1. Union, Intersection, and Compound Inequalities
MATH 017
Intermediate Algebra
S. Rook

2. Overview Section 2.5 in the textbook Intersection of sets
Solving compound inequalities involving intersection:
Using the word and
Having two inequalities
Union of sets
Solving compound inequalities involving union

3. Intersection of Sets

4. Intersection of Sets
Intersection (∩) [of 2 sets]: the elements common to both sets.
Usually easier to start with the set containing the least number of elements

5. Intersection of Sets (Example)
Ex 1: Given A = {x | x is a whole number},
B = {x | -2 ≤ x < 5}, find A ∩ B

6. Intersection of Sets (Example)
Ex 2: Given A = {x | x is a natural number}, B = {-3, -2, -1, 0, 1, 2, 3}, find A ∩ B

7. Intersection of Sets (Examples)
Ex 3: Given A = {x | x ≤ -7}, B = {x | x < -2}, find A ∩ B

8. Solving Compound Inequalities Using Intersection

9. Solving Compound Inequalities Using Intersection
Can be found in two formats:
Two linear inequalities separated by the word and
A statement containing two inequality symbols
-4 < x < 7

10. Compound Inequalities Separated by and
Solve each linear inequality as normal
Graphing is somewhat trickier:
Draw 3 number lines with equal intervals
On the first number line, graph the solution to the first inequality
On the second number line, graph the solution to the second inequality
On the third number line, lay the first two number lines on top of each other
The intersection is the area between the left ( or [ and the right ) or ]
Obtain the interval notation from the intersection

11. Compound Inequalities Separated by and (Example)
Ex 4: Solve, graph, and put into interval notation: 2x – 3 ≤ 11 and 2x < 3x – 4

12. Compound Inequalities Separated by and (Example)
Ex 5: Solve, graph, and put into interval notation: 2(x – 3) – 3x ≤ 3(x + 1) and
8x – 2(x – 3) > 24

13. Compound Inequalities with Two Inequality Symbols
Most common way to see an intersection compound inequality
Somewhat trickier to solve
Goal is to isolate the variable between the 2 inequalities
Perform Algebraic operations on 3 sides instead of 2
Simple to graph
Once the variable is isolated, the intersection is obtained

14. Compound Inequalities with Two Inequality Symbols (Example)
Ex 6: Solve, graph, and put into interval notation: -9 < 2x – 7 ≤ 7

15. Compound Inequalities with Two Inequality Symbols (Example)
Ex 7: Solve, graph, and put into interval notation:

16. Union of Sets

17. Union of Sets
Union (U) [of 2 sets]: the distinct elements from both sets
In other words, dump the elements of both sets together and remove the duplicates

18. Union of Sets (Example)
Ex 8: Given A = {-2, 0, 1, 3, 4} and
B = {1, 2, 3, 4, 5}, find A U B

19. Union of Sets (Example)
Ex 9: Given A = {x | x ≥ 0} and
B = {x | x ≥ 5}, find A U B

20. Solving Compound Inequalities Using Union

21. Solving Compound Inequalities Using Union
Two inequalities separated by the word or
Solve each linear inequality as normal
Graphing is somewhat trickier:
Draw 3 number lines with equal intervals
On the first number line, graph the solution to the first inequality
On the second number line, graph the solution to the second inequality
On the third number line, lay the first two number lines on top of each other – this represents the union
Remove any parentheses or brackets that have shading to the left and right
Obtain the interval notation from the union

22. Solving Compound Inequalities Using Union (Example)
Ex 10: Solve, graph, and put into interval notation: 6(x – 3) – 5(x – 2) > -4 or
3(1 – x) – 6(2 – x) ≥ 0

23. Solving Compound Inequalities Using Union (Example)
Ex 11: Solve, graph, and put into interval notation:

24. Summary
After studying these slides, you should know how to do the following:
Find the intersection of [2] sets
Solve compound inequalities involving intersection when:
The keyword and is used
The statement contains two inequalities
Find the union of [2] sets
Solve compound inequalities involving union