Chapter 2 Section 5 Copyright © 2011 Pearson Education, Inc.

Set Operations and Compound Inequalities Find the intersection of two sets. Solve compound inequalities with the word and. Find the union of two sets. Solve compound inequalities with the word or

Slide Copyright © 2011 Pearson Education, Inc. Objective 1 Find the intersection of two sets.

Slide Copyright © 2011 Pearson Education, Inc. Intersection of Sets For any two sets A and B, the intersection of A and B, symbolized A ∩ B, is defined as follows: A B

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 1 Let A = {3, 4, 5, 6} and B = {5, 6, 7}. Find A ∩ B. The set A ∩ B, the intersection of A and B, contains those elements that belong to both A and B; that is, the number s 5 and 6. A ∩ B = {3, 4, 5, 6} ∩ {5, 6, 7} Therefore, A ∩ B = {5,6}.

Slide Copyright © 2011 Pearson Education, Inc. A compound inequality consists of two inequalities linked by a consecutive word such as and or or. Examples of compound inequalities are

Slide Copyright © 2011 Pearson Education, Inc. Objective 2 Solve compound inequalities with the word and.

Slide Copyright © 2011 Pearson Education, Inc. Solving a Compound Inequality with and Step 1Solve each inequality individually. Step 2Since the inequalities are joined with and, the solution set of the compound inequality will include all numbers that satisfy both inequalities in Step 1 (the intersection of the solution sets).

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 2 Solve the compound inequality and graph the solution set. x + 3 –12 Step 1 Solve each inequality individually. x + 3 –12 x + 3 – 3 < 1 – 3 x < –2 x – > – x > –8

Slide Copyright © 2011 Pearson Education, Inc. continued Step 2Because the inequalities are joined with the word and, the solution set will include all numbers that satisfy both inequalities. ) x < –2 ( x > –8 The solution set is (–8, –2). ) (

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 3 Solve the compound inequality and graph the solution set. Step 1 Solve each inequality individually. Remember to reverse the inequality symbol.

Slide Copyright © 2011 Pearson Education, Inc. continued Step 2The overlap of the graphs consists of the numbers that are greater than or equal to –4 and are also greater than or equal to –3. [ The solution set is [–3,∞). [ [

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 4 Solve x + 2 > 3 and 2x + 1 < –3. Solve each inequality individually. x + 2 > 3 and 2x + 1 < –3 x > 1 2x < –4 x < –2 ( ) x > 1

Slide Copyright © 2011 Pearson Education, Inc. continued There is no number that is both greater than 1 and less than –2, so the given compound inequality has no solution. The solution set is

Slide Copyright © 2011 Pearson Education, Inc. Objective 3 Find the union of two sets.

Slide Copyright © 2011 Pearson Education, Inc. Union of Sets For any two sets A and B, the union of A and B, symbolized A B, is defined as follows: Slide AB

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 5 Let A = {3, 4, 5, 6} and B = {5, 6, 7}. Find A B. The set A B, the intersection of A and B, consists of all elements in either A or B (or both). Start by listing the elements of set A: 3,4,5,6. Then list any additional elements from set B. In this case, the elements 5 and 6 are already listed, so the only additional element is 7. Therefore, A B = {3, 4, 5, 6, 7}.

Slide Copyright © 2011 Pearson Education, Inc. Objective 4 Solve compound inequalities with the word or.

Slide Copyright © 2011 Pearson Education, Inc. Solving a Compound Inequality with or Step 1Solve each inequality individually. Step 2Since the inequalities are joined with or, the solution set of the compound inequality includes all numbers that satisfy either one of the two inequalities in Step 1 (the union of the solution sets).

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 6 Solve Solve each inequality individually. ( ) x < 1 Step 1 x > 3

Slide Copyright © 2011 Pearson Education, Inc. continued The graph of the solution set consists of all numbers greater than 3 or less than1. The solution set is ) ( Step 2

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 7 Solve Solve each inequality individually. ] or ]

Slide Copyright © 2011 Pearson Education, Inc. continued The solution set is all numbers that are either less than or equal to 5 or less than or equal to 2. All real numbers less than or equal to 5 are included. The solution set is ]

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 8 Solve Solve each inequality individually. ] or [

Slide Copyright © 2011 Pearson Education, Inc. continued The solution set is all numbers that are either less than or equal to 5 or greater than or equal to 2. All real numbers are included. The solution set is

Slide Copyright © 2011 Pearson Education, Inc. EXAMPLE 9 The five highest grossing domestic films as of July 2005 are listed in this table. List the elements that satisfy each set. FilmAdmissionsGross Income Gone with the Wind202,044,569$1,293,085,000 Star Wars178,119,595$1,139,965,000 The Sound of Music142,415,376$911,458,000 E.T.141,925,359$908,322,298 The Ten Commandments 131,000,000$838,400,000 Source: Exhibitor Relations Co., Inc.

Slide Copyright © 2011 Pearson Education, Inc. a. The set of films with admissions greater than 130,000,000 and gross income less than $800,000,000. b. The set of films with admissions greater than 130,000,000 or gross income less than $500,000,000. a. All films had admissions greater than 130,000,000, but no films had a gross income of less than $800,000,000. Thus, there are no elements (films) that satisfy both conditions, so the required set is the empty set, symbolized by b. Since all the films had admissions greater than 130,000,000 and we have an or statement, the second condition doesn’t have an effect on the solution set. The required set is the set of all films that is {Gone with the Wind, Stars Wars, The Sound of Music, E.T., The Ten Commandments}. continued Solution