset.

[ ]: The square brackets are employed 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 employed when the interval is half-open, or when only the left endpoint is excluded in the set, and the right endpoint is not excluded. Also called 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 included in the set, while the right endpoint is not included. This is also called a right-open interval.

## Number Line Representations for the Different Interval Types

Aside from being denoted with symbols, the various 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 display all the different types of intervals as they are described in the number line.

## Practice Examples for Interval Notation

Now that you know 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 problem is a easy conversion; simply use 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 written as (-6, 9].

### Example 2

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

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

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

Plus, because no maximum number was stated regarding the number of teams a school can send to the debate competition, this number should be positive to infinity.

Thus, the interval notation should be written 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 regular calorie intake. For the diet to be successful, they must have at least 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 lowest while the value 2000 is the maximum value.

The question suggest that both 1800 and 2000 are inclusive 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 confined to a variation 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 Do You Graph an Interval Notation?

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

There are laws to writing an interval notation to the number line: a closed interval is written with a filled circle, and an open integral is written with an unfilled 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 basically a different 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 lower than or equal to, then the interval is denoted 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 written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the number is ruled out from the combination.

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