Inequalities and Absolute Value 8 Chapter Inequalities and Absolute Value

Set Operations and Compound Inequalities 8.2 Set Operations and Compound Inequalities 1. Recognize set intersection and union. 2. Find the intersection of two sets. 3. Solve compound inequalities with the word and. 4. Find the union of two sets. 5. Solve compound inequalities with the word or.

Recognize set intersection and union. Objective 1 Recognize set intersection and union.

Recognize set intersection and union. Consider the two sets A and B defined as follows. A = {1, 2, 3}, B = {2, 3, 4} The set of all elements that belong to both A and B, called their intersection and symbolized is given by = {2, 3}. Intersection The set of all elements that belong to either A or B or both, called their union and symbolized is given by = {1, 2, 3, 4}. Union

Find the intersection of two sets. Objective 2 Find the intersection of two sets.

Intersection of Sets

Finding the Intersection of Two Sets Classroom Example 1 Finding the Intersection of Two Sets 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 numbers 5 and 6. A ∩ B = {3, 4, 5, 6} ∩ {5, 6, 7} = {5, 6}

Solve compound inequalities with the word and. Objective 3 Solve compound inequalities with the word and.

Compound Inequalities A compound inequality consists of two inequalities linked by a consecutive word such as and or or.

Solving a Compound Inequality with and Step 1 Solve each inequality individually. Step 2 Because 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).

Solving a Compound Inequality with and Classroom Example 2 Solving a Compound Inequality with and Solve the compound inequality, and graph the solution set. x + 3 < 1 and x – 4 > –12 Solve each inequality individually. Because the inequalities are joined with the word and, the solution set will include all numbers that satisfy both inequalities.

Solving a Compound Inequality with and (cont.) Classroom Example 2 Solving a Compound Inequality with and (cont.) Graph each solution set. The solution set is (–8, –2).

Solving a Compound Inequality with and Classroom Example 3 Solving a Compound Inequality with and Solve and graph. The solution set is [–3, ∞).

Solving a Compound Inequality with and Classroom Example 4 Solving a Compound Inequality with and Solve and graph. There is no number that is both greater than 1 and less than –2, so the given compound inequality has no solution.

Find the union of two sets. Objective 4 Find the union of two sets.

Union of Sets

Finding the Union of Two Sets Classroom Example 5 Finding the Union of Two Sets Let A = {3, 4, 5, 6} and B ={5, 6, 7}. Find 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.

Solve compound inequalities with the word or. Objective 5 Solve compound inequalities with the word or.

Solving a Compound Inequality with or Step 1 Solve each inequality individually. Step 2 Because 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).

Solving a Compound Inequality with or Classroom Example 6 Solving a Compound Inequality with or Solve and graph. Solve each inequality individually.

Solving a Compound Inequality with or Classroom Example 7 Solving a Compound Inequality with or Solve and graph. Solve each inequality individually. 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.

Solving a Compound Inequality with or Classroom Example 8 Solving a Compound Inequality with or Solve and graph. Solve each inequality individually. 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.

Applying Intersection and Union Classroom Example 9 Applying Intersection and Union The top five U.S. trading partners for 2012 are listed in the table. Amounts are in millions of dollars. (Source: U.S. Census Bureau.) List the elements of each set.

Applying Intersection and Union (cont.) Classroom Example 9 Applying Intersection and Union (cont.) a. The set of countries to which exports were greater than $150,000 million and from which imports were less than $150,000 million There is no country that satisfies both conditions.

Applying Intersection and Union (cont.) Classroom Example 9 Applying Intersection and Union (cont.) b. The set of countries to which exports were between $50,000 million and $150,000 million or from which imports were greater than $300,000 million China, Japan, and Canada