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5.6 Notes Alg1.notebook January 10, 2013 56 Solving Systems of Linear Inequalities Skills we've learned 1. Graph y < x + 4 and y ≤ x + 1 2. Write the inequality to represent the graph: 3. You can spend at most $12.00 for drinks at a picnic. Iced tea costs $1.50 a gallon, and lemonade costs $2.00 per gallon. Write a system to represent this situation. Graph the solutions, describe reasonable solutions, and then give two possible combinations of drinks you could buy. Warmup Answers 1. Graph y < x + 4 and y ≤ x + 1 2. Write the inequality to represent the graph: 3. You can spend at most $12.00 for drinks at a picnic. Iced tea costs $1.50 a gallon, and lemonade costs $2.00 per gallon. Write a system to represent this situation. Graph the solutions, describe reasonable solutions, and then give two possible combinations of drinks you could buy. x ≥ 0 y ≥ 0 Only whole number solutions are reasonable. Possible answer: (2 gal tea, 3 gal lemonade) and (4 gal tea, 1 gal lemonade) 5.6 Notes Alg1.notebook January 10, 2013 56 Solving Systems of Linear Inequalities Graph and solve systems of linear inequalities in two variables To help us make decisions based on numerous variables. Identifying Solutions for Linear Inequalities 1. Tell whether the ordered pair is a solution of the system. Substitute (1,3) for (x, y) in both inequalities to see if they are solutions to the inequalities. y ≤ –3x + 1 y < 2x + 2 5.6 Notes Alg1.notebook January 10, 2013 2. Tell whether the ordered pair is a solution of the system. y < –2x – 1 y ≥ x + 3 Solving a System of Inequalities by Graphing To show all the solutions of a system of linear inequalities, graph the solutions of each inequality. The solutions of the system are represented by the overlapping shaded regions. The first graph is what we did in the warmup. 5.6 Notes Alg1.notebook January 10, 2013 Solving a System of Inequalities by Graphing 3. Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. { y ≤ 3 y > x + 5 Solutions: _____________ _____________ Not Solutions _____________ _____________ 4. Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. { 3x + 2y ≥ 2 y < 4x + 3 Solutions: _____________ _____________ Not Solutions _____________ _____________ 5.6 Notes Alg1.notebook January 10, 2013 More if Needed. Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. Solutions: _____________ _____________ Not Solutions _____________ _____________ Graphing Systems with Parallel Boundaries In Lesson 54, you saw that in systems of linear equations, if the lines are parallel, there are no solutions. With systems of linear inequalities, that is not always true. This system has no solutions. The solutions are all points between the parallel lines but not on the dashed lines. The solutions are the same as the solutions of y ≥ 4x + 6. 5.6 Notes Alg1.notebook January 10, 2013 5. Graph the system of linear inequalities. Describe the solutions. B) A) This system has no solutions. C) The solutions are all points between the parallel lines including the solid lines. The solutions are the same as the solutions of y > –2x + 3. Application In one week, Ed can mow at most 9 times and rake at most 7 times. He charges $20 for mowing and $10 for raking. He needs to make more than $125 in one week. Show and describe all the possible combinations of mowing and raking that Ed can do to meet his goal. List two possible combinations. Step 1 Write a system of inequalities. Let x represent the number of mowing jobs and y represent the number of raking jobs. x ≤ 9 y ≤ 7 20x + 10y > 125 Step 2 Graph the system. Shade Step 3 Describe all possible combinations. The graph should be in only the first quadrant because the number All possible combinations of jobs cannot be negative. represented by ordered pairs of whole numbers in the solution region will meet Ed’s requirement of mowing, raking, and earning more than $125 in one week. Answers must be whole numbers because he cannot work a portion of a job. Step 4 List the two possible combinations. Two possible combinations are: 7 mowing and 4 raking jobs 8 mowing and 1 raking jobs 5.6 Notes Alg1.notebook January 10, 2013 5.6 p.370 #3, 5, 7, 10, 15, 18 33 (x3), 37 Ch. 2 Study Guide p.150 #7, 8, 13, 15, 20, 30 32, 42, 49, 53, 63, 68, 74, 79 5.5 Review Ch. 1 5.6 Review Ch. 2 Review Review Ch. 5 Ch. 5 and Ch. 3 & 4 Test Ch. 5 Final Exam Review 5.4 No school Review for Final Day 1: Review for Chapter 2 Skills we've learned 21 Graphing and Writing Inequalities 22 Solving Inequalities by Adding or Subtracting 23 Solving Inequalities by Multiplying or Dividing 24 Solving TwoStep and MultiStep Inequalities 25 Solving Inequalities with Variables on Both 26 Solving Compound Inequalities 27 Solving AbsoluteValue Inequalities 5.6 Notes Alg1.notebook January 10, 2013 21 Graphing and Writing Inequalities 1) Graph a. x > 4 b. 2) Write an equation from a graph a. b. 22 Solving Inequalities by Addition/Subtraction 1) Solve and Graph a. 7 + x < 4 b. 6 > 3 + x 23 Solving Inequalities by Multiply/Divide a. 4x > 9 b. 24 Solving TwoStep and MultiStep Inequalities a. b. 2 < 3 + 7x 25 Solving Inequalities with Variables on Both a. b. 3x + 1 5x < 4 2x 5.6 Notes Alg1.notebook January 10, 2013 26 Solving Compound Inequalities a. b. 2x > 8 OR x + 4 > 10 27 Solving AbsoluteValue Inequalities a. b.