Describe Sets as Intersections or Unions
Learning Outcomes
- Use interval notation to describe intersections and unions
- Use graphs to describe intersections and unions
Use Interval Notation to Describe Sets of Numbers as Intersections and Unions
When two inequalities are joined by the word and, the solution of the compound inequality occurs when both inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word or, the solution of the compound inequality occurs when either of the inequalities is true. The solution is the combination, or union, of the two individual solutions. In this section we will learn how to solve compound inequalities that are joined with the words AND and OR. First, it will help to see some examples of inequalities, intervals, and graphs of compound inequalities. This will help you describe the solutions to compound inequalities properly. Venn diagrams use the concept of intersections and unions to compare two or more things. For example, this Venn diagram shows the intersection of people who are breaking your heart and those who are shaking your confidence daily. Apparently Cecilia has both of these qualities; therefore, she is the intersection of the two.![Two circles. One is people who are breaking my heart. The other is people who are shaking my confidence daily. The area where the circles overlap is labeled Cecilia.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01182824/Screen-Shot-2016-05-06-at-3.25.21-PM-300x234.png)
![x> 2 and x< 6](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01182826/Screen-Shot-2016-05-10-at-4.43.10-PM-300x46.png)
![Open circle on 2 and open circle on 6 with a line through all numbers between 2 and 6.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01182827/Screen-Shot-2016-05-11-at-4.53.25-PM-300x46.png)
![Two circles, one the Internet and the other Privacy.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01182829/Screen-Shot-2016-05-06-at-3.26.52-PM-300x150.png)
![Open circle on 2 and line through all numbers less than 2. Open circle on 6 and line through all numbers grater than 6.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/06/01182831/Screen-Shot-2016-05-10-at-4.53.44-PM-300x39.png)
It is common convention to construct intervals starting with the value that is furthest left on the number line as the left value, such as , where is less than . The number on the right should be greater than the number on the left.
Example
Draw the graph of the compound inequality or and describe the set of x-values that will satisfy it with an interval.Answer:
The graph of has an open circle on and a blue arrow drawn to the right to contain all the numbers greater than .
The graph of has a closed circle at 4 and a red arrow to the left to contain all the numbers less than .
What do you notice about the graph that combines these two inequalities?
Since this compound inequality is an or statement, it includes all of the numbers in each of the solutions. In this case, the solution is all the numbers on the number line. (The region of the line greater than and less than or equal to is shown in purple because it lies on both of the original graphs.) The solution to the compound inequality or is the set of all real numbers and can be described in interval notation as
Examples
Draw a graph of the compound inequality: and , and describe the set of x-values that will satisfy it with an interval.Answer:
The graph of each individual inequality is shown in color.
Since the word and joins the two inequalities, the solution is the overlap of the two solutions. This is where both of these statements are true at the same time.
The solution to this compound inequality is shown below.
Notice that this is a bounded inequality. You can rewrite as since the solution is between and , including . You read as “x is greater than or equal to and less than .” You can rewrite an and statement this way only if the answer is between two numbers. The set of solutions to this inequality can be written in interval notation like this:
Examples
Draw the graph of the compound inequality and , and describe the set of x-values that will satisfy it with an interval.Answer:
First, draw a graph. We are looking for values for x that will satisfy both inequalities since they are joined with the word and.
In this case, there are no shared x-values, and therefore there is no intersection for these two inequalities. We can write "no solution," or DNE.