1. Chapter 3 Linear Programs
Section 3.1 Linear Inequalities in Two Variables
Section 3.2 Solutions of Systems of Inequalities: A Geometric Picture
Section 3.3 Linear Programming: A Geometric Approach
Section 3.4 Applications

2. Graph the linear inequality.

3. Graph the linear inequality.

4. Graph the linear inequality.

5. Graph the linear inequality.

6. Graphing a Linear Inequality
Section 3.1
Graphing a Linear Inequality
Graph the inequality of the form ax + by < c (The procedure also applies if the inequality symbols are <, > or >.)
Select a point that is not on the line from one half plane. The point (0,0) is usually a good choice when it is not on the line. If (0,0) is on the line. If (0,0) is on the line, use a point that is not on the line.
Continued on next slide

7. Continued
Substitute the coordinates of the point for x and y in the inequality.
If the selected point satisfies the inequality, then shade the half plane where the point lies. These points are on the graph.
If the selected point does not satisfy the inequality, shade the half plane opposite the point.
If the inequality symbol is < or >, use a dotted line for the graph of ax + by = c. This indicates that the points on the line are not a part of the graph.
If the inequality symbol is < or >, use a solid line for the graph of ax + by = c. This indicates that the line is a part of the graph.

8. Example
An automobile assembly plant has an assembly line that produces the Hatchback Special and the Sportster. Each Hatchback requires 2.5 hours of assembly line time, and each Sportster requires 3.5 hours. The assembly line has a maximum operating time of 140 hours per week. Graph the number of cars of each type that can be produced in one week.

10. A bakery is making whole-wheat bread and apple bran muffins
A bakery is making whole-wheat bread and apple bran muffins. The bread takes 4 hours to prepare. The muffins take 0.5 hour to prepare. The maximum preparation time available is 16 hours. Graph the number of of each type that can be prepared in one day.

11. Acme Manufacturing has two product lines
Acme Manufacturing has two product lines. Line A can produce 200 gadgets per hour and line B can produce 350 widgets per hour. Because of warehouse limitations, the total number of gadgets and widgets produced must not exceed 75,000. Write an inequality that describes the number of each that can produced and graph it.

12. A service club agrees to donate at least 500 hours of community service. A full member is to give 4 hours and a pledge is to give 6 hours. Write an inequality that expresses this information and graph it.

13. On a typical long distance call you talk for 30 minutes
On a typical long distance call you talk for 30 minutes. On a typical local call you talk for 10 minutes. Your phone company offers a special low rate of $0.08 per minute for long distance calls and $0.03 per minute for local calls, for customers who spend at least 240 minutes on the phone per month. Your parents have set a limit of no more than 15 long distance calls per month and 30 local calls per month. Write some inequalities that describe this situation.

14. Two manufacturing plants make the same kind of bicycle
Two manufacturing plants make the same kind of bicycle. The table gives the hours of general labor, machine time, and technical labor required to make one bicycle in each plant. For the two plants combined, the manufacturer can afford to use up to 4000 hours of general labor, up to 1500 hours of machine time, and up to 2300 hours of technical labor per week. Write some linear inequalities that describe this situation.

15. HW 3.1 Pg

16. 3.2 Systems of Linear Inequalities

17. Graph the system of linear inequalities.

18. Graph the system of linear inequalities.

19. Graph the system of linear inequalities.

20. Graph the system of linear inequalities.

21. Graph the system of linear inequalities.

22. Graph the system of linear inequalities.

23. Graph the system of linear inequalities.

24. Graph the system of linear inequalities.

25. Your Turn
Graph the system of linear inequalities.

26. A Geometric Picture Section 3.2
_______________ problems are described by systems of linear inequalities. The ________________ is the region of intersection on a graph of a system of inequalities and is the ___________ to the system of linear inequalities.
_____________________ problems generally have a _____________________ on the variables that states that some or all of the variables can never be ____________ because the quantities they measure (number of items, weight of materials) can never be negative.

27. Example Graph the solutions (feasible region) to the following system
The lines x + y = 4, –3x + 2y = 3 and x = 0 determine _____________ of the solution set. The intersection of these half plane solutions of the boundaries forms the __________________________. -6 -4 -2 2 4 6 8

28. Example continued
The points A and B on the graph are called the _______________________________ because they are the points in the feasible region where the boundaries ___________. -6 -4 -2 2 4 6 8 A B
Corners are the _______________ solution to a linear programming problem. Corners can be found by solving pairs of simultaneous equations of the lines forming the ________.
The corner A is found by finding the intersection of x = 0 and –3x +2y = 3.

29. Types of Solutions
The system of inequalities has __________________________ if the feasible region can be enclosed in a region where all points are a finite distance apart.

30. Types of Solutions
A system of inequalities has ___________________________ because some of the points in the feasible region are infinitely apart.

31. Example
Find the solution set (feasible region) of the system

32. Summary Graphing a System of Inequalities
Replace each inequality symbol with an equals sign to obtain a linear equation.
Graph each line. Use a solid line if it is a part of the solution. Use a dotted line if it is not a part of the solution. The line is a part of the solution when < or > is used. The line is not a part of the solution when < or > is used.
Select a test point not on the line.

33. continued
If the test point satisfies the original inequality, it is in the correct half plane. If it does not satisfy the inequality, the other half plane is the correct one.
Shade the correct half plane.
When the above steps are completed for each inequality, determine where the shaded half planes overlap. This region is the graph of the system of inequalities.

34. Write a system of linear inequalities that has the given graph

35. Write a system of linear inequalities that has the given graph

36. Write a system of linear inequalities that has the given graph

37. Write a system of linear inequalities that has the given graph

38. A retired couple has up to $50,000 to invest
A retired couple has up to $50,000 to invest. As their financial adviser, you recommend that they place at least $35,000 in Treasury bills yielding 7% and at most $10,000 in corporate bonds yielding 10%.
Using x to denote the amount of money invested in Treasury bills and y the amount invested in corporate bonds, write a system of linear inequalities that describes the possible amounts of each investment.
Graph the system and label the corner points.

39. Mike’s Toy Truck Company manufacturers two models of toy trucks, a standard model and a deluxe model. Each standard model requires 2 hours for painting and 3 hours for detail work; each deluxe model requires 3 hours for painting and 4 hours for detail work. Two painters and three detail workers are employed by the company, and each works 40 hours per week.
Using x to denote the number of standard model trucks and y to denote the number of deluxe model trucks, write a system of linear inequalities that describes the possible number of each model of truck that can be manufactured in a week.
Graph the system and label the corner points.

40. Bill’s Coffee House, a store that specializes in coffee, has available 75 pounds of A grade coffee and 120 pounds of B grade coffee. These will be blended into 1 pound packages as follows: An economy blend that contains 4 ounces of A grade coffee and 12 ounces of B grade coffee and a superior blend that contains 8 ounces of A grade coffee and 8 ounces of B grade coffee.
Using x to denote the number of packages of the economy blend and y to denote the number of packages of the superior blend, write a system of linear inequalities that describes the possible number of packages of each kind of blend.
Graph the system and label the corner points.

41. Nola’s Nuts, a store that specializes in selling nuts, has available 90 pounds of cashews and 120 pounds of peanuts. These are to be mixed in 12-ounce packages as follows: a lower-priced package containing 8 ounces of peanuts and 4 ounces of cashews and a quality package containing6 ounces of peanuts and 6 ounces of cashews.
Using x to denote the number of ounces of cashews and y to denote the number of peanuts, write a system of linear inequalities that describes the possible number of packages of each kind of blend.
Graph the system and label the corner points.

42. A small truck can carry no more than 1600 pounds of cargo nor more than 150 cubic feet of cargo. A printer weighs 20 pounds and occupies 3 cubic feet of space. A microwave oven weighs 30 pounds and occupies 2 cubic feet of space.
Using x to represent the number of microwave ovens and y to represent the number of printers, write a system of linear inequalities that describes the number of ovens and printers that can be hauled by the truck.
Graph the system and label the corner points.

43. Your Turn
A theater wishes to book a musical group that requires a guarantee $ Tickets prices are $10 for students and $15 for adults, and the theater’s maximum capacity is 550 seats.
State the inequalities that represent this information.
Graph the system of linear inequalities
Find the corner points

44. Hw 3.2 Pg

45. Linear Programming Constraints and Objection Function Section 3.3
A linear inequality of the form
a1x + a2y < b or a1x + a2y > b
is called a __________ of a linear programming problem. The restrictions x > 0 and y > 0 are __________________.
THEOREM
Given a linear _____________________ subject to linear inequality constraints, if the objection function has an ____________________ (maximum or minimum), it must occur at a corner point of the feasible region.

46. Example Find the maximum value of the objective function z = 10x + 15y
subject to the constraints
SOLUTION
First, ______________________ of the system of inequalities and locate the __________________. The corner points can be found by solving the system of equations of the lines that intersect at the point.

47. Example continued
The corner points of the feasible region are ________, _________, __________, and ____________.

48. Example continued Now, find the value of z at each corner point.
z = 10x + 15y
(0,90)
(0,0)
(150,0)
(120,60)
The maximum value of z is _______________ and occurs at the corner ________________.

50. Find the minimum and maximum values of the objective function subject to the given constraints.

51. Your Turn
Find the minimum and maximum values of the objective function subject to the given constraints.

52. Theorems Bounded Feasible Region
When the feasible region is not __________ and is __________, the objective function has both a ______________ and a _______________ value, which must occur at corner points.
Unbounded Feasible Region
When a feasible region with ______________ conditions is _________________, an objective function assumes a ___________ at a corner point of the feasible region. However, the objective function can be arbitrarily large for points in the feasible region, so no optimal ______________ solution exists.

53. Example Find the maximum value of the objective function z = 4x + 6y
subject to the constraints

54. Example continued
Now, find the value of ______ at each corner _________.
Corner
z = 4x + 6y
The maximum value of z is ___ and occurs at the corner _____.
The minimum value of z is ____ and occurs at the corner ______.

55. Maximizing Profit A manufacturer of skis produces two types: downhill and cross-country. Use the following table to determine how many of each kind of ski should be produced to achieve a maximum profit. What is the maximum profit? What would the maximum profit be if the maximum time available for manufacturing is increased to 48 hours?

56. Farm Management A farmer has 70 acres of land available or planting either soybeans or wheat. The cost of preparing the soil, the workdays required, and the expected profit per acre planted for each type of crop are given in the following table:
The farmer cannot spend more than $1800 in preparation costs nor use more than a total of 120 workdays. How many acres of each crop should be planted to maximize the profit? What is the maximum profit? What is the maximum profit if the farmer is willing to spend no more than $2400 on preparation?

57. Farm Management A small farm in Illinois has 100 acres of land available on which to grow corn and soybeans. The following table shows the cultivation cost per acre, the labor cost per acre, and the expected profit per acre. The column on the right shows the amount of money available for each of these expenses. Find the number of acres of each crop that should be planted to maximize profit.

58. Dietary Requirements A certain diet requires at least 60 units of carbohydrates, 45 units of protein, and 30 units of fat each day. Each ounce of Supplement A provides 5 units of carbohydrates, 3 units of protein, and 4 units of fat. Each ounce of Supplement B provides 2 units of carbohydrates, 2 units of protein, and 1 unit of fat. If Supplement A costs $1.50 per ounce and Supplement B costs $1.00 per ounce, how many ounces of each supplement should be taken daily to minimize the cost of the diet?

59. Production Scheduling In a factory, machine 1 produces 8-inch pliers at the rate of 60 units per hour and 6-inch pliers at the rate of 70 units per hour. Machine 2 produces 8-inch pliers at the rate of 40 units per hour and 6-inch pliers at the rate of 20 units per hour. It costs $50 per hour to operate machine 1, and machine 2 costs $30 per hour to operate. The production schedule requires that at least 240 units of 8-inch pliers and at least 140 units of 6-inch pliers be produced during each 10-hour day. Which combination of machines will cost the least money to operate?

60. Farm Management An owner of a fruit orchard hires a crew of workers to prune at least 25 of his 50 fruit trees. Each newer tree requires one hour to prune, while each older tree needs one-and-a-half hours. The crew contracts to work for at least 30 hours and charge $15 for each newer tree and $20 for each older tree. To minimize his cost, how many of each kind of tree will the orchard owner have pruned? What will be the cost?

61. Groups
On your way into class you picked up one of 8 different problems.
Find the other people in class who have your problem and form a group.
When you have a solution to your problem let me know and I will check your solution.
One person from your group will need to explain your problem to the class
Everyone is responsible to know how to do every problem presented. Quiz tomorrow

70. Discussion and Writing
Explain in your own words what a linear programming problem is and how it can be solved.

71. HW 3.3
Pg odd

72. Section 3.4
Applications
Solving linear programming problems geometrically works well when there are only two variables and a few constraints.
Typically though, linear programming problems will require dozens of variables with several constraints.
A correct analysis and description of the problem is essential before applying any method. An erroneous constraint will yield an erroneous solution.
This section works on correctly setting up linear programming problems in more than two variables so that the methods in Chapter 4 can be utilized to solve these larger systems.

79. Number of Condos needed
Example
Adventure Time offers one-week summer vacations during the month of August. The package includes round-trip transportation and a week’s accommodations at the Lodge. The Lodge gives a discount to Adventure
Time if they rent two- or three-week blocks
of condos, rent of $1000 per condo for a
two-week period and $1300 per condo for a
3-week period. Adventure Time expects
to need the number of condos shown.
Week
Number of Condos needed
First 30
Second 42
Third 21
Fourth 32
How many condos should Adventure Time rent for two weeks, and how many should be rented for three weeks, to meet the needed number and to minimize Adventure Time’s rental costs?

80. Number of condos needed
Example continued
SOLUTION
First, determine all possible ways to schedule two- and three-week blocks in August. The table below helps to “visualize” the possible ways to schedule the blocks.
Two-week periods
Three-week periods
Number of condos needed
Week 1
Week 2
Week 3
Week 4

81. Example continued
Since Adventure Time wants to minimize their rental costs, we need to minimize
The relationship between the number of condos needed and the five time periods can be written as

82. HW 3.4 Pg