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

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

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}

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.

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

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.

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

Example 3 ) ) [ 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

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

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

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

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= ∅

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

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

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

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}

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.

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 )

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.

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

Example 8 ) ) [ 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

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.

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

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

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

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