Intersections, Unions, and Compound Inequalities

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 Sentences A B A 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} Solution: The numbers 1, 2, 3, are common to both set, 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. -7 -7 -7 -2 -2 -2 -1 -1 -1 1 1 1 3 3 3 5 5 5 7 7 7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 0 0 0 4 4 4 6 6 6 8 8 8 2 2 2 Example 2 Graph and write interval notation for the conjunction ) ) ) )

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. 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. -7 -7 -7 -2 -2 -2 -1 -1 -1 1 1 1 3 3 3 5 5 5 7 7 7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 0 0 0 4 4 4 6 6 6 8 8 8 2 2 2 Example 3 Graph and write interval notation for the conjunction [ ) ) [

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 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. 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. -7 -7 -7 -2 -2 -2 -1 -1 -1 1 1 1 3 3 3 5 5 5 7 7 7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 0 0 0 4 4 4 6 6 6 8 8 8 2 2 2 Example 4 Graph and write interval notation for the conjunction [ [ [

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

13. 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. -7 -7 -7 -2 -2 -2 -1 -1 -1 1 1 1 3 3 3 5 5 5 7 7 7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 0 0 0 4 4 4 6 6 6 8 8 8 2 2 2 Example 5 Graph and write interval notation for the disjunction ) ) ) )

15. Unions of sets and disjunctions of sentences A B A The unionof 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 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. disjunctionsof sentences When two or more sentences are joined by the word orto make a compound sentence, the new sentence is called a disjunctionof 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. -7 -7 -7 -2 -2 -2 -1 -1 -1 1 1 1 3 3 3 5 5 5 7 7 7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 0 0 0 4 4 4 6 6 6 8 8 8 2 2 2 Example 7 Graph and write interval notation for the conjunction ) ) ) )

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. Example 8 Graph and write interval notation for the conjunction 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. -7 -7 -7 -2 -2 -2 -1 -1 -1 1 1 1 3 3 3 5 5 5 7 7 7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 0 0 0 4 4 4 6 6 6 8 8 8 2 2 2 Example 8 Graph and write interval notation for the conjunction ) [ ) [

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. Example 9 Graph and write interval notation for the conjunction 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. -7 -7 -7 -2 -2 -2 -1 -1 -1 1 1 1 3 3 3 5 5 5 7 7 7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 0 0 0 4 4 4 6 6 6 8 8 8 2 2 2 Example 9 Graph and write interval notation for the conjunction ) ) ) )

25. Example 10 Graph and write interval notation for the conjunction 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 4 Divide both sides by 3

26. -7 -7 -7 -2 -2 -2 -1 -1 -1 1 1 1 3 3 3 5 5 5 7 7 7 -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 0 0 0 4 4 4 6 6 6 8 8 8 2 2 2 Example 10 Graph and write interval notation for the conjunction ) [