Solve systems of inequalities Use linear programming to solve problems

1. 9.6 Systems of Inequalities
Chapter 9 Systems and Matrices

2. Concepts and Objectives
 Systems of Inequalities
 Graph systems of inequalities
 Identify solutions to systems of inequalities

3. Graphing Review
 To graph a linear inequality, put the inequality into
slope-intercept form.
 Plot the y-intercept and count the slope from there
(rise over run)
Symbol Line Shade
< below
> above
 below
 above

4. Graphing Review
 Example: Graph the solution to  
2 3 6
x y

5. Graphing Review
 Example: Graph the solution to  
2 3 6
x y
  
3 2 6
y x
  
2
2
3
y x

6. Graphing Review
 Example: What is the solution to ?
 
2
2
y x

7. Graphing Review
 Example: What is the solution to ?
Vertex: 0, –2
 
2
2
y x

8. Graphing Review
 Example: What is the solution to ?
Vertex: 0, –2
 
2
2
y x

9. Graphing Review
 Example: What is the solution to ?
Vertex: 0, –2
 
2
2
y x

10. Graphing Review
 Example: What is the solution to ?
Vertex: 0, –2
 
2
2
y x

11. Systems of Inequalities
 The solution to a system of inequalities will be the graph
of the overlap between the two (or more) inequalities.
 You must graph the system to show the solution.
 If the inequalities do not overlap, then there is no
solution to the system.

12. Systems of Inequalities
 Example: What is the solution to the system?
  


 

2 4
3 2
x y
x y

13. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #1:
  


 

2 4
3 2
x y
x y
   
2 4
y x
 
1
2
2
y x

14. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #1:
  


 

2 4
3 2
x y
x y
   
2 4
y x
 
1
2
2
y x

15. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #2:
  


 

2 4
3 2
x y
x y
  
3 2
y x

16. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #2:
  


 

2 4
3 2
x y
x y
  
3 2
y x

17. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #2:
  


 

2 4
3 2
x y
x y
  
3 2
y x

18. Systems of Inequalities
 Example: What is the solution to the system?
  

 

2
1
3 2 4
x y
x y

19. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #1
  

 

2
1
3 2 4
x y
x y
   
2
1
y x
 
2
1
y x

20. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #2
  

 

2
1
3 2 4
x y
x y
  
2 3 4
y x
  
3
2
2
y x

21. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #2
  

 

2
1
3 2 4
x y
x y
  
2 3 4
y x
  
3
2
2
y x

22. Systems of Inequalities
 Example: What is the solution to the system?
Inequality #2
  

 

2
1
3 2 4
x y
x y
  
2 3 4
y x
  
3
2
2
y x

23. Linear Programming
 An important application of mathematics is called linear
programming, and it is used to find an optimum value
(for example, minimum cost or maximum profit). It was
first developed to solve problems in allocating supplies
for the U.S. Army Air Corps during World War II.
Solving a Linear Programming Problem
1. Write the objective function and all necessary constraints.
2. Graph the region of feasible solutions.
3. Identify all vertices or corner points.
4. Find the value of the objective function at each vertex.
5. The solution is given by the vertex producing the optimal value
of the objective function.

24. Linear Programming (cont.)
Example: The Charlson Company makes two products –
headphones and microphones. Each headphone gives a
profit of $30, while each microphone produces $70 profit.
The company must manufacture at least 10 headphones
per day to satisfy one of its customers, but no more than 50
because of production problems. The number of
microphones produced cannot exceed 60 per day, and the
number of headphones cannot exceed the number of
microphones. How many of each should the company
manufacture to obtain maximum profit?

25. Linear Programming (cont.)
Example (cont.)
 First, we translate the statements of the problem into
symbols:
 Let x = number of headphones produced daily
 Let y = number of microphones produced daily
 At least 10 headphones per day: x  10
 No more than 50 headphones per day: x  50
 No more than 60 microphones per day: y  60
 Headphones cannot exceed microphones: x y
 No negative numbers: x  0, y  0

26. Linear Programming (cont.)
Example (cont.)
 Each headphone gives a profit of $30, and each
microphone produces a profit of $70, so the the total
daily profit is:
Profit = 30x + 70y
 This equation defines the function to be maximized,
called the objective function.
 To find the maximum possible profit, subject to these
constraints, we graph each constraint.

27. Linear Programming (cont.)
 Here’s what the graph looks like:
To make the
intersection
clearer, I did not
graph the x  0
and y  0, since
it did not affect
the overlap.

28. Linear Programming (cont.)
 To make the intersection clearer, I outlined it:
Region of feasible
solutions

29. Linear Programming (cont.)
 The vertices of the region of feasible solutions are where
we will find the maximum profit.

30. Linear Programming (cont.)
 To find the maximum profit, we find the value of the
objective function for each vertex and see which one
produces the biggest number.
 The maximum profit is made for 50 headphones and 50
microphones.
x y Profit = 30x + 70y
10 10 30(10)+70(10)=1000
10 60 30(10)+70(60)=4500
50 60 30(50)+70(60)=5000
50 50 30(50)+70(50)=5700

31. A Note About Desmos
 Now that we have worked through all of this graphing
manually, how about using Desmos?
 Pros: Desmos will graph the system without having to
manipulate the inequalities.
 Cons: At this time, Desmos will not easily let us just
shade the intersection.
 On classwork checks, quizzes, and tests, I will assume
that you will use Desmos to graph these – but it’s up to
you to know how to interpret the results.

32. A Note About Desmos (cont.)
 For example, here’s the Desmos graph of the last system:
 The solution is the area inside the purple region.
(To get the  symbol, type <=)

33. Classwork
 9.6 Assignment (College Algebra)
 Page 905: 32-48 (4), page 894: 48-54 (even), page
877: 62-68, 74 (even)
 9.6 Classwork Check
 Quiz 9.5