Find the intersection of 2 sets

1
Section 4.2

• Find the intersection of 2 sets
• Solve compound inequalities involving and
• Find the union of 2 sets
• Solve compound inequalities involving or

2
Intersection of Sets

• The intersection of sets A and B, written A n
  B, is the set of elements common to both set A
  and set B.
• A n B x x is in A and x is in B

3

• Find the intersection
• 1,3,4,7,8,9,10 n 2,4,5,6,9,10

4

• Find the intersection
• 1,3,4,7,8,9,10 n 2,4,5,6,9,10
• The elements that occur in both sets are

5

• Find the intersection
• 1,3,4,7,8,9,10 n 2,4,5,6,9,10
• The elements that occur in both sets are
• 4,9,10

6

• Find the intersection
• 1,3,4,7,8,9,10 n 2,4,5,6,9,10
• The elements that occur in both sets are
• 4,9,10
• 1,3,4,7,8,9,10 n 2,4,5,6,9,10 4,9,10

7

• Find the intersection
• 1,3,4,7,8,9,10 n 2,4,5,6,9,10
• The elements that occur in both sets are
• 4,9,10
• 1,3,4,7,8,9,10 n 2,4,5,6,9,10 4,9,10
• The intersection of the two sets is 4,9,10

8

• Find the intersection
• 1,3,5,7,9,11,13 n 0,2,4,6,8,10,12

9

• Find the intersection
• 1,3,5,7,9,11,13 n 0,2,4,6,8,10,12
• The elements that occur in both sets are

10

• Find the intersection
• 1,3,5,7,9,11,13 n 0,2,4,6,8,10,12
• The elements that occur in both sets are
• none

11

• Find the intersection
• 1,3,5,7,9,11,13 n 0,2,4,6,8,10,12
• The elements that occur in both sets are
• none
• 1,3,5,7,9,11,13 n 0,2,4,6,8,10,12 Ø

12

• Find the intersection
• 1,3,5,7,9,11,13 n 0,2,4,6,8,10,12
• The elements that occur in both sets are
• none
• 1,3,5,7,9,11,13 n 0,2,4,6,8,10,12 Ø
• The intersection of the two sets is Ø.

13
Solving Compound Inequalities Involving AND

• Solve each inequality separately.
• Graph the solution set for each inequality on a
  number line and take the intersection of these
  solution sets. The intersection is where the 2
  solutions overlap.

14
Solve x 3 lt 5 and 2x 1 lt 5
15
Solve x 3 lt 5 and 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 and 2x 1 lt 5

16
Solve x 3 lt 5 and 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 and 2x 1 lt 5
• x lt 8

17
Solve x 3 lt 5 and 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 and 2x 1 lt 5
• x lt 8
  2x lt 4
• 
  

18
Solve x 3 lt 5 and 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 and 2x 1 lt 5
• x lt 8
  2x lt 4
• 
  x lt 2

19
Solve x 3 lt 5 and 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 and 2x 1 lt 5
• x lt 8
  2x lt 4
• 
  x lt 2
• Now graph both inequalities and find the
  intersection.

20
Solve 2x 1 gt 9 and 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 and 3x 1 lt 7

21
Solve 2x 1 gt 9 and 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 and 3x 1 lt 7
• 2x gt 10

22
Solve 2x 1 gt 9 and 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 and 3x 1 lt 7
• 2x gt 10
• x gt 5

23
Solve 2x 1 gt 9 and 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 and 3x 1 lt 7
• 2x gt 10
  3x lt 6
• x gt 5

24
Solve 2x 1 gt 9 and 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 and 3x 1 lt 7
• 2x gt 10
  3x lt 6
• x gt 5
  x lt 2
• 
  

25
Solve 2x 1 gt 9 and 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 and 3x 1 lt 7
• 2x gt 10
  3x lt 6
• x gt 5
  x lt 2
• 
  
• Now graph both inequalities and find the
  intersection.

26

• If a lt b and a lt x and x lt b
• then a lt x lt b

27

• If a lt b and a lt x and x lt b
• then a lt x lt b
• So if 3 lt 2x 1 and 2x 1 lt 3
• then 3 lt 2x 1 lt 3

28

• If a lt b and a lt x and x lt b
• then a lt x lt b
• So if 3 lt 2x 1 and 2x 1 lt 3
• then 3 lt 2x 1 lt 3
• This shortened (mathematical) way of writing the
  statement leaves off the and, but has the same
  meaning.

29

• If a lt b and a lt x and x lt b
• then a lt x lt b
• So if 3 lt 2x 1 and 2x 1 lt 3
• then 3 lt 2x 1 lt 3
• This shortened (mathematical) way of writing the
  statement leaves off the and, but has the same
  meaning.
• Now solve by performing the same operations on
  all parts of the inequality and solve for x in
  the middle.

30
Solve and graph 3 lt 2x 1 3

• 3 lt 2x 1 3 original equation

31
Solve and graph 3 lt 2x 1 3

• 3 lt 2x 1 3 original equation
• 4 lt 2x 2 subtract 1 from all
  parts

32
Solve and graph 3 lt 2x 1 3

• 3 lt 2x 1 3 original equation
• 4 lt 2x 2 subtract 1 from all
  parts
• 2 lt x 1 divide all parts by 2

33
Solve and graph 3 lt 2x 1 3

• 3 lt 2x 1 3 original equation
• 4 lt 2x 2 subtract 1 from all
  parts
• 2 lt x 1 divide all parts by 2
• Our solution is x 2lt x 1.

34
Solve and graph 3 lt 2x 1 3

• 3 lt 2x 1 3 original equation
• 4 lt 2x 2 subtract 1 from all
  parts
• 2 lt x 1 divide all parts by 2
• Our solution is x 2lt x 1.
• Now graph the solution.

35
Union of Sets

• The union of sets A and B, written
  A U B, is the set of elements that are in either
  set A or set B (or both).
• A U B x x is in A or x is in B

36

• Find the union
• 1,3,4,7 U 1,4,5,6

37

• Find the union
• 1,3,4,7 U 1,4,5,6
• The elements that occur either set are

38

• Find the union
• 1,3,4,7 U 1,4,5,6
• The elements that occur either set are
• 1,3,4,5,6,7

39

• Find the union
• 1,3,4,7 U 1,4,5,6
• The elements that occur either set are
• 1,3,4,5,6,7
• 1,3,4,7 U 1,4,5,6 1,3,4,5,6,7

40

• Find the union
• 1,3,4,7 U 1,4,5,6
• The elements that occur either set are
• 1,3,4,5,6,7
• 1,3,4,7 U 1,4,5,6 1,3,4,5,6,7
• The union of the two sets is 1,3,4,5,6,7

41
Solving Compound Inequalities Involving OR

• Solve each inequality separately.
• Graph the solution set for each inequality on a
  number line and take the union of these solution
  sets.

42
Solve x 3 lt 5 or 2x 1 lt 5
43
Solve x 3 lt 5 or 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 or 2x 1 lt
  5

44
Solve x 3 lt 5 or 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 or 2x 1 lt
  5
• x lt 8

45
Solve x 3 lt 5 or 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 or 2x 1 lt
  5
• x lt 8
  2x lt 4
• 
  

46
Solve x 3 lt 5 or 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 or 2x 1 lt
  5
• x lt 8
  2x lt 4
• 
  x lt 2

47
Solve x 3 lt 5 or 2x 1 lt 5

• Solve each inequality separately
• x 3 lt 5 or 2x 1 lt
  5
• x lt 8
  2x lt 4
• 
  x lt 2
• Now graph both inequalities and find the union.

48
Solve 2x 1 gt 9 or 3x 1 lt 7
49
Solve 2x 1 gt 9 or 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 or 3x 1 lt
  7

50
Solve 2x 1 gt 9 or 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 or 3x 1 lt
  7
• 2x gt 10

51
Solve 2x 1 gt 9 or 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 or 3x 1 lt
  7
• 2x gt 10
• x gt 5

52
Solve 2x 1 gt 9 or 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 or 3x 1 lt
  7
• 2x gt 10
  3x lt 6
• x gt 5

53
Solve 2x 1 gt 9 or 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 or 3x 1 lt
  7
• 2x gt 10
  3x lt 6
• x gt 5
  x lt 2
• 
  

54
Solve 2x 1 gt 9 or 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 or 3x 1 lt
  7
• 2x gt 10
  3x lt 6
• x gt 5
  x lt 2
• 
  
• Now graph both inequalities and find the union.

55
Solve 2x 1 gt 9 or 3x 1 lt 7

• Solve each inequality separately
• 2x 1 gt 9 or 3x 1 lt
  7
• 2x gt 10
  3x lt 6
• x gt 5
  x lt 2
• 
  
• Now graph both inequalities and find the union.