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9.7 Systems of Inequalities and Linear Programming

Graph linear inequalities.
Graph systems of linear inequalities.
Solve linear programming problems.

A graph of an inequality is a drawing that represents its solutions. We have already seen that an inequality in one variable can be graphed on the number line. An inequality in two variables can be graphed on a coordinate plane.

Graphs of Linear Inequalities

A statement like $5 x - 4 y < 20$ $5 x - 4 y < 20$ is a linear inequality in two variables.

A solution of a linear inequality in two variables is an ordered pair $(x, y)$ $(x, y)$ for which the inequality is true. For example, $(1, 3)$ $(1, 3)$ is a solution of $5 x - 4 y < 20$ $5 x - 4 y < 20$ because $5 \cdot 1 - 4 \cdot 3 < 20$ $5 \cdot 1 - 4 \cdot 3 < 20$ , or $- 7 < 20$ $- 7 < 20$ , is true. On the other hand, $(2, - 6)$ $(2, - 6)$ is not a solution of $5 x - 4 y < 20$ $5 x - 4 y < 20$ because $5 \cdot 2 - 4 \cdot (- 6) ∡ 20$ $5 \cdot 2 - 4 \cdot (- 6) ∡ 20$ , or $34 ∡ 20$ $34 ∡ 20$ .

The solution set of an inequality is the set of all ordered pairs that make it true. The graph of an inequality represents its solution set.

Example 1

Graph: $y < x + 3$ $y < x + 3$ .

Solution

We begin by graphing the related equation $y = x + 3$ $y = x + 3$ . We use a dashed line because the inequality symbol is $<$ $<$ . This indicates that the line itself is not in the solution set of the inequality.

Note that the line divides the coordinate plane into two regions called half-planes. One of these half-planes satisfies the inequality. Either all points in a half-plane are in the solution set of the inequality or none is.

To determine which half-plane satisfies the inequality, we try a test point in either region. The point (0,0) is usually a convenient choice so long as it does not lie on the line.

Since $(0, 0)$ $(0, 0)$ satisfies the inequality, so do all points in the half-plane that contains $(0, 0)$ $(0, 0)$ . We shade this region to show the solution set of the inequality.

Now Try Exercise 13.

In general, we use the following procedure to graph linear inequalities in two variables by hand.

Example 2

Graph: $3 x + 4 y \geq 12$ $3 x + 4 y \geq 12$ .

Solution

First, we graph the related equation $3 x + 4 y = 12$ $3 x + 4 y = 12$ . We use a solid line because the inequality symbol is ≥. This indicates that the line is included in the solution set.
To determine which half-plane to shade, we test a point in either region. We choose $(0, 0)$ $(0, 0)$ .

Because $(0, 0)$ $(0, 0)$ is not a solution, all the points in the half-plane that does not contain $(0, 0)$ $(0, 0)$ are solutions. We shade that region, as shown in the following figure.

Now Try Exercise 17.

Example 3

Graph $x > - 3$ $x > - 3$ on a plane.

Solution

First, we graph the related equation $x = - 3$ $x = - 3$ . We use a dashed line because the inequality symbol is $>$ $>$ . This indicates that the line is not included in the solution set.
The inequality tells us that all points $(x, y)$ $(x, y)$ for which $x > - 3$ $x > - 3$ are solutions. These are the points to the right of the line. We can also use a test point to determine the solutions. We choose $(5, 1)$ $(5, 1)$ .

Because $(5, 1)$ $(5, 1)$ is a solution, we shade the region containing that point—that is, the region to the right of the dashed line.

Now Try Exercise 23.

Example 4

Graph $y \leq 4$ $y \leq 4$ on a plane.

Solution

First, we graph the related equation $y = 4$ $y = 4$ . We use a solid line because the inequality symbol is $\leq$ $\leq$ .
The inequality tells us that all points $(x, y)$ $(x, y)$ for which $y \leq 4$ $y \leq 4$ are solutions of the inequality. These are the points on or below the line. We can also use a test point to determine the solutions. We choose $(- 2, 5)$ $(- 2, 5)$ .

Because $(- 2, 5)$ $(- 2, 5)$ is not a solution, we shade the half-plane that does not contain that point.

Now Try Exercise 25.

Systems of Linear Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. For example,

\begin{matrix} x + y \leq 4, \\ x - y \geq 2 \end{matrix}

$\begin{matrix} x + y \leq 4, \\ x - y \geq 2 \end{matrix}$

is a system of two linear inequalities in two variables.

A solution of a system of inequalities is an ordered pair that is a solution of each inequality in the system. To graph a system of linear inequalities, we graph each inequality and determine the region that is common to all the solution sets.

Example 5

Graph the solution set of the system

\begin{matrix} x + y \leq 4, \\ x - y \geq 2 . \end{matrix}

$\begin{matrix} x + y \leq 4, \\ x - y \geq 2 . \end{matrix}$

Solution

We graph $x + y \leq 4$ $x + y \leq 4$ by first graphing the equation $x + y = 4$ $x + y = 4$ using a solid line. Next, we choose $(0, 0)$ $(0, 0)$ as a test point and find that it is a solution of $x + y \leq 4$ $x + y \leq 4$ , so we shade the half-plane containing $(0, 0)$ $(0, 0)$ using red. Next, we graph $x - y = 2$ $x - y = 2$ using a solid line. We find that $(0, 0)$ $(0, 0)$ is not a solution of $x - y \geq 2$ $x - y \geq 2$ , so we shade the half-plane that does not contain $(0, 0)$ $(0, 0)$ using green. The arrows near the ends of each line help to indicate the half-plane that contains each solution set.

The solution set of the system of equations is the region shaded both red and green, or brown, including parts of the lines $x + y = 4$ $x + y = 4$ and $x - y = 2$ $x - y = 2$ .

A system of inequalities may have a graph that consists of a polygon and its interior. As we will see later in this section, in many applications we will need to know the vertices of such a polygon.

Example 6

Graph the following system of inequalities and find the coordinates of any vertices formed:

\begin{matrix} 3 x - y \leq 6, (1) \\ y - 3 \leq 0, (2) \\ x + y \geq 0 . (3) \end{matrix}

$\begin{matrix} 3 x - y \leq 6, (1) \\ y - 3 \leq 0, (2) \\ x + y \geq 0 . (3) \end{matrix}$

Solution

We graph the related equations $3 x - y = 6$ $3 x - y = 6$ , $y - 3 = 0$ $y - 3 = 0$ , and $x + y = 0$ $x + y = 0$ using solid lines. The half-plane containing the solution set for each inequality is indicated by the arrows near the ends of each line. We shade the region common to all three solution sets.

To find the vertices, we solve three systems of equations. The system of equations from inequalities (1) and (2) is

\begin{matrix} 3 x - y & = & 6, \\ y - 3 & = & 0 . \end{matrix}

$\begin{matrix} 3 x - y & = & 6, \\ y - 3 & = & 0 . \end{matrix}$

Solving, we obtain the vertex $(3, 3)$ $(3, 3)$ .

The system of equations from inequalities (1) and (3) is

\begin{matrix} 3 x - y = 6, \\ x + y = 0 . \end{matrix}

$\begin{matrix} 3 x - y = 6, \\ x + y = 0 . \end{matrix}$

Solving, we obtain the vertex $(\frac{3}{2}, - \frac{3}{2})$ $(\frac{3}{2}, - \frac{3}{2})$ .

The system of equations from inequalities (2) and (3) is

\begin{matrix} y - 3 = 0, \\ x + y = 0 . \end{matrix}

$\begin{matrix} y - 3 = 0, \\ x + y = 0 . \end{matrix}$

Solving, we obtain the vertex $(- 3, 3)$ $(- 3, 3)$ .

Now Try Exercise 55.

Applications: Linear Programming

In many applications, we want to find a maximum value or a minimum value. In business, for example, we might want to maximize profit and minimize cost. Linear programming can tell us how to do this.

In our study of linear programming, we will consider linear functions of two variables that are to be maximized or minimized subject to several conditions, or constraints. These constraints are expressed as inequalities. The solution set of the system of inequalities made up of the constraints contains all the feasible solutions of a linear programming problem. The function that we want to maximize or minimize is called the objective function.

It can be shown that the maximum and minimum values of the objective function occur at a vertex of the region of feasible solutions. Thus we have the following procedure.

Example 7

Maximizing Profit Dovetail Carpentry Shop makes bookcases and desks. Each bookcase requires 5 hr of woodworking and 4 hr of finishing. Each desk requires 10 hr of woodworking and 3 hr of finishing. Each month the shop has 600 hr of labor available for woodworking and 240 hr for finishing. The profit on each bookcase is $40 and on each desk is $75. How many of each product should be made each month in order to maximize profit? What is the maximum profit?

Solution

We let x = the number of bookcases to be produced and y = the number of desks. Then the profit P is given by the function

\begin{matrix} P = 40 x + 75 y. & To emphasize that P is a function of two variables, we sometimes write P (x, y) = 40 x + 75 y. \end{matrix}

$\begin{matrix} P = 40 x + 75 y. & To emphasize that P is a function of two variables, we sometimes write P (x, y) = 40 x + 75 y. \end{matrix}$

We know that x bookcases require 5x hr of woodworking and y desks require 10y hr of woodworking. Since there is no more than 600 hr of labor available for woodworking, we have one constraint:

5 x + 10 y \leq 600 .

$5 x + 10 y \leq 600 .$

Similarly, the bookcases and the desks require 4x hr and 3y hr of finishing, respectively. There is no more than 240 hr of labor available for finishing, so we have a second constraint:

4 x + 3 y \leq 240 .

$4 x + 3 y \leq 240 .$

We also know that $x \geq 0$ $x \geq 0$ and $y \geq 0$ $y \geq 0$ because the carpentry shop cannot make a negative number of either product.

Thus we want to maximize the objective function.

P = 40 x + 75 y

$P = 40 x + 75 y$

subject to the constraints

\begin{matrix} 5 x + 10 y & \leq & 600, \\ 4 x + 3 y & \leq & 240, \\ x & \geq & 0, \\ y & \geq & 0 . \end{matrix}

$\begin{matrix} 5 x + 10 y & \leq & 600, \\ 4 x + 3 y & \leq & 240, \\ x & \geq & 0, \\ y & \geq & 0 . \end{matrix}$

We graph the system of inequalities and determine the vertices, as shown in the figure at left.

Next, we evaluate the objective function P at each vertex.

The carpentry shop will make a maximum profit of $4560 when 24 bookcases and 48 desks are produced and sold.

Now Try Exercise 65.

9.7 Exercise Set

In Exercises 1–8, match the inequality with one of the graphs (a)–(h) that follow.

1. $y > x$ $y > x$
2. $y < - 2 x$ $y < - 2 x$
3. $y \leq x - 3$ $y \leq x - 3$
4. $y \geq x + 5$ $y \geq x + 5$
5. $2 x + y < 4$ $2 x + y < 4$
9. $3 x + y < - 6$ $3 x + y < - 6$
7. $2 x - 5 y > 10$ $2 x - 5 y > 10$
8. $3 x - 9 y < 9$ $3 x - 9 y < 9$

Graph.

9. $y > 2 x$ $y > 2 x$
10. $2 y < x$ $2 y < x$
11. $y + x \geq 0$ $y + x \geq 0$
12. $y - x < 0$ $y - x < 0$
13. $y > x - 3$ $y > x - 3$
14. $y \leq x + 4$ $y \leq x + 4$
15. $x + y < 4$ $x + y < 4$
16. $x - y \geq 5$ $x - y \geq 5$
17. $3 x - 2 y \leq 6$ $3 x - 2 y \leq 6$
18. $2 x - 5 y < 10$ $2 x - 5 y < 10$
19. $3 y + 2 x \geq 6$ $3 y + 2 x \geq 6$
20. $2 y + x \leq 4$ $2 y + x \leq 4$
21. $3 x - 2 \leq 5 x + y$ $3 x - 2 \leq 5 x + y$
22. $2 x - 6 y \geq 8 + 2 y$ $2 x - 6 y \geq 8 + 2 y$
23. $x < - 4$ $x < - 4$
24. $y > - 3$ $y > - 3$
25. $y \geq 5$ $y \geq 5$
26. $x \leq 5$ $x \leq 5$
27. $- 4 < y < - 1$ $- 4 < y < - 1$ (Hint: Think of this as $- 4 < y$ $- 4 < y$ and $y < - 1 .$ $y < - 1 .$ )
28. $- 3 \leq x \leq 3$ $- 3 \leq x \leq 3$
29. $y \geq | x |$ $y \geq | x |$
30. $y \leq | x + 2 |$ $y \leq | x + 2 |$

In Exercises 31–36, match the system of inequalities with one of the graphs (a)–(f) that follow.

31. $\begin{matrix} y > x + 1, \\ y \leq 2 - x \end{matrix}$ $\begin{matrix} y > x + 1, \\ y \leq 2 - x \end{matrix}$
32. $\begin{matrix} y < x - 3, \\ y \geq 4 - x \end{matrix}$ $\begin{matrix} y < x - 3, \\ y \geq 4 - x \end{matrix}$
33. $\begin{matrix} 2 x + y < 4, \\ 4 x + 2 y > 12 \end{matrix}$ $\begin{matrix} 2 x + y < 4, \\ 4 x + 2 y > 12 \end{matrix}$
34. $\begin{matrix} x \leq 5, \\ y \geq 1 \end{matrix}$ $\begin{matrix} x \leq 5, \\ y \geq 1 \end{matrix}$
35. $\begin{matrix} x + y & \leq & 4, \\ x - y & \geq & - 3, \\ x & \geq & 0, \\ y & \geq & 0 \end{matrix}$ $\begin{matrix} x + y & \leq & 4, \\ x - y & \geq & - 3, \\ x & \geq & 0, \\ y & \geq & 0 \end{matrix}$
36. $\begin{matrix} x - y & \geq & - 2, \\ x + y & \leq & 6, \\ x & \geq & 0, \\ y & \geq & 0 \end{matrix}$ $\begin{matrix} x - y & \geq & - 2, \\ x + y & \leq & 6, \\ x & \geq & 0, \\ y & \geq & 0 \end{matrix}$

Find a system of inequalities with the given graph. Answers may vary.

Graph the system of inequalities. Then find the coordinates of the vertices.

43. $\begin{matrix} y \leq x, \\ y \geq 3 - x \end{matrix}$ $\begin{matrix} y \leq x, \\ y \geq 3 - x \end{matrix}$
44. $\begin{matrix} y \leq x, \\ y \geq 5 - x \end{matrix}$ $\begin{matrix} y \leq x, \\ y \geq 5 - x \end{matrix}$
45. $\begin{matrix} y \geq x, \\ y \leq 4 - x \end{matrix}$ $\begin{matrix} y \geq x, \\ y \leq 4 - x \end{matrix}$
46. $\begin{matrix} y \geq x, \\ y \leq 2 - x \end{matrix}$ $\begin{matrix} y \geq x, \\ y \leq 2 - x \end{matrix}$
47. $\begin{matrix} y \geq - 3, \\ x \geq 1 \end{matrix}$ $\begin{matrix} y \geq - 3, \\ x \geq 1 \end{matrix}$
48. $\begin{matrix} y \leq - 2, \\ x \geq 2 \end{matrix}$ $\begin{matrix} y \leq - 2, \\ x \geq 2 \end{matrix}$
49. $\begin{matrix} x \leq 3, \\ y \geq 2 - 3 x \end{matrix}$ $\begin{matrix} x \leq 3, \\ y \geq 2 - 3 x \end{matrix}$
50. $\begin{matrix} x \geq - 2, \\ y \leq 3 - 2 x \end{matrix}$ $\begin{matrix} x \geq - 2, \\ y \leq 3 - 2 x \end{matrix}$
51. $\begin{matrix} x + y \leq 1, \\ x - y \leq 2 \end{matrix}$ $\begin{matrix} x + y \leq 1, \\ x - y \leq 2 \end{matrix}$
52. $\begin{matrix} y + 3 x \geq 0, \\ y + 3 x \leq 2 \end{matrix}$ $\begin{matrix} y + 3 x \geq 0, \\ y + 3 x \leq 2 \end{matrix}$
53. $\begin{matrix} 2 y - x \leq 2, \\ y + 3 x \geq - 1 \end{matrix}$ $\begin{matrix} 2 y - x \leq 2, \\ y + 3 x \geq - 1 \end{matrix}$
54. $\begin{matrix} y & \leq & 2 x + 1, \\ y & \geq & - 2 x + 1, \\ x - 2 & \leq & 0 \end{matrix}$ $\begin{matrix} y & \leq & 2 x + 1, \\ y & \geq & - 2 x + 1, \\ x - 2 & \leq & 0 \end{matrix}$
55. $\begin{matrix} x - y & \leq & 2, \\ x + 2 y & \geq & 8, \\ y - 4 & \leq & 0 \end{matrix}$ $\begin{matrix} x - y & \leq & 2, \\ x + 2 y & \geq & 8, \\ y - 4 & \leq & 0 \end{matrix}$
56. $\begin{matrix} x + 2 y & \leq & 12, \\ 2 x + y & \leq & 12, \\ x & \geq & 0, \\ y & \geq & 0 \end{matrix}$ $\begin{matrix} x + 2 y & \leq & 12, \\ 2 x + y & \leq & 12, \\ x & \geq & 0, \\ y & \geq & 0 \end{matrix}$
57. $\begin{matrix} 4 y - 3 x & \geq & - 12, \\ 4 y + 3 x & \geq & - 36, \\ y & \leq & 0, \\ x & \leq & 0 \end{matrix}$ $\begin{matrix} 4 y - 3 x & \geq & - 12, \\ 4 y + 3 x & \geq & - 36, \\ y & \leq & 0, \\ x & \leq & 0 \end{matrix}$
58. $\begin{matrix} 8 x + 5 y & \leq & 40, \\ x + 2 y & \leq & 8, \\ x & \geq & 0, \\ y & \geq & 0 \end{matrix}$ $\begin{matrix} 8 x + 5 y & \leq & 40, \\ x + 2 y & \leq & 8, \\ x & \geq & 0, \\ y & \geq & 0 \end{matrix}$
59. $\begin{matrix} 3 x + 4 y & \geq & 12, \\ 5 x + 6 y & \leq & 30, \\ 1 \leq x & \leq & 3 \end{matrix}$ $\begin{matrix} 3 x + 4 y & \geq & 12, \\ 5 x + 6 y & \leq & 30, \\ 1 \leq x & \leq & 3 \end{matrix}$
60. $\begin{matrix} y - x & \geq & 1, \\ y - x & \leq & 3, \\ 2 \leq x & \leq & 5 \end{matrix}$ $\begin{matrix} y - x & \geq & 1, \\ y - x & \leq & 3, \\ 2 \leq x & \leq & 5 \end{matrix}$

Find the maximum value and the minimum value of the function and the values of x and y for which they occur.

61. $\begin{matrix} P = 17 x - 3 y + 60, subject to \\ 6 x + 8 y \leq 48, \\ 0 \leq y \leq 4, \\ 0 \leq x \leq 7 . \end{matrix}$ $\begin{matrix} P = 17 x - 3 y + 60, subject to \\ 6 x + 8 y \leq 48, \\ 0 \leq y \leq 4, \\ 0 \leq x \leq 7 . \end{matrix}$
62. $\begin{matrix} Q = 28 x - 4 y + 72, subject to \\ 5 x + 4 y \geq 20, \\ 0 \leq y \leq 4, \\ 0 \leq x \leq 3 . \end{matrix}$ $\begin{matrix} Q = 28 x - 4 y + 72, subject to \\ 5 x + 4 y \geq 20, \\ 0 \leq y \leq 4, \\ 0 \leq x \leq 3 . \end{matrix}$
63. $\begin{matrix} F = 5 x + 36 y, subject to \end{matrix}$ $\begin{matrix} F = 5 x + 36 y, subject to \end{matrix}$

$\begin{matrix} 5 x + 3 y & \leq & 34, \\ 3 x + 5 y & \leq & 30, \\ x & \geq & 0, \\ y & \geq & 0 . \end{matrix}$ $\begin{matrix} 5 x + 3 y & \leq & 34, \\ 3 x + 5 y & \leq & 30, \\ x & \geq & 0, \\ y & \geq & 0 . \end{matrix}$
64. $G = 16 x + 14 y$ $G = 16 x + 14 y$ , subject to

$\begin{matrix} 3 x + 2 y & \leq & 12, \\ 7 x + 5 y & \leq & 29, \\ x & \geq & 0, \\ y & \geq & 0 . \end{matrix}$ $\begin{matrix} 3 x + 2 y & \leq & 12, \\ 7 x + 5 y & \leq & 29, \\ x & \geq & 0, \\ y & \geq & 0 . \end{matrix}$
65. Maximizing Mileage. Jazmin owns a pickup truck and a moped. She can afford 12 gal of gasoline to be split between the truck and the moped. Jazmin’s truck gets 20 mpg and, with the fuel currently in the tank, can hold at most an additional 10 gal of gas. Her moped gets 100 mpg and can hold at most 3 gal of gas. How many gallons of gasoline should each vehicle use if Jazmin wants to travel as far as possible on the 12 gal of gas? What is the maximum number of miles that she can travel?
66. Maximizing Income. Golden Harvest Foods makes jumbo biscuits and regular biscuits. The oven can cook at most 200 biscuits per day. Each jumbo biscuit requires 2 oz of flour, each regular biscuit requires 1 oz of flour, and there are 300 oz of flour available. The income from each jumbo biscuit is $0.45 and from each regular biscuit is $0.30. How many of each size biscuit should be made in order to maximize income? What is the maximum income?
67. Maximizing Profit. Waterbrook Farm includes 240 acres of cropland. The farm owner wishes to plant this acreage in both corn and soybeans. The profit per acre in corn production is $325 and in soybeans is $180. A total of 320 hr of labor is available. Each acre of corn requires 2 hr of labor, whereas each acre of soybean requires 1 hr of labor. How should the land be divided between corn and soybeans in order to yield the maximum profit? What is the maximum profit?
68. Maximizing Profit. Norris Mill can convert logs into lumber and plywood. In a given week, the mill can turn out 400 units of production, of which at least 100 units of lumber and at least 150 units of plywood are required by regular customers. The profit is $25 per unit of lumber and $38 per unit of plywood. How many units of each should the mill produce in order to maximize the profit? What is the maximum profit?
69. Minimizing Cost. An animal feed to be mixed from soybean meal and oats must contain at least 120 lb of protein, 24 lb of fat, and 10 lb of mineral ash. Each 100-lb sack of soybean meal costs $20 and contains 50 lb of protein, 8 lb of fat, and 5 lb of mineral ash. Each 100-lb sack of oats costs $8 and contains 15 lb of protein, 5 lb of fat, and 1 lb of mineral ash. How many sacks of each should be used in order to satisfy the minimum requirements at minimum cost? What is the minimum cost?
70. Minimizing Cost. Suppose that in the preceding exercise the oats were replaced by alfalfa, which costs $10 per 100-lb sack and contains 20 lb of protein, 6 lb of fat, and 8 lb of mineral ash. How much of each would now be required in order to minimize the cost? What is the minimum cost?
71. Maximizing Income. Francisco is planning to invest up to $40,000 in corporate and municipal bonds. The least he is allowed to invest in corporate bonds is $6000, and he does not want to invest more than $22,000 in corporate bonds. He also does not want to invest more than $30,000 in municipal bonds. The interest is 3% on corporate bonds and $4 \frac{1}{4} %$ $4 \frac{1}{4} %$ on municipal bonds. This is simple interest for one year. How much should he invest in each type of bond in order to maximize his income? What is the maximum income?
72. Maximizing Income. Mila is planning to invest up to $22,000 in certificates of deposit at City Bank and People’s Bank. She wants to invest at least $2000 but no more than $14,000 at City Bank. People’s Bank does not insure more than a $15,000 investment, so she will invest no more than that in People’s Bank. The interest is $2 \frac{1}{2} %$ $2 \frac{1}{2} %$ at City Bank and $1 \frac{3}{4} %$ $1 \frac{3}{4} %$ at People’s Bank. This is simple interest for one year. How much should she invest in each bank in order to maximize her income? What is the maximum income?
73. Minimizing Transportation Cost. An airline with two types of airplanes, $P_{1}$ $P_{1}$ and $P_{2}$ $P_{2}$ , has contracted with a tour group to provide transportation for a minimum of 2000 first-class, 1500 tourist-class, and 2400 economy-class passengers. For a certain trip, airplane $P_{1}$ $P_{1}$ costs $12 thousand to operate and can accommodate 40 first-class, 40 tourist-class, and 120 economy-class passengers, whereas airplane $P_{2}$ $P_{2}$ costs $10 thousand to operate and can accommodate 80 first-class, 30 tourist-class, and 40 economy-class passengers. How many of each type of airplane should be used in order to minimize the operating cost? What is the minimum operating cost?
74. Minimizing Transportation Cost. Suppose that in the preceding exercise a new airplane $P_{3}$ $P_{3}$ becomes available, having an operating cost for the same trip of $15 thousand and accommodating 40 first-class, 40 tourist-class, and 80 economy-class passengers. If airplane $P_{1}$ $P_{1}$ were replaced by airplane $P_{3}$ $P_{3}$ , how many of $P_{2}$ $P_{2}$ and $P_{3}$ $P_{3}$ should be used in order to minimize the operating cost? What is the minimum operating cost?
75. Maximizing Profit. It takes Fena Tailoring 3 hr of cutting and 6 hr of sewing to make a tiered silk organza bridal dress. It takes 6 hr of cutting and 3 hr of sewing to make a lace sheath bridal dress. The shop has at most 27 hr per week available for cutting and at most 36 hr per week for sewing. The profit is $320 on an organza dress and $305 on a lace dress. How many of each kind of bridal dress should be made each week in order to maximize profit? What is the maximum profit?
76. Maximizing Profit. Cambridge Metal Works manufactures two sizes of gears. The smaller gear requires 4 hr of machining and 1 hr of polishing and yields a profit of $45. The larger gear requires 1 hr of machining and 1 hr of polishing and yields a profit of $30. The firm has available at most 24 hr per day for machining and 9 hr per day for polishing. How many of each type of gear should be produced each day in order to maximize profit? What is the maximum profit?
77. Minimizing Nutrition Cost. Suppose that it takes 12 units of carbohydrates and 6 units of protein to satisfy Jacob’s minimum weekly requirements. A particular type of meat contains 2 units of carbohydrates and 2 units of protein per pound. A particular cheese contains 3 units of carbohydrates and 1 unit of protein per pound. The meat costs $3.50 per pound and the cheese costs $4.60 per pound. How many pounds of each are needed in order to minimize the cost and still meet the minimum requirements? What is the minimum cost?
78. Minimizing Salary Cost. The Spring Hill school board is analyzing education costs for Hill Top School. It wants to hire teachers and teacher’s aides to make up a faculty that satisfies its needs at minimum cost. The average annual salary for a teacher is $53,000 and for a teacher’s aide is $23,600. The school building can accommodate a faculty of no more than 50 but needs at least 20 faculty members to function properly. The school must have at least 12 aides, but the number of teachers must be at least twice the number of aides in order to accommodate the expectations of the community. How many teachers and teacher’s aides should be hired in order to minimize salary costs? What is the minimum salary cost?
79. Maximizing Animal Support in a Forest. A certain area of forest is populated by two species of animal, which scientists refer to as A and B for simplicity. The forest supplies two kinds of food, referred to as $F_{1}$ $F_{1}$ and $F_{2} .$ $F_{2} .$ For one year, each member of species A requires 1 unit of $F_{1}$ $F_{1}$ and 0.5 unit of $F_{2} .$ $F_{2} .$ Each member of species B requires 0.2 unit of $F_{1}$ $F_{1}$ and 1 unit of $F_{2} .$ $F_{2} .$ The forest can normally supply at most 600 units of $F_{1}$ $F_{1}$ and 525 units of $F_{2}$ $F_{2}$ per year. What is the maximum total number of these animals that the forest can support?
80. Maximizing Animal Support in a Forest. Refer to Exercise 79. If there is a wet spring, then supplies of food increase to 1080 units of $F_{1}$ $F_{1}$ and 810 units of $F_{2} .$ $F_{2} .$ In this case, what is the maximum total number of these animals that the forest can support?

Skill Maintenance

Solve.

81. $- 5 \leq x + 2 < 4$ $- 5 \leq x + 2 < 4$ [1.6]
82. $| x - 3 | \geq 2$ $| x - 3 | \geq 2$ [3.5]
83. $x^{2} - 2 x \leq 3$ $x^{2} - 2 x \leq 3$ [4.6]
84. $\frac{x - 1}{x + 2} > 4$ $\frac{x - 1}{x + 2} > 4$ [4.6]

Synthesis

Graph the system of inequalities.

85. $\begin{matrix} y \geq x^{2} - 2, \\ y \leq 2 - x^{2} \end{matrix}$ $\begin{matrix} y \geq x^{2} - 2, \\ y \leq 2 - x^{2} \end{matrix}$
86. $\begin{matrix} y < x + 1, \\ y \geq x^{2} \end{matrix}$ $\begin{matrix} y < x + 1, \\ y \geq x^{2} \end{matrix}$

Graph the inequality.

87. $| x + y | \leq 1$ $| x + y | \leq 1$
88. $| x | + | y | \leq 1$ $| x | + | y | \leq 1$
89. $| x | > | y |$ $| x | > | y |$
90. $| x - y | > 0$ $| x - y | > 0$
91. Allocation of Resources. Comfort-by-Design Furniture produces chairs and sofas. Each chair requires 20 ft of wood, 1 lb of foam rubber, and 2 ${yd}^{2}$ ${yd}^{2}$ of fabric. Each sofa requires 100 ft of wood, 50 lb of foam rubber, and 20 ${yd}^{2}$ ${yd}^{2}$ of fabric. The manufacturer has in stock 1900 ft of wood, 500 lb of foam rubber, and 240 ${yd}^{2}$ ${yd}^{2}$ of fabric. The chairs can be sold for $200 each and the sofas for $750 each. How many of each should be produced in order to maximize income? What is the maximum income?

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Table of Contents for 9.7 Systems of Inequalities and Linear Programming

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Table of Contents for
9.7 Systems of Inequalities and Linear Programming