1. Chapter 2.5 – Compound Inequalities

2. Objectives I will find the intersection of two sets.
I will solve compound inequalities containing and.
I will find the union of two sets.
I will solve compound inequalities containing or.

3. What’s the difference?
You get a discount if you are at least 18 years old and no more than 60 years old.
18 < x < 60
You get a discount if you are less than 18 years old or at least 60 years old.
18 > x > 60

4. Compound Inequalities
Two inequalities joined by the words and or or are called compound inequalities.
Compound Inequalities
x + 3 < 8 and x > 2

5. Intersection of Two Sets
The intersection of two sets. A and B, is the set of all elements common to both sets. A intersect B is denoted by, B A

6. The numbers 4 and 6 are in both sets.
Example 1
If A = { x / x is an even number greater than 0 and less than 10} and B = {3, 4, 5, 6} find A ∩ B.
List the elements in sets A and B
A = {2, 4, 6, 8}
B = {3, 4, 5, 6}
The numbers 4 and 6 are in both sets.
The intersection is {4, 6}

7. You try it!  Find the intersection: {1, 2, 3, 4, 5} ∩ {3,4,5,6}
{3,4,5}

8. Solutions to compound inequalities
A value is a solution of a compound inequality formed by the word and if it is a solution of both inequalities.
For example, the solution set of the compound inequality x ≤ 5 and x ≥ 3 contains all values of x that make both inequalities true.

9. Compound Inequalities
A compound inequality such as x ≥ 3 and x ≤ 5 can be written more compactly.
3 ≤ x ≤ 5

10. Graphing compound inequalities
{ x / x ≤ 5} ] 1 5 3 4 2 6
{x / x ≥ 3} [ 1 5 3 4 2 6
{ x / 3 ≤ x ≤ 5 1 5 3 4 2 6 [ ]

11. Example 2 Solve x – 7 < 2 and 2x + 1 < 9
Step 1: Solve each inequality separately
x – 7 < 2 and 2 x + 1 < 9
x < 9 and x < 8
x < 9 and x < 4

12. Example 2
Step 2: Graph the two intervals on two number lines to find their intersection.
x < 9
x < 4 4 8 6 7 5 9 ) 4 8 6 7 5 9 )

13. Example 2 The solution set is ( - ∞, 4)
Step 3: Graph the compound inequality
{ x / x < 9 and x < 4} = { x / x < 4} 3 7 5 6 4 8 )
The solution set is ( - ∞, 4)

14. Solve: x + 5 < 9 and 3x -1 < 2
Give it a try! 
Solve: x + 5 < 9 and 3x -1 < 2

15. Example 3 Solve 2 x ≥ 0 and 4 x – 1 ≤ -9
First Step: Solve each inequality separately.

16. Example 3
Step 2: Graph the intervals to find their intersection.

17. Example 3
Step 3: Graph the intersection

18. You try it! 
Solve: 4x ≥ 0 and 2x + 4 ≤ 2

19. Example 4 Solve 2 < 4 – x < 7
Step 1: Isolate the x in the middle
2 < 4 – x < 7
2 – 4 < 4 – x – 4 < 7 – 4
-2 < -x < 3
-2 < - x < 3
2 > x > 3
Subtract 4 from
all 3 parts
Divide all three parts
by -1
Must reverse symbol
because dividing by a
negative.

20. 2 > x > -3 is equivalent to -3 < x < 2
Example 4
2 > x > -3 is equivalent to -3 < x < 2
Interval Notation (-3, 2)
Graph the inequality -3 1 -1 -2 2 ( )

21. Give it a try! 
Solve: 5 < 1 – x < 9

22. Example 5
Solve

23. Example 5
Graph the solution
Interval Notation

24. Give it a try! 

25. Union of two sets
The solution set of a compound inequality formed by the word or is the union of the solution set of two inequalities.
The union of two sets, A and B, is the set of elements that belong to either of the sets. A union B is denoted by
A U B

26. Example 6
If A = {x / x is an even number greater than 0 and less than 10} and B = {3, 4, 5, 6} Find A U B.
List the elements in Set A and Set B
A: {2, 4, 6, 8}
B: {3, 4, 5, 6}
Union : {2,3, 4, 5, 6, 8}

27. Union
A value is a solution of a compound inequality formed by the word or if it is a solution of either inequality.
Graph of {x / x ≤ 1}
Graph of { x / x ≥ 3}

28. Unions
Graph of { x/ x ≤ 1 or x ≥ 3}

29. Give it a try!  Find the union: {1, 2, 3, 4, 5} U {3, 4, 5, 6}
{1,2,3,4,5,6}

30. Give it a try!  Solve: 5 x – 3 ≤ 10 or x + 1 ≥ 5
First solve each inequality separately.

31. Example 7
Now we can graph each interval and find their union.

32. Example 7:
Solution set:

33. Give it a try! 
Solve: 3x – 2 ≥ 10 or x – 6 ≤ -4

34. Example 8 Solve: – 2x – 5 < -3 or 6x < 0
Step 1: Solve each inequality separately

35. Example 8
Graph each interval and find their union

36. Give it a try! 
Solve: x – 7 ≤ -1 or 2x – 6 ≥ 2