Section 2.5 Compound Inequalities 101 SECTION 2.5 Compound Inequalities Body mass index (BMI) is a measure of how much body fat a person has. A normal BMI is between 18.5 and 24.9. The formula to calculate BMI is BMI = 703w h2 , where w is weight (in pounds) and h is height (in inches). Find the weight necessary for a person with a height of 5 ft 4 in. to have a normal BMI. To solve this problem, we need to find the solution of the inequality 18.5 ≤ 703w 642 ≤ 24.9 This is an example of a compound inequality, which we will discuss in this section. The Intersection of Sets In Chapter 1, we discussed the concept of sets and their intersection and union. We will explore this concept further as we deal with sets which are intervals of the real numbers. Knowing how to find the intersection and union of sets will enable us to solve special types of inequalities, namely, compound inequalities. We will first investigate the intersection of two sets. To do this, think about the intersection of two streets. The intersection is where the streets meet one another or where they cross. The intersection is the part of the road that the two streets have in common. This idea extends to sets, as discussed in Section 1.1. Recall, the intersection of two sets is the set of elements that the two sets have in common. Definition: The Intersection of Two Sets Let A and B be sets. The intersection of A and B is denoted by A B and is the set of elements that are members of both sets A and B. In the figure, the intersection is represented by the purple piece. Procedure: Finding the Intersection of Two Intervals A and B Step 1: Draw the graph of each interval. Step 2: Find the interval of the real number line that contains members from both sets, that is, where the two sets overlap. ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Determine the intersection of two sets. 2. Determine the union of two sets. 3. Solve compound inequalities involving “and.” 4. Solve compound inequalities involving “or.” 5. Solve applications of compound inequalities. 6. Troubleshoot common errors. Objective 1 Examples Find the intersection of the sets. Draw the graph of the intersection and write each solution set in interval notation and set-builder notation. 1a. A = (-∞, 7 and B = 1, ∞) 1b. A = (3, ∞) and B = (0, ∞) 1c. A = 4, ∞) and B = (-∞, 2 Objective 1 ▶ Determine the intersection of two sets. A A B B
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