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General Studies Math: Help & Review8 chapters | 85 lessons

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Lesson Transcript

Instructor:
*Joshua White*

Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

This lesson defines interval notation and explains how to write sets in interval notation. It also explores the various types of intervals, including open and closed intervals.

There are lots of little tricks that are used to simplify problems and information in math. **Interval notation** is one of them. When we express a set of real numbers using start and endpoints in addition to brackets and parentheses, we are using interval notation. Let's take a closer look at how this works.

Say you're shopping around for new cell phone service. You have two options: you can renew your contract with your current provider and get a new phone for free, or you can purchase a phone for the full price and use a prepaid carrier with a cheaper monthly plan. You do the math and find out that for months 1-15, the contract plan is cheaper, but after 15 months, the prepaid carrier becomes the cheaper option.

Now, if you were to share this information with one of your friends, you wouldn't say:

*If I use my new phone service for one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, and fifteen months, it's cheaper to stay with my current provider, but if I plan to use it for sixteen, seventeen, eighteen, nineteen, twenty, etc. months, the prepaid carrier is cheaper.*

That's crazy. Nobody talks like that. Instead, you would simplify it for your friend and say something like:

*If I keep the phone for up to 15 months, the contract plan is cheaper, but if I use it for 16 months or more, the prepaid plan would be cheaper.*

In other words, you wouldn't list every single possibility that applies to both situations.

A whole range of possibilities with a start and/or endpoint is called a **set** in math. A set is a group of unique, non-repeating elements, or in the case of the phone example, numbers representing months. One set might be the number of months in which it is cheaper to extend your contract. Another set might be the number of months in which it is cheaper to pay for the phone and switch to a prepaid carrier. A third set could be months where the costs are exactly even. Interval notation is a good way to express these sets in math. Let's take a look at how interval notation actually works.

Before you can write a set in interval notation, you need to determine two things. First, you need to determine the **endpoints** of the interval. Endpoints are the interval of numbers between one point and another. In our contract plan phone example, the endpoints were 1 and 15. Endpoints can be numbers, positive infinity, or negative infinity. The second thing you need to determine is what type of interval you have.

- In an
**open interval**, both endpoints are not included in the interval. It is written in the format (a,b) where a and b are the endpoints. - In a
**closed interval**, the endpoints are included in the interval. It is written in the form shown below where a and b are the endpoints.

Thus, a parenthesis indicates an open endpoint (endpoint not included), and a bracket indicates a closed endpoint (endpoint is included). When there is a mix of one included endpoint and one non-included endpoint, the interval will have each side identified by the type that its endpoint is. For example, (2,6] would be a left-open, right-closed interval since the left endpoint is not included in the interval but the right endpoint is. The following is an example of a left-closed right-open interval since the left endpoint, -1, is included in the interval, and the right endpoint, 10, is not included.

When one end of the interval goes on forever to positive or negative infinity, then the interval will be identified as right- or left-open or closed depending upon which side has a numerical endpoint (this will determine whether it is right or left) and whether that endpoint is included or not (open or closed). The following interval is a left-closed interval because there is a numerical endpoint only on the left side, and it is closed because -1 is included in the interval.

However, the following example is a right-open interval because only the right side has a numerical endpoint, 0, and it is not included in the interval.

Once you have determined the endpoints and type of interval, you can express any inequality or set of numbers in an interval notation.

Going back to the phone example, the sets we found can each be written as inequalities. In months 1-15, it is cheaper to extend your contract and get a free phone. This can be written as:

In the example, *x* represents the number of months you plan to use the phone.

For 16 months and beyond, it is cheaper to purchase the phone at full cost and switch to a prepaid carrier. This can be written as:

Now, we're going to express those inequalities in interval notation. For the first set, which represents all the months where it is cheaper to extend the contract, we have endpoints of 1 and 15. Are they included in the set? Yes, since you can have cell phone service for only 1 month, 1 will be included in the set and have a bracket next to it. The endpoint 15 should be included if you can have service for 15 months (yes, you can), and it is cheaper for 15 months of service to extend your contract (yes, it is). Therefore, 15 will be included and have a bracket as well.

So, the interval notation of the set representing the months where it is cheaper to extend your contract would be:

Note that it's a closed interval since both endpoints are included in it. Now, let's take a look at the other set.

For the second interval, which represents all the months where it is cheaper to purchase the phone and use the prepaid carrier, one endpoint is 16, since at 16 months it becomes cheaper to purchase the phone. But what is the other endpoint of the inequality? If you think about it, there really is no limit on the number of months that you can stay with the new prepaid provider. In other words, it will be cheaper after 16 months, 17 months, 18 months, 19 months, 20 months, etc., and you can keep counting the number of months forever. Thus, there is only one numerical endpoint for this interval (16) since it can continue on to positive infinity. This matches with the inequality we wrote, because there is no upper limit that *x* must be less than:

The set representing the months where it is cheaper to purchase your phone and switch providers is a left-closed interval since 16 is included in it but positive infinity isn't by definition:

Let's look at some example problems.

**1. Write the following inequalities in interval notation and identify the type.**

*x* < -5

This inequality represents all numbers that are less than -5 going to negative infinity. Thus, in interval notation it would be written as:

There is a parenthesis on -5 because it is not included in the interval. The interval type is right-open.

**2. Write the following inequalities in interval notation and identify the type.**

This inequality represents all numbers greater than 0 and less than or equal to 20. Thus, the left endpoint, 0, is open, but the right endpoint, 20, is closed. The interval notation is (0,20] and the interval type is left-open, right-closed.

**3. Write the following compound inequality in interval notation: x < 2 or x > 7**

The first inequality represents all numbers less than, but not including, 2. Thus, it would go from negative infinity to 2, which will have a parenthesis:

The second inequality represents all numbers from 7, not included, up to positive infinity:

So the full answer for this example problem looks like this:

**4. The following graph shows an inequality graphed on a number line. Write an interval representing the inequality shown:**

The left endpoint, -1, has a closed circle, which indicates that it is included in the set. Thus, -1 will have a bracket on it. The right endpoint, 3, has an open circle, indicating it is not included in the set and will have a parenthesis on it. So, it would be written like this:

Let's review. Any set of real numbers can be written in **interval notation**. There are two types of intervals: **open intervals**, where neither of the endpoints are included in the set and the interval will have parentheses on both sides, and **closed intervals**, where the endpoints are included as indicated by brackets on both sides.

When there is a mix of one included endpoint and one non-included endpoint, the interval will have each side identified by the type that its endpoint is. For example, (2,6] would be a left-open, right-closed interval since the left endpoint is not included in the interval but the right endpoint is. Interval notation can be used when you are given an inequality, a graph on a number line, or even a statement describing a set of numbers.

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