Section 9.1: Set Operations And Compound Inequalities
§1 Intersection Of Two Sets
Imagine the inequality x  3 . What does this mean or represent? This means any number greater than or equal
to 3 is a part of the solution set. For example, if the original inequality was 2 x  5  1 , then when we solve this
we end up with x  3 . Hence any value of x greater than or equal to three is a solution.
Similarly, if you were asked to solve 3x  1  17 , we can isolate the x term to get that x  6 . Hence any value
of x less than or equal to 6 is a solution to this inequality.
What if you were asked, then to solve 2 x  5  1 AND 3x  1  17 . What does this mean? You need to find the
values of x that are solutions to both inequalities. We also call this the intersection. This type of example is a
compound inequality. Think about this carefully. We actually already found the solution to each of the
inequalities separately. How can we use these to find the answer?
We know that x  3 is the solution to the first inequality and x  6 is the solution to the second inequality.
On the number line, the graph of x  3 represents any value of x that is greater than or equal to 3. Hence we
mark the number line at 3 with a closed circle or a bracket and points to the right. It is graphed on the number
line as follows:
Similarly, the graph of x  6 represents any value of x that is less than or equal to 6. We mark the number line at
6 with a closed circle or bracket and point to the left. It is graphed on the number line as follows:
So then, what is the compound inequality asking for?
It is asking for the values of x that are in the solution for BOTH inequalities. We can see here that any value of x
greater than or equal to 3 AND less than or equal to 6 is in the solution set for both. Hence the answer becomes
3  x  6, or 3, 6 . Graphically, it would be represented by
So we can see that the intersection of two sets are the values of x that are in both sets! The best approach is to
solve each inequality separately and then draw each solution on the number line. The answer then is the values
of x that occur in both sets.
PRACTICE
1) Solve the compound inequality. Express the solution set in interval notation and graph the solution set on the
number line: 2 x  4 x  8 and 4 x  5  3x  2
2) Solve the compound inequality. Express the solution set in interval notation and graph the solution set on the
number line: x  3  2 and 2 x  4  10
3) Solve the compound inequality. Express the solution set in interval notation and graph the solution set on the
number line: x  3  2 and x  4  1
§2 Union Of Two Sets
The union of two sets is represented by the word OR. The union represents the values of x that are in either of
the two inequalities. So the final solution represents the values of x that are in either one of the solution set of
both inequalities.
For example, say you were asked to wear blue jeans and a white shirt. Then that means both conditions have to
be satisfied. You can’t just wear one or the other. You have to wear both. However, how does this change if you
were asked to wear blue jeans or a blue shirt? As long as you wore one or the other, then you would be ok.
So it’s sort of the same thing here, except of course we are doing math and talking about solutions to
inequalities. Remember, the intersection (used with the word AND) represents the values of x that are in both
solutions. The union (used with the word OR) represents any x value in either solution.
Try the following: Solve x  1  2 or 3x  5  2 x  6
First, let’s solve each one separately. For the first inequality, we get x  3 . For the second inequality, we
combine like terms first solve for x. You should end up with x  1. What is the answer?
The answer is any value of x that is greater than 3 OR less than 1. We can graph this on the number line as:
We see that in interval notation, the solution is  ,1   3,   . It may be easier to graph the solution on the
number line first; then the solution in interval notation be may a little more clear.
Note that this means any value between 1 and 3 is not a solution to either inequality. However, any number
greater than 3 is a solution to the first inequality but not the second; similarly, any number less than 1 is a
solution to the second inequality but not the first.
PRACTICE
4) Solve the compound inequality. Express the solution in interval notation and graph the solution set on the
number line: x  4 or x  8
5) Solve the compound inequality. Express the solution in interval notation and graph the solution set on the
number line: 3x  2  13 or x  5  7
6) Solve the compound inequality. Express the solution in interval notation and graph the solution set on the
number line x  1  3 or x  4  2