1. System of linear inequalities

2. System of linear inequalities
Notes:
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

3. System of linear inequalities

4. Graphing a System of linear inequalities
y  2x – 3
2x + y  2
What is the graph of the system?
Step 1: graph y  2x – 3
Step 2: graph 2x + y  2 1 2 3 4 5 -1 -2 -3
The blue region
represents solution of
2x + y  2
The green region
represents solution of
both inequalities
The yellow region
represents solution of
y  2x - 3
The system’s solutions lie in the green region where the graphs overlap
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

5. 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

6. 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?
What do you know?
What do you need?
What do have to do?
At most 6 h playing basketball and doing homework
Less than 2 h playing basketball
At least 2 h doing homework
To find different ways you can spend your time
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
Let y = the number of hours doing homework
Write a system of inequalities
x+ y  At most 6 h of basketball and homework
x Less than 2 h of basketball
y  At least 2 h of homework

7. 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?
x+ y  At most 6 h of basketball and homework
x Less than 2 h of basketball
After-School Activities
y  At least 2 h of homework 1 2 3 4 5 6 7
Graph the system
x 2
Important:
Because the time cannot be negative the graph makes sense only in the first quadrant.
Hours of Homework, y
y  2
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
Hours of Basketball, x

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

9. Using a System of linear inequalities
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  At most 126 ft of fence
x At most 50 feet long
Dog Run fence
y  At least 10 feet wide 10 20 30 40 50 60 70
Graph the system
x 50
x+ y  6
Important:
Because the distance cannot be negative the graph makes sense only in the first quadrant.
Dog run wide, y
Remember:
The solutions of the system are all the points in the shaded region including the points on the solid boundary lines
y  10
Dog run long, x

10. Using a System of linear inequalities
More practice
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?
What do you know?
What do you need?
What do have to do?
At most 1200 seats
At most 50 feet long
At least 500 tickets at $10
Possible ways to price the tickets
Write and graph an inequality for each restriction.
Find the region where all three restrictions are met
Let x = $ 12 tickets
Let y = $ 10 tickets
Write a system of inequalities
2x+ 2y  At most 126 ft of fence
x At most 50 feet long
y  At least 10 feet wide