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Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan

Chapter 6

Linear Programming - all with Video Answers

Educators

Section 1

Graphing Systems of Linear Inequalities in Two Variables

Problem 1

Find the graphical solution of each inequality.
$$4 x-\frac{8<0}{0}$$

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Problem 2

Find the graphical solution of each inequality.
$$3 y+2>0$$

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Problem 3

Find the graphical solution of each inequality.
$$x-y \leq 0$$

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Problem 4

Find the graphical solution of each inequality.
$$3 x+4 y \leq-2$$

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Problem 5

Find the graphical solution of each inequality.
$$x \leq-3$$

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Problem 6

Find the graphical solution of each inequality.
$$y \geq-1$$

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Problem 7

Find the graphical solution of each inequality.
$$2 x+y \leq 4$$

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Problem 8

Find the graphical solution of each inequality.
$$-3 x+6 y \geq 12$$

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Problem 9

Find the graphical solution of each inequality.
$$4 x-3 y \leq-24$$

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Problem 10

Find the graphical solution of each inequality.
$5 x-3 y \geq 15$$

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Problem 11

Write a system of linear inequalities that describes the shaded region.

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Problem 12

Write a system of linear inequalities that describes the shaded region.

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Problem 13

Write a system of linear inequalities that describes the shaded region.

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Problem 14

Write a system of linear inequalities that describes the shaded region.

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Problem 15

Write a system of linear inequalities that describes the shaded region.

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Problem 16

Write a system of linear inequalities that describes the shaded region.

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Problem 17

Write a system of linear inequalities that describes the shaded region.

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Problem 18

Write a system of linear inequalities that describes the shaded region.

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Problem 19

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{r}
2 x+4 y>16 \\
-x+3 y \geq 7
\end{array}
$$

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Problem 20

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{r}
3 x-2 y>-13 \\
-x+2 y>5
\end{array}
$$

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Problem 21

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{r}
x-y \leq 0 \\
2 x+3 y \geq 10
\end{array}
$$

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Problem 22

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{rr}
x+y & \geq-2 \\
3 x-y & \leq 6
\end{array}
$$

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Problem 23

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
x+2 y & \geq 3 \\
2 x+4 y & \leq-2
\end{aligned}
$$

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Problem 24

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{l}
2 x-y \geq 4 \\
4 x-2 y<-2
\end{array}
$$

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Problem 25

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{r}
x+y \leq 6 \\
0 \leq x \leq 3 \\
y \geq 0
\end{array}
$$

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Problem 26

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{l}
4 x-3 y \leq 12 \\
5 x+2 y \leq 10 \\
x \geq 0, y \geq 0
\end{array}
$$

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Problem 27

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
3 x-6 y & \leq 12 \\
-x+2 y & \leq 4 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

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Problem 28

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
x+y & \geq 20 \\
x+2 y & \geq 40 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

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Problem 29

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
3 x-7 y & \geq-24 \\
x+3 y & \geq 8 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

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Problem 30

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
3 x+4 y & \geq 12 \\
2 x-y & \geq-2 \\
0 \leq y & \leq 3 \\
x & \geq 0
\end{aligned}
$$

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Problem 31

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{array}{rr}
x+2 y \geq & 3 \\
5 x-4 y \leq & 16 \\
0 \leq y \leq & 2 \\
x & \geq 0
\end{array}
$$

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Problem 32

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
x+y & \leq 4 \\
2 x+y & \leq 6 \\
2 x-y & \geq-1 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

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Problem 33

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
6 x+5 y & \leq 30 \\
3 x+y & \geq 6 \\
x+y & \geq 4 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

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Problem 34

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
6 x+7 y & \leq 84 \\
12 x-11 y & \leq 18 \\
6 x-7 y & \leq 28 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

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Problem 35

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
x-y & \geq-6 \\
x-2 y & \leq-2 \\
x+2 y & \geq 6 \\
x-2 y & \geq-14 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

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Problem 36

Determine graphically the solution set for each system of inequalities and indicate whether the solution set is bounded or unbounded.
$$
\begin{aligned}
x-3 y & \geq-18 \\
3 x-2 y & \geq 2 \\
x-3 y & \leq-4 \\
3 x-2 y & \leq 16 \\
x \geq 0, y & \geq 0
\end{aligned}
$$

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Problem 37

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
The solution set of a linear inequality involving two variables is either a half-plane or a straight line.

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Problem 38

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
The solution set of the inequality $a x+b y+c \leq 0$ is either a left half-plane or a lower half-plane.

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Problem 39

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
The solution set of a system of linear inequalities in two variables is bounded if it can be enclosed by a rectangle.

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Problem 40

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
The solution set of the system
$$
\begin{array}{l}
a x+b y \leq e \\
c x+d y \leq f \\
x \geq 0, y \geq 0
\end{array}
$$
where $a, b, c, d, e$, and $f$ are positive real numbers, is a bounded set.

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