Math 95 Section 9.1 Notes Compound Inequalities Objectives: Students will be able to  Perform set operations  Solve compound inequalities with the connectors “AND” and “OR” Definition: Union: This will represent a set that contains the elements that are in one set or the other or both Symbol:  Intersection: This will represent a set that contains the elements that are in both sets Symbol:  Example A  1, 2,3, 4,5
B  2, 4, 6
Find A  B and A  B
A  B= 1, 2,3, 4,5, 6
This set is made up of all elements in both sets, but you only count each element once.
A  B= 2, 4
This set is made up of the elements that are shared by both sets.
Compound Inequalities: These are inequalities that are connected by the words “AND” or “OR”. In the compound inequalities,  If it is connected by “AND”, you are looking at the intersection.  If it is connected by “OR”, you are looking at the union. Example Solve the compound inequality. Graph the solution set on a number line and write the solution set in interval notation. a  6  2
and
a  6  6  2  6
and
a  8
and
5a  30
5a 30

5
5
a6
So both inequalities have been solve. Now represent them on a number line. ) 6 ( ‐8 Since the connecting word is “AND” in the compound inequality, you want to look at the overlapping portions of the number line. You should notice that it overlaps between ‐8 and 6. So this is what the solution should look like: ) ( 6 ‐8 You should notice that ‐8 and 6 are not included based on the inequality signs in the problem. Interval Notation:  8, 6  Example: Solve the compound inequality. Graph the solution set on a number line and write the solution set in interval notation. 8  3 y  2 and 3  y  7   16  4 y Solve each piece  8  3 y  3 y  3 y  2
3 y  8  2
3 y  8  8  2  8
3y  6
3y 6

3 3
y2
3 y  21  16  4 y
3y  5  4 y
3y  4 y  5  4 y  4 y
y 5  0
y 55  05
y  5
y 5

1 1
y  5
So look at both solutions on a number line. [ ) 2 ‐5 You should notice that since our connector word is “AND”, you should be looking for the overlapping, but there is not any. So you would say that there is no solution. You will represent it this way. Interval Notation: { } Example Solve the compound inequality. Graph the solution set on a number line and write the solution set in interval notation. x  0 or 3x  1  7 So you will solve each piece. On the left side, you don’t have anything to solve. So you have only the right side. 3x  1  7
3x  1  1  7  1
x  0 3x  6
3x 6

3 3
x2
So let’s look at the solution on the number line. ) [ 0 2 In this problem, since our connector word is “OR”, we will be looking for the solution to come from one set or the other. This is represented on the number line. So write the solution set, you would have this: Interval Notation:  , 0    2,   You should notice that the arrow keeps going to the left. So that means that you are going toward negative infinity. In the other direction, you should notice that the arrow keeps going to the right. This means that it is going toward positive infinity. Example Solve the compound inequality. Graph the solution set on a number line and write the solution set in interval notation. p  7  10 or 3  p  1  12 So you will solve each piece. p  7  7  10  7
p3
3 p  3  12
3 p  3  3  12  3
3 p  15
3 p 15

3
3
p5
So let’s look at the solution on the number line: ] ] 3 5 You notice in this problem that both arrows are going the same way and we are working the connector word “OR”. So you know that the solution has to be in one or the other or both. So the solution set would be represented like this: ] 5 Since you want everything to the left of three and you want everything between 3 and 5 that is what is represented on the number line. You will notice that 5 is included because it is included in the inequality. Interval Notation:  ,5