# Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental principle that pupils are required grasp due to the fact that it becomes more critical as you advance to more complex math.

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

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 simply a method to express a subset of all real numbers along 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 essentially composed of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such effortless utilization.

Despite that, intervals are typically employed to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become complicated as the functions become further complex.

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

x is higher than negative 4 but less than two

So far we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. However, 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 principles that help writing and comprehending intervals on the number line less difficult.

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

Various types of intervals lay the foundation for denoting the interval notation. These kinds of interval are essential to get to know due to the fact they underpin the complete notation process.

### Open

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

The inequality notation {x | -4 < x < 2} express x as being higher than negative four but less than two, meaning that it does not contain 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 contrary of the previous 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 negative four 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 states that the interval includes those two boundary values: -4 and 2.

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

### Half-Open

A half-open interval is a blend of previous 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 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 written 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 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 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 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 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 known as a left open interval.

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

## Number Line Representations for the Different Interval Types

Aside from being written with symbols, the different interval types can also be represented 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 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 simple conversion; simply utilize 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 written as (-6, 9].

### Example 2

For a school to take part in a debate competition, they require minimum of three teams. Represent 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 value 3 is consisted in the set, which implies that 3 is a closed value.

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

Thus, 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 do a diet program constraining 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 value 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.

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 also known as a bounded interval.

## Interval Notation FAQs

### How To Graph an Interval Notation?

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

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

### How Do You 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 greater than or less a value (not equal to), then the value should be written with parentheses () in the notation.

If x is greater 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 see how these symbols are employed.

### How To Rule Out Numbers in Interval Notation?

Values 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 combination.

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