Linear Inequalities, Interval and Set Builder Notation

1. Interval Notation

Before we move on and introduce inequalities, let’s take a look at interval notation to get a better understanding of endpoints. An endpoint is where a range of values starts or ends.

On a number line, we define specific intervals on the number line using symbols.

For example, let’s look at the interval on the number line below that shows a highlighted range of values:

If we examine the highlighted portion of the number line, we can see that the highlighted portion starts at negative 2 and ends at positive 3. These are the endpoints of this interval: -2 and +3. Note there is an open circle at -2. This means that the interval includes the numbers between -2 and 3, but the interval does not include -2. So, -1, -1.7, and -1.99 are within the interval, but -2 is not within the interval.

We can also see that there is a filled-in circle or a closed circle at the positive 3 on the number line. This means that the interval ends at 3 and the endpoint 3 is included in the interval. So, 1, 2.95, and 3 are all within the interval.

Open and Closed Circles for Intervals on a Number Line
Endpoint is included in interval (closed circle)	Endpoint is not included in interval (open circle)

So, how would we write this interval between -2 and +3 without showing a picture on the number line? We need a few more symbols to express whether the endpoint is included within the interval or not. These include square brackets and parentheses.

Square brackets correspond to the closed circles on the number line. They indicate that the endpoint is included in the interval. The parentheses (or rounded brackets) are used when there are open circles on the number line, and the endpoint is not included in the interval.

1a. One Interval with Two Endpoints

EXAMPLE

Write the highlighted range on the number line in interval notation.



First, we note the endpoints of the highlighted range. There is an endpoint (or circle) at -2 and at 3. These will be the numbers written in our interval.

We will just need to put the correct bracket in front of the -2 and behind the 3. We see that there is an open circle at -2; this will correspond to a parenthesis in front of the -2. There is a closed circle at the positive 3. This will correspond to a square bracket after the 3. So, our interval is (-2, 3].

1b. One Interval with One Endpoint

Intervals with one endpoint are written quite differently.

watch

Here is a video to help introduce the notations that are used.

try it

Use your understanding from the video to answer the following questions.

Write the highlighted number line using interval notation.

Write the highlighted number line using interval notation.

1c. More Than One Interval

Sometimes, we want to describe numbers that fall into two or more non-overlapping intervals. For example, a person’s resting heart rate per minute might be of concern if it is below 40 beats per minute or above 100 beats per minute.

Here is the shaded number line for this situation. (Note that the left-hand interval starts at 0 since we can’t have a negative number of heartbeats).

How would we represent this using interval notation?

To do so, we need to define the union of two intervals, which can also be thought of as sets of numbers.

If there is more than one interval, we use the union symbol (⋃) between the intervals in interval notation.

In our problem, note that the first interval is written and the second is written . Then, the interval notation that includes all the numbers is .

EXAMPLE

Use the number line below and write the interval in interval notation.



This number line has two ranges highlighted. How do we write this in interval notation? We have one interval that includes all values less than -4 (but not including -4), and another interval that includes all values greater than or equal to 1.

In interval notation, we define two intervals: the first interval is and the second interval is . To accept both intervals as solutions for x, we use the symbol for union, , to connect or join the two intervals together. To complete our solution in interval notation, we have .

big idea

Interval notation describes a range of values from a starting point to a stopping point. We use either square brackets or parentheses, depending on if we are including or excluding exact values of the endpoints. Square brackets [ ] are used to include endpoints, and parentheses (   ) are used to exclude endpoints. We always use parentheses with the infinities.

hint

The interval is used to represent the entire real number line.

term to know

Union

The collection of all numbers across two or more sets of numbers.

2. What Is an Inequality?

So far in this course, we have only talked about equalities; in other words, or . In this lesson, we will cover inequalities. An inequality is a mathematical statement to show that one quantity is greater than or less than another quantity. The following outlines the different inequality symbols and their meanings:

Symbol	Translation	Example
	Less than	3 is less than 5
	Greater than	7 is greater than 5
	Less than or equal to	4 is less than or equal to 5
	Greater than or equal to	8 is greater than or equal to five

Inequalities are used in statistics, business, economics, and optimization when comparing values to each other. As you likely know, young children seem to master inequalities at an irritatingly early age. What happens if you give one child three candy bars and the other, two? Of course, the child with the two candy bars will complain that the other child “has more.”

hint

Can’t remember which way the inequality symbol should be facing? Imagine it’s a hungry animal’s mouth and it always wants to eat the bigger value. The wide-open side of the inequality points to the larger value.

try it

Replace each blank with the appropriate inequality symbol. Choose from "<" and ">".

4 _ 7

4 is smaller/less than 7. Have the inequality "mouth" face toward the larger value: 7.

-5 _ -12

Negative 5 (-5) is greater than negative 12 (-12) because it is further to the right on the number line. Have the inequality “mouth” face toward the larger value: -5.

6 _ 8

6 is smaller/less than 8. Have the inequality “mouth” face toward the larger value: 8.

-4 _ -3

-4 is smaller/less than negative 3 (-3). Have the inequality “mouth” face toward the larger value: - 3.

Inequalities are also used in algebraic problems. For example, if we have an expression such as , this means our variable can be any number smaller than 4 such as -2, 0, 3, 3.9 or even 3.999999999, as long as it is smaller than 4.

term to know

Inequality

A mathematical statement that two quantities are not equal in value.

3. Set Builder Notation

When looking at number lines with a range of highlighted values, we can write the range of values a couple of different ways: in a format called set builder notation, and in interval notation. We will use inequalities to help us write the intervals in set builder notation. Let’s look at the symbols so far and add in the inequality symbols also.

Endpoint Symbols for Notating Intervals
Endpoint IS included in interval	Endpoint IS NOT included in interval
[   or   ]	(   or   )
or	or

If we add inequalities to the chart of endpoint symbols, we see that the less than (<) and the greater than (>) symbols indicate that the endpoint is not included in the interval. We see that the less than or equal to (≤) or greater than or equal to (≥) symbols do include the endpoints in the interval. If we have an expression such as greater than or equal to -2, this means our variable can be any number greater than or equal to -2, such as 5, 0, -1, -1.9999, or even -2.

In set builder notation, we can use inequalities to help us write the interval in set builder notation. In set builder notation, we define the range of values as a set of numbers, and we use curly brackets to define the set, with a description of what is to be included in the set.

3a. One Interval with Two Endpoints

EXAMPLE

Write the highlighted range in set builder notation.



We see that the highlighted values range has endpoints at -2 to +3. There is an open circle at -2 which corresponds to the greater than symbol, because -2 is not included in the interval. Also, there is 3, which corresponds to a less than or equal to sign, because the interval goes up to 3 and can equal 3 also. For now, we will write this range of values as two intervals each corresponding to an endpoint. We first have that x is between -2 and 3. We can say that and .

In set builder notation, we write . This reads as “all x-values, such that x is greater than (but not including) -2 but x is also less than and can include 3.”

So, that means that x can be any number in the highlighted range that is between the values of -2 and 3, but x cannot equal -2 and can equal any value up to and including 3. In a later section, we will learn to write this interval as one inequality called a compound inequality.

try it

Write the highlighted range in set builder notation.

We first determine our endpoints. There are endpoints at -3 and 1. There are open circles at both endpoints meaning the endpoints are not included in the intervals. The open circles correspond to strictly less than (<) symbols. We use curly brackets and write the interval as .

This is read as, "all x-values, such that x is greater than (but not including) -3, and less than (but not including) 1." So, that means that x can be any number in the highlighted range that is between the values of -3 and 1, but x cannot equal -3 or 1.

hint

In a future lesson, you will learn another way to express this type of inequality that is joined by an “and”.

3b. One Interval with One Endpoint

EXAMPLE

Consider the interval highlighted on the number line below.



How would you write the highlighted range in set builder notation?

We first determine our endpoints. There is only one endpoint at 3 with a closed circle. The line extends to positive infinity. Let’s think about the x values that would be included in this interval. In this interval, this would be any number greater than 3 or equal to 3. We would use the greater than or equal to symbol. So, the highlighted range is: .

This is read as, "all x-values, such that x is greater than or equal to 3.” So, that means that can be any number in the highlighted range that is greater than or equal to 3.

3c. More Than One Interval

Let’s consider the same example from before where there were two intervals on the number line.

EXAMPLE

Use the number line below and write the interval in set builder notation.



Again, we have two inequality statements to include in our set. First, we have . We also have .

To write this in set builder notation, we write . We use the connecting word "or" because values that fit within either inequality statement will fit the number line solution.

big idea

Set builder notation uses curly braces to define the number line solution as a set of values. Open circles correspond to the strict inequalities < and >, while closed circles correspond to non-strict inequalities ≥ and ≤. In set builder notation, the vertical bar is read, "such that...".

4. Writing a Solution Set from a Number Line

Let's practice writing in set builder notation and interval notation, using number line solutions.

EXAMPLE

Use the number line below and write the interval in set builder notation and interval notation.

Notation	Explanation	Solution
Set Builder Notation	We notice that all values from 2 to positive infinity are highlighted. In set builder notation, we would write this as . Note that our inequality symbol does not include the exact value of 2.
Interval Notation	In interval notation, this is written as . Note that we use a parenthesis to the left of 2, and also a parenthesis to the right of infinity symbol.

try it

Consider the interval highlighted on the number line below.

How would you write the interval in set builder notation?

This number line has two ranges highlighted. First, we have . We also have . Note we are using less than or equal to and greater than or equal to symbols in our ranges. To write this in set builder notation, we write .

How would you write the interval in interval notation?

In interval notation, we define two intervals: the first interval is and the second interval is . Note the square brackets at -5 and 8, because these correspond to closed circles. To complete our solution in interval notation, we have .

hint

Note again, if there is more than one interval, we use “or” between the intervals in set builder notation and the union symbol (⋃) between the intervals in interval notation.