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Finite Mathematics and Calculus with Applications

Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Chapter 3

Linear Programming:The Graphical Method

Educators


Problem 1

Graph each linear inequality.
$$x+y \leq 2$$

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Problem 2

Graph each linear inequality.
$$y \leq x+1$$

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Problem 3

Graph each linear inequality.
$$x \geq 2-y$$

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Problem 4

Graph each linear inequality.
$$y \geq x-3$$

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Problem 5

Graph each linear inequality.
$$4 x-y<6$$

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Problem 6

Graph each linear inequality.
$$4 y+x>6$$

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Problem 7

Graph each linear inequality.
$$4 x+y<8$$

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Problem 8

Graph each linear inequality.
$$2 x-y>2$$

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Problem 9

Graph each linear inequality.
$$x+3 y \geq-2$$

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Problem 10

Graph each linear inequality.
$$2 x+3 y \leq 6$$

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Problem 11

Graph each linear inequality.
$$x \leq 3 y$$

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Problem 12

Graph each linear inequality.
$$2 x \geq y$$

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Problem 13

Graph each linear inequality.
$$x+y \leq 0$$

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Problem 14

Graph each linear inequality.
$$3 x+2 y \geq 0$$

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Problem 15

Graph each linear inequality.
$$y< x$$

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Problem 16

Graph each linear inequality.
$$y >5 x$$

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Problem 17

Graph each linear inequality.
$$x <4$$

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Problem 18

Graph each linear inequality.
$$y> 5$$

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Problem 19

Graph each linear inequality.
$$y \leq-2$$

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Problem 20

Graph each linear inequality.
$$x \geq-4$$

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Problem 21

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{array}{l}{x+y \leq 1} \\ {x-y \geq 2}\end{array}$$

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Problem 22

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{array}{l}{4 x-y<6} \\ {3 x+y<9}\end{array}$$

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Problem 23

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{aligned} x+3 y & \leq 6 \\ 2 x+4 y & \geq 7 \end{aligned}$$

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Problem 24

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{array}{l}{-x-y<5} \\ {2 x-y<4}\end{array}$$

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Problem 25

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{aligned} x+y & \leq 7 \\ x-y & \leq-4 \\ 4 x+y & \geq 0 \end{aligned}$$

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Problem 26

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{aligned} 3 x-2 y & \geq 6 \\ x+y & \leq-5 \\ y & \leq-6 \end{aligned}$$

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Problem 27

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{array}{l}{-2< x <3} \\ {-1 \leq y \leq 5} \\ {2 x+y <6}\end{array}$$

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Problem 28

$$
\begin{array}{l}{1<x<4} \\ {y>2} \\ {x>y}\end{array}
$$

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Problem 29

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{aligned} y-2 x & \leq 4 \\ y & \geq 2-x \\ x & \geq 0 \\ y & \geq 0 \end{aligned}$$

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Problem 30

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{aligned} 2 x+3 y & \leq 12 \\ 2 x+3 y &>3 \\ 3 x+y &<4 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}$$

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Problem 31

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{aligned} 3 x+4 y &>12 \\ 2 x-3 y &<6 \\ 0 \leq y & \leq 2 \\ x & \geq 0 \end{aligned}$$

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Problem 32

Graph the feasible region for each system of inequalities. Tell whether each region is bounded or unbounded.
$$\begin{aligned} 0 \leq x & \leq 9 \\ x-2 y & \geq 4 \\ 3 x+5 y & \leq 30 \\ y & \geq 0 \end{aligned}$$

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Problem 33

Use a graphing calculator to graph the following.
$$2 x-6 y>12$$

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Problem 34

Use a graphing calculator to graph the following.
$$4 x-3 y<12$$

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Problem 35

Use a graphing calculator to graph the following.
$$\begin{array}{l}{3 x-4 y<6} \\ {2 x+5 y>15}\end{array}$$

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Problem 36

Use a graphing calculator to graph the following.
$$\begin{array}{l}{6 x-4 y>8} \\ {2 x+5 y<5}\end{array}$$

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Problem 37

The regions A through G in the figure can be described by the inequalities
$$\begin{array}{r}{x+3 y ? 6} \\ {x+y ? 3} \\ {x-2 y ? 2} \\ {x \geq 0} \\ {y \geq 0}\end{array}$$
where? can be either $\leq$ or $\geq$ . For each region, tell what the ? should be in the three inequalities. For example, for region A, the ? should be $\geq, \leq,$ and $\leq,$ because region $\mathrm{A}$ is described by the inequalities
$$\begin{aligned} x+3 y & \geq 6 \\ x+y & \leq 3 \\ x-2 y & \leq 2 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}$$

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Problem 38

A small pottery shop makes two kinds of planters, glazed and unglazed. The glazed type requires 1$/ 2$ hour to throw on the wheel and 1 hour in the kiln. The unglazed type takes 1 hour to throw on the wheel and 6 hours in the kiln. The wheel is available for at most 8 hours per day, and the kiln for at most 20 hours per day.
a. Complete the following table.

(Table cant Copy)

b. Set up a system of inequalities and graph the feasible region.
c. Using your graph from part b, can 5 glazed and 2 unglazed planters be made? Can 10 glazed and 2 unglazed planters be made?

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Problem 39

Carmella and Walt produce handmade shawls and afghans. They spin the yarn, dye it, and then weave it. A shawl requires 1 hour of spinning, 1 hour of dyeing, and 1 hour of weaving. An afghan needs 2 hours of spinning, 1 hour of dyeing, and 4 hours of weaving. Together, they spend at most 8 hours spinning, 6 hours dyeing, and 14 hours weaving.
a. Complete the following table.

(Table cant Copy)

b. Set up a system of inequalities and graph the feasible region.
c. Using your graph from part b, can 3 shawls and 2 afghans be made? Can 4 shawls and 3 afghans be made?

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Problem 40

Perform the following steps.
a. Write a system of inequalities to express the conditions of the problem.
b. Graph the feasible region of the system.

Southwestern Oil supplies two distributors located in the Northwest. One distributor needs at least 3000 barrels of oil, and the other needs at least 5000 barrels. South-western can send out at most $10,000$ barrels. Let $x=$ the number of barrels of oil sent to distributor 1 and $y=$ the number sent to distributor 2 .

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Problem 41

Perform the following steps.
a. Write a system of inequalities to express the conditions of the problem.
b. Graph the feasible region of the system.

The loan department in a bank will use at most $\$ 30$ million for commercial and home loans. The bank's policy is to allocate at least four times as much money to home loans as to commercial loans. The bank's return is 6$\%$ on a home loan and 8$\%$ on a commercial loan. The manager of the loan department wants to earn a return of at least $\$ 1.6$ million on these loans. Let $x=$ the amount (in millions) for home loans and $y=$ the amount (in millions) for commercial loans.

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Problem 42

Perform the following steps.
a. Write a system of inequalities to express the conditions of the problem.
b. Graph the feasible region of the system.

The California Almond Growers have at most 2400 boxes of almonds to be shipped from their plant in Sacramento to Des Moines and San Antonio. The Des Moines market needs at least 1000 boxes, while the San Antonio market must have at least 800 boxes. Let $x=$ the number of boxes to be shipped to Des Moines and $y=$ the number of boxes to be shipped to San Antonio.

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Problem 43

Perform the following steps.
a. Write a system of inequalities to express the conditions of the problem.
b. Graph the feasible region of the system.

The Gillette Company produces two popular battery-operated razors, the M3 Power $^{TM}$ and the Fusion Power$^{TM}$. Because of demand, the number of M3 Power$^{TM}$ razors is never more than one-half the number of Fusion Power$^{TM}$ razors. The factory’s production cannot exceed 800 razors per day. Let $x=$ the number of M3 Power$^{TM}$ razors and $y=$ the number of Fusion Power" razors produced per day.

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Problem 44

Perform the following steps.
a. Write a system of inequalities to express the conditions of the problem.
b. Graph the feasible region of the system.

A cement manufacturer produces at least 3.2 million barrels of cement annually. He is told by the Environmental Protection Agency (EPA) that his operation emits 2.5 lb of dust for each barrel produced. The EPA has ruled that annual emissions must be reduced to no more than 1.8 million lb. To do this, the manufacturer plans to replace the present dust collectors with two types of electronic precipitators. One type would reduce emissions to 0.5 lb per barrel and operating costs would be 16$\notin$ per barrel. The other would reduce the dust to 0.3 lb per barrel and operating costs would be 20$\notin$ per barrel. The manufacturer does not want to spend more than 0.8 million dollars in operating costs on the precipitators. He needs to know how many barrels he could produce with each type. Let $x=$ the number of barrels (in millions) produced with the first type and $y=$ the number of barrels (in millions) produced with the second type.

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Problem 45

A dietician is planning a snack package of fruit and nuts. Each ounce of fruit will supply 1 unit of protein, 2 units of carbohydrates, and 1 unit of fat. Each ounce of nuts will supply 1 unit of protein, 1 unit of carbohydrates, and 1 unit of fat. Every package must provide at least 7 units of protein, at least 10 units of carbohydrates, and no more than 9 units of fat. Let $x=$ the ounces of fruit and $y=$ the ounces of nuts to be used in each package.

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