Chapter 8 Graphs, Relations, and Functions

1. Chapter 8 Graphs, Relations, and Functions Section 6 CompoundInequalities

2. Section 8.6 Objectives 1 Determine the Intersection or Union of Two Sets 2 Solve Compound Inequalities Involving “and” 3 Solve Compound Inequalities Involving “or” 4 Solve Problems Using Compound Inequalities

3. Implies intersection. 6 1 4 7 2 5 8 3 A B = {4, 5} Intersection of Sets The intersection of two sets A and B, denoted A B, is the set of all elements that belong to both set Aand set B. Let A = {1, 2, 3, 4, 5} and let B = {4, 5, 6, 7, 8}. A B

4. Implies union. 8 1 2 4 3 10 5 6 Union of Sets The union of two sets A and B, denoted A B, is the set of all elements that are in the set Aor in the set B or in both A and B. Let A = {1, 2, 3, 4, 5, 6} and let B = {2, 4, 6, 8, 10}. A B AB = {1, 2, 3, 4, 5, 6, 8, 10}

5. [ ) -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 5 5 Empty set Intersection and Union Example: Let A = {x| x 4} and B = {x| x < 1}. a.) Graph sets A and B on a number line. b.) Find A B and A  B. Write the solutions in interval notation. x 4 x< 1 A B =  or { } A  B = (– , 1) or [4, )

6. – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 x≤ –1 ] x≤ 4 ] ] Compound Inequalities Using “and” A compound inequalityis formed by joining two inequalities with the word “and” or “or”. Example: Solve the compound inequality and graph the solution set. x+ 3 ≤ 7 andx – 2 ≤ –3 x+ 3 ≤ 7 x – 2 ≤ –3 x≤ 4 x≤ –1 The solution set is {x| x≤ –1} or (– , –1].

7. – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 ( ) ) ( Compound Inequalities Using “or” Example: Solve the compound inequality and graph the solution set. 6x 10 < 8 or 2x + 1 > 9 6x 10 < 8 2x + 1 > 9 6x < 18 2x > 8 x < 3 x > 4 The solution set is {x| x< 3 or x > 4} or (– , 3) or (4, ).

8. Solving Compound Inequalities Steps for Solving Compound Inequalities Involving “and” Step 1: Solve each inequality separately. Step 2: Find the INTERSECTION of the solution sets of each inequality. Steps for Solving Compound Inequalities Involving “or” Step 1: Solve each inequality separately. Step 2: Find the UNION of the solution sets of each inequality.

9. Get the variable by itself in the middle. The inequality signs are reversed. Solving Compound Inequalities Example: Solve the compound inequality.  3 <  4x + 1 < 17  4 <  4x < 16 Subtract 1 from all three parts. 1 > x >  4 Divide each part by  4. The solution set is {x|– 4 < x< 1} or (–4, 1).

10. Compound Inequality Applications Example: Jeni needs to earn a C in her Statistics class. Her current test scores are 67, 72, 81, and 75. Her final exam is worth three test scores. In order to earn a C, Jeni’s average must lie between 70 and 79, inclusive. What range of scores can Jeni receive on the final exam and earn a C in the course? Step 1: Identify.We need to find the range of possible score for the final exam. This is a direct translation problem involving inequalities. Step 2: Name.Let x represent the final exam score. Continued.

11. The final exam is worth 3 test scores. Compound Inequality Applications Example continued: Step 3: Translate.The average is calculated by adding up all of the test scores and dividing by the total number of scores. Step 4: Solve. Simplify. Multiply each part by 7. Subtract 295 from each part. Continued.

12. Compound Inequality Applications Example continued: Divide each part by 3. Step 5: Check.Since x represents the final exam score, any score between 65 and 86, inclusive, should yield an average score between 70 and 79. Step 6: Answer the Question. Jeni must score between 65 and 86, inclusive on her final exam to earn a C average for the course.