Lesson 3-6: Compound Inequalities Name:
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1 Lesson 3-6: Compound Inequalities Name: W hen people plan a house, they often have many requirements in mind that can be written as inequalities. Such requirements could be the dimensions of rooms or the overall size of the house. In this activity, we will investigate what happens when we combine requirements. First, we will graph each requirement separately, then we will graph combinations of those requirements. Here are the basic requirements we will work with: Dimension Requirements: Area Requirements: Requirement A: d 5 ft Requirement D: A 1500 sq ft Requirement B: d 12 ft Requirement E: A 1200 sq ft Requirement C: d 10 ft Requirement F: A 1800 sq ft 1. First, let s investigate what happens when we combine requirements A & B with and : b. Graph the numbers that meet requirement B: c. Graph only the numbers that meet both A and B: 2. Now let s investigate what happens when we combine requirements A & B with or : b. Graph the numbers that meet requirement B: c. Graph all the numbers that meet either A or B: 3. Now investigate what happens when we combine requirements A & C with and : b. Graph the numbers that meet requirement C: c. Graph only the numbers that meet both A and C: 4. Investigate what happens when we combine requirements A & C with or : b. Graph the numbers that meet requirement C: c. Graph all the numbers that meet either A or C: Mastery Algebra 1 OBJ: Graphing compound inequalities 3-6.A
2 Each graph that you have drawn on a number line represents the solution set or truth set. Solution Set (or Truth Set) A solution set is all the numbers that make a statement true. 5. Next, combine requirements D & E: a. Graph the numbers that meet requirement D: b. Graph the numbers that meet requirement E: c. Graph the numbers that meet requirements D and E: d. Graph all the numbers that meet requirement D or E: 6. Now combine requirements E & F: a. Graph the numbers that meet requirement E: b. Graph the numbers that meet requirement F: c. Graph the numbers that meet requirements E and F: d. Graph all the numbers that meet requirement E or F: 7. Now combine requirements D & F: a. Graph the numbers that meet requirement D: b. Graph the numbers that meet requirement F: c. Graph the numbers that meet requirements D and F: d. Graph all the numbers that meet requirement D or F: 8. Now combine requirements B & C: a. Graph the numbers that meet requirement B: b. Graph the numbers that meet requirement C: c. Graph the numbers that meet requirements B and C: d. Graph all the numbers that meet requirement B or C: 9. Now write some notes to help you remember what you have learned about combining solution sets using the words and & or. Mastery Algebra 1 OBJ: Graphing compound inequalities 3-6.B
3 More 3-6: Lines Dancing Graphing Compound Inequalities Combining two inequalities with the word or creates the union of the two solution sets. The symbol for the union of two sets is. Any value that is in the solution set to either of the original inequalities is in the solution set of compound inequality. Combining two inequalities with the word and creates the intersection of the two solution sets. The symbol for the intersection of two sets is. Only values that are in the solution set to both original inequalities may be in the solution set of the compound inequality. Use the table of inequality requirements below to find each union or intersection. Sketch each individual requirement first, and then sketch the compound inequality on the third number line. Requirement A: x 3 Requirement B: x 2 Requirement C: x 0 Requirement D: x 1 Requirement E: x 4 Requirement F: x 2 Requirement G: x 0 1. Requirements A and C. A: Mastery Algebra 1 OBJ: Graphing compound inequalities 3-6.C C: A C: 2. Requirements A or C. A: C: A C: 3. Requirements B and D. 4. Requirements B or D. B: B: D: D: B D: B D: 5. Requirements A and B. 6. Requirements A or B. A: A: B: B: A B: A B:
4 Graph each set of requirements as indicated and write the solution as an inequality. Remember to label each number line! 7. Requirements E and G. 8. Requirements E or G. 9. Requirements C and D. 10. Requirements C or D. 11. Requirements B and C. 12. Requirements D or F. 13. Requirements A and F. 14. Requirements B or F. 15. Based on these graphs, what generalizations can you make about the differences between these two types of compound inequalities (unions and intersections)? Mastery Algebra 1 OBJ: Graphing compound inequalities 3-6.D
5 Even More 3-6: Compounding Inequities S olve each compound inequality. Graph the solutions on the number line AND state 3 numbers in the solution set. Show all your work! 1. 4m 5 7 or 4m x y 6 1 or y (5 x ) 12 and 7x 4x x y or 2y 5y 12 Algebra 1 OBJ: Practice solving and graphing compound inequalities 3-6.E
6 7. 3( c 4) 12 and c k 7 or 2k (3w 5) 7 or 2w 8 5w ( x 3) 4 14 Algebra 1 OBJ: Practice solving and graphing compound inequalities 3-6.F
7 Practice 3-6 Solve each compound inequality and graph the solution: Name: Date: Period: 1. 5 x k 3 1 or k d 8 and 2d y x 7 and 2 3(x 1) ( 3x 6) 24 or 2( 3b 2) 4b ( 4x 1) 10 and 5 ( x 9) 2x 7 Algebra 1 Obj: practice solving compound inequalities Practice 3-6
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MATH 60 Course Notebook Chapter #1 Integers and Real Numbers Before we start the journey into Algebra, we need to understand more about the numbers and number concepts, which form the foundation of Algebra.
a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0. (x 1) 2 > 0.
For some problems, several sample proofs are given here. Problem 1. a. See the textbook for examples of proving logical equivalence using truth tables. b. There is a real number x for which f (x) < 0.
Lesson 18: Problem Set Sample Solutions
Problem Set Sample Solutions Problems 5 7 serve to review the process of computing f(g(x)) for given functions f and g in preparation for work with inverses of functions in Lesson 19. 1. Sketch the graphs