1. Intersections, Unions, and Compound Inequalities
Two inequalities joined by the word “and” or the word “or” are called compound inequalities.

2. Intersections of Sets and Conjunctions of SentencesB
A ∩𝑩
The intersection of two set A and B is the set of all elements that are common in both A and B.

3. Example 1
Find the intersection. {1, 2, 3, 4, 5} ∩ −2, −1, 0, 1, 2, 3
Solution: The numbers 1, 2, 3, are common to both sets, so the intersection is {1, 2, 3}

4. Conjunction of the intersection
When two or more sentences are joined by the word and to make a compound sentence, the new sentence is called a conjunction of the intersection. The following is a conjunction of inequalities.
A number is a solution of a conjunction if it is a solution of both of the separate parts.
The solution set of a conjunction is the intersection of the solution sets of the individual sentences.

5. Example 2
Graph and write interval notation for the conjunction ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8

6. Mathematical Use of the Word “and”
The word “and” corresponds to “intersection” and to the symbol “∩“. Any solution of a conjunction must make each part of the conjunction true.

7. Graph and write interval notation for the conjunction
Example 3
Graph and write interval notation for the conjunction
SOLUTION: This inequality is an abbreviation for the conjunction true
Subtracting 5 from both sides of each inequality
Dividing both sides of each inequality by 2

8. Example 3 ) ) [ Graph and write interval notation for the conjunction2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 [ ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8

9. The steps in example 3 are often combined as follows
Subtracting 5 from all three regions
Dividing by 2 in all three regions
Caution: The abbreviated form of a conjunction, like -3 ≤x <4
can be written only if both inequality symbols point in the same direction. It is not acceptable to write a sentence like -1 > x < 5 since doing so does not indicate if both -1 > x and x < 5 must be true or if it is enough for one of the separate inequalities to be true

10. Graph and write interval notation for the conjunction
Example 4
Graph and write interval notation for the conjunction
SOLUTION: We first solve each inequality retaining the word and
Add 5 to both sides
Subtract 2 from both sides
Divide both sides by 2
Divide both sides by 5

11. Example 4 [ Graph and write interval notation for the conjunction [ [2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 [ 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 [ 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8

12. Sometimes there is no way to solve both parts of a conjunction at once
Example 5
Sometimes there is no way to solve both parts of a conjunction at once
When A ∩𝑩=∅
A and B are said to be disjoint. A B
A ∩𝑩= ∅

13. Graph and write interval notation for the conjunction
Example 5
Graph and write interval notation for the conjunction
SOLUTION: We first solve each inequality separately
Add 3 to both sides of this inequality
Add 1 to both sides of this inequality
Divide by 2
Divide by 3

14. Example 5
Graph and write interval notation for the disjunction 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8

15. Unions of sets and disjunctions of sentencesB
A ∪𝑩
The union of two set A and B is the collection of elements that belong to A and / or B.

16. Example 6
Find the union. {2, 3, 4}∪ 3, 5, 7
Solution: The numbers in either or both sets are 2, 3, 4, 5, and 7, so the union is {2, 3, 4, 5, 7}

17. disjunctions of sentences
When two or more sentences are joined by the word or to make a compound sentence, the new sentence is called a disjunction of the sentences. Here is an example.
A number is a solution of a disjunction if it is a solution of at least one of the separate parts.
The solution set of a disjunction is the union of the solution sets of the individual sentences.

18. Example 7
Graph and write interval notation for the conjunction ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 )

19. Mathematical Use of the Word “or”
The word “or” corresponds to “union” and to the symbol “∪“. For a number to be a solution of a disjunction, it must be in at least one of the solution sets of the individual sentences.

20. Graph and write interval notation for the disjunction
Example 8
Graph and write interval notation for the disjunction
SOLUTION: We first solve each inequality separately
Subtract 7 from both sides of inequality
Subtract 13 from both sides of inequality
Divide both sides by 2
Divide both sides by -5

21. Example 8 ) ) [ Graph and write interval notation for the disjunction2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 [ 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) [ 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8

22. Caution: A compound inequality like:
As in Example 8, cannot be expressed as because to do so would be to day that x is simultaneously less than -4 and greater than or equal to 2. No number is both less than -4 and greater than 2, but many are less than -4 or greater than 2.

23. Graph and write interval notation for the disjunction
Example 9
Graph and write interval notation for the disjunction
SOLUTION: We first solve each inequality separately
Add 5 to both sides of this inequality
Add 3 to both sides of this inequality
Divide both sides by -2

24. Example 9
Graph and write interval notation for the conjunction ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 ) ) 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8

25. Graph and write interval notation for the disjunction
Example 10
Graph and write interval notation for the disjunction
SOLUTION: We first solve each inequality separately
Add 11 to both sides of this inequality
Subtract 9 from both sides of this inequality
Divide both sides by 3
Divide both sides by 4

26. Example 10 ) Graph and write interval notation for the disjunction [ 2-7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 [ 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8