System of linear inequalities

A system of linear inequalities is made up of two or more inequalities A solution of a System of linear inequalities is an ordered pair that makes all the inequalities in the system true The graph of a system of inequalities is the set of the points that represent all of the solutions of the system Notes:

System of linear inequalities

Graphing a System of linear inequalities What is the graph of the system? y  2x – 3 2x + y  2 Step 1: graph y  2x – 3 Step 2: graph 2x + y  2 The system’s solutions lie in the green region where the graphs overlap The green region represents solution of both inequalities The green region represents solution of both inequalities The yellow region represents solution of y  2x - 3 The yellow region represents solution of y  2x - 3 The blue region represents solution of 2x + y  2 The blue region represents solution of 2x + y  2 Check: (3,0) is in green region. See if (3,0) satisfies both inequalities y  2x – 3 0  2(3) – 3 0  3 2x + y  2 2(3) + 0  2 6  2

Writing a System of linear inequalities from a graph What system of linear inequalities is represented by graph ? Write the inequality that represents the yellow region. Then Write the inequality that represents the blue region

Using a System of linear inequalities You are planning what to do after school. You can spend at most 6 h daily playing basketball and doing homework. You want to spend less than 2 h playing basketball. You must spend at least 2 h on homework. What is a graph showing how you can spend your time? At most 6 h playing basketball and doing homework Less than 2 h playing basketball At least 2 h doing homework At most 6 h playing basketball and doing homework Less than 2 h playing basketball At least 2 h doing homework What do you know? What do you need? To find different ways you can spend your time To find different ways you can spend your time What do have to do? Write and graph an inequality for each restriction. Find the region where all three restrictions are met Write and graph an inequality for each restriction. Find the region where all three restrictions are met Let x = the number of hours playing basketball Write a system of inequalities Write a system of inequalities x+ y  6 At most 6 h of basketball and homework Let y = the number of hours doing homework x x 2 Less than 2 h of basketball y  2 At least 2 h of homework

Using a System of linear inequalities Graph the system Graph the system less than You are planning what to do after school. You can spend at most 6 h daily playing basketball and doing homework. You want to spend less than 2 h playing basketball. You must spend at least 2 h on homework. What is a graph showing how you can spend your time? x+ y  6 x+ y  6 At most 6 h of basketball and homework x  2 x  2 Less than 2 h of basketball y  2 y  2 At least 2 h of homework Important: Because the time cannot be negative the graph makes sense only in the first quadrant. Remember: The solutions of the system are all the points in the shaded region including the points on the solid boundary lines x+ y  6 x  2 y  2 Hours of Basketball, x Hours of Homework, y After-School Activities

Using a System of linear inequalities You want to build a fence for a rectangular dog run. You want the run be at least 10 feet wide. The run can be at most 50 feet long. You have 126 ft of fencing. What is a graph showing the possible dimensions of the dog run? At most 126 ft of fence At most 50 feet long At least 10 feet wide At most 126 ft of fence At most 50 feet long At least 10 feet wide What do you know? What do you need? Possible dimensions of the dog run Possible dimensions of the dog run What do have to do? Write and graph an inequality for each restriction. Find the region where all three restrictions are met Write and graph an inequality for each restriction. Find the region where all three restrictions are met Write a system of inequalities Write a system of inequalities 2x+ 2y  126 At most 126 ft of fence Let y = the wide of the dog run x x 50 At most 50 feet long y  10 At least 10 feet wide You turn Let x = the long of the dog run RECTAGLE PERIMETER FORMULA 2l + 2w RECTAGLE PERIMETER FORMULA 2l 2l + 2w2w

Using a System of linear inequalities Graph the system Graph the system Important: Because the distance cannot be negative the graph makes sense only in the first quadrant. Remember: The solutions of the system are all the points in the shaded region including the points on the solid boundary lines x+ y  6 x  50 y  10 Dog run long, x Dog run wide, y Dog Run fence You want to built a fence for a rectangular dog run. You want the run be at least 10 feet wide. The run can be at most 50 feet long. You have 126 ft of fencing. What is a graph showing the possible dimensions of the dog run? 2x+ 2y  126 At most 126 ft of fence x  50 At most 50 feet long y  10 At least 10 feet wide

Using a System of linear inequalities An arena contains 1200 seats. For an upcoming concert, tickets will be priced $12.00 for some seats and $10.00 for others. At least 500 tickets are to be priced at $10.00, and the total sales must be at least $7200 to make a profit. What are the possible ways to price the tickets? At most 1200 seats At most 50 feet long At least 500 tickets at $10 At most 1200 seats At most 50 feet long At least 500 tickets at $10 What do you know? What do you need? Possible ways to price the tickets Possible ways to price the tickets What do have to do? Write and graph an inequality for each restriction. Find the region where all three restrictions are met Write and graph an inequality for each restriction. Find the region where all three restrictions are met Write a system of inequalities Write a system of inequalities 2x+ 2y  126 At most 126 ft of fence Let y = $ 10 tickets x x 50 At most 50 feet long y  10 At least 10 feet wide More practice Let x = $ 12 tickets