Chapter 5

Write the partial fraction decomposition of each rational expression. $$\frac{3 x+50}{(x-9)(x+2)}$$

In Exercises 1–26, graph each inequality. $$y>1$$

Solve each system by the substitution method. \(\left\\{\begin{array}{c}5 x+2 y=0 \\ x=3 y-0\end{array}\right.\)

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{c} 2 x-4 y+3 z=17 \\ x+2 y-z=0 \\ 4 x-y-z=6 \end{array}\right. $$

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned}&z=2 x+4 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\x+3 y \geq 6 \\\x+y \geq 3 \\\x+y \leq 9\end{array}\right.\end{aligned}$$

Write the partial fraction decomposition of each rational expression. $$\frac{5 x-1}{(x-2)(x+1)}$$

In Exercises 1–26, graph each inequality. $$y>-3$$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}4 x+3 y=0 \\ 2 x-y=0\end{array}\right.\)

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{c} x+\quad z=3 \\ x+2 y-z=1 \\ 2 x-y+z=3 \end{array}\right. $$

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned}&z=10 x+12 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\x+y \leq 7 \\\2 x+y \leq 10 \\\2 x+3 y \leq 18\end{array}\right.\end{aligned}$$

Write the partial fraction decomposition of each rational expression. $$\frac{7 x-4}{x^{2}-x-12}$$

In Exercises 1–26, graph each inequality. $$x^{2}+y^{2} \leq 1$$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}2 x+5 y=-4 \\ 3 x-y=11\end{array}\right.\)

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{c} 2 x+y=-2 \\ x+y-z=4 \\ 3 x+2 y+z=0 \end{array}\right. $$

In Exercises 5–14, an objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part ( \(b\) ) to determine the maximum value of the objective function and the values of \(x\) and \(y\) for which the maximum occurs. Objective Function Constraints $$\begin{aligned}&z=5 x+6 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+y \geq 10 \\\x+2 y \geq 10 \\\x+y \leq 10\end{array}\right.\end{aligned}$$

Write the partial fraction decomposition of each rational expression. $$\frac{9 x+21}{x^{2}+2 x-15}$$

In Exercises 1–26, graph each inequality. $$x^{2}+y^{2} \leq 4$$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}2 x+5 y=1 \\ -x+6 y=8\end{array}\right.\)

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is \(\$ 125\) for the rear-projection televisions and \(\$ 200\) for the plasma televisions. a. Let \(x=\) the number of rear-projection televisions manufactured in a month and let \(y=\) the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: \(\cdot\) Equipment in the factory allows for making at most 450 rear-projection televisions in one month. \(\cdot\) Equipment in the factory allows for making at most 200 plasma televisions in one month. \(\cdot\) The cost to the manufacturer per unit is \(\$ 600\) for the rear- projection televisions and \(\$ 900\) for the plasma televisions. Total monthly costs cannot exceed \(\$ 360,000\). Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at \((0,0),(0,200),(300,200),(450,100),\) and \((450,0) .]\) e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing- rear-projection televisions each month and maximum monthly profit is $\$$

Write the partial fraction decomposition of each rational expression. $$\frac{4}{2 x^{2}-5 x-3}$$

In Exercises 1–26, graph each inequality. $$x^{2}+y^{2}>25$$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}2 x-3 y=8-2 x \\ 3 x+4 y=x+3 y+14\end{array}\right.\)

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{ccc} x+y & = & -4 \\ y-z & = & 1 \\ 2 x+y+3 z & = & -21 \end{array}\right. $$

a. A student earns \(\$ 10\) per hour for tutoring and \(\$ 7\) per hour as a teacher's aide. Let \(x=\) the number of hours each week spent tutoring and let \(y=\) the number of hours each week spent as a teacher's aide. Write the objective function that models total weekly earnings. b. The student is bound by the following constraints: \(\cdot\) To have enough time for studies, the student can work no more than 20 hours per week. \(\cdot\) The tutoring center requires that each tutor spend at least three hours per week tutoring. \(\cdot\) The tutoring center requires that each tutor spend no more than eight hours per week tutoring. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because \(x\) and \(y\) are nonnegative. d. Evaluate the objective function for total weekly earnings at each of the four vertices of the graphed region. [The vertices should occur at \((3,0),(8,0),(3,17), \text { and }(8,12) .]\) Complete the missing portions of this statement: The student can earn the maximum amount per week by tutoring for hours _____ per week and working as a teacher’s aide for _____ hours per week. The maximum amount that the student can earn each week is $_____.

Write the partial fraction decomposition of each rational expression. $$\frac{x}{x^{2}+2 x-3}$$

In Exercises 1–26, graph each inequality. $$x^{2}+y^{2}>36$$

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{l} x+y=4 \\ x+z=4 \\ y+z=4 \end{array}\right. $$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}3 x-4 y=x-y+4 \\ 2 x+6 y=5 y-4\end{array}\right.\)

Use the two steps for solving a linear programming problem, given in the box on page 577 , to solve the problems in Exercises 17–23. A manufacturer produces two models of mountain bicycles. The times (in hours) required for assembling and painting each model are given in the following table: $$\begin{array}{lcc}\hline & \text { Model } A & \text { Model } B \\\\\hline \text { Assembling } & 5 & 4 \\\\\text { Painting } & 2 & 3\end{array}$$ The maximum total weekly hours available in the assembly department and the paint department are 200 hours and 108 hours, respectively. The profits per unit are \(25 for model A and \)15 for model B. How many of each type should be produced to maximize profit?

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}+13 x-9}{x(x-1)(x+3)}$$

In Exercises 1–26, graph each inequality. $$(x-2)^{2}+(y+1)^{2}<9$$

Solve each system by the substitution method. $$ \left\\{\begin{array}{l}y-\frac{1}{3} x+\frac{2}{3} \\ y-\frac{5}{7} x-2\end{array}\right. $$

Use the two steps for solving a linear programming problem, given in the box on page 577 , to solve the problems in Exercises 17–23. A large institution is preparing lunch menus containing foods A and B. The specifications for the two foods are given in the following table: $$\begin{array}{cccc}\hline & \text { Units of Fat } & \text { Units of } & \text { Units of } \\\\\text { Food } & \text { per Ounce } & \text { Carbohydrates } & \text { Protein } \\\\\hline \mathrm{A} & 1 & \text { per Ounce } & \text { per Ounce } \\\\\mathrm{B} & 1 & 1 & 1 \\\\\hline\end{array}$$ Each lunch must provide at least 6 units of fat per serving, no more than 7 units of protein, and at least 10 units of carbohydrates. The institution can purchase food A for \(\$ 0.12\) per ounce and food \(\mathrm{B}\) for \(\$ 0.08\) per ounce. How many ounces of each food should a serving contain to meet the dietary requirements at the least cost?

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}-5 x-15}{x(x+1)(x-5)}$$

In Exercises 1–26, graph each inequality. $$(x+2)^{2}+(y-1)^{2}<16$$

Solve each system by the substitution method. $$\left\\{\begin{array}{l} 2 x+y-4 \\ (x+1)^{2}+(y-2)^{2}-4 \end{array}\right.$$

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{aligned} 7 z-3 &=2(x-3 y) \\ 5 y+3 z-7 &=4 x \\ 4+5 z &=3(2 x-y) \end{aligned}\right. $$

Solve each system by the addition method. \(\left\\{\begin{array}{l}y--\frac{1}{2} x+2 \\ y-\frac{3}{4} x+7\end{array}\right.\)

Write the partial fraction decomposition of each rational expression. $$\frac{4 x^{2}-7 x-3}{x^{3}-x}$$

In Exercises 1–26, graph each inequality. $$y < x^{2}-1$$

In Exercises \(19-22,\) find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$ (-1,6),(1,4),(2,9) $$

Solve each system by the addition method. \(\left\\{\begin{array}{l}x+y-1 \\ x-y-3\end{array}\right.\)

On June \(24,1948,\) the former Soviet Union blocked all land and water routes through East Germany to Berlin. A gigantic airlift was organized using American and British planes to bring food, clothing, and other supplies to the more than 2 million people in West Berlin. The cargo capacity was \(30,000\) cubic feet for an American plane and \(20,000\) cubic feet for a British plane. To break the Soviet blockade, the Western Allies had to maximize cargo capacity but were subject to the following restrictions: \(\cdot\) No more than 44 planes could be used. "The larger American planes required 16 personnel per flight, double that of the requirement for the British planes. The total number of personnel available could not exceed 512 \(\cdot\) The cost of an American flight was \(\$ 9000\) and the cost of a British flight was \(\$ 5000 .\) Total weekly costs could not exceed \(\$ 300,000\) Find the number of American and British planes that were used to maximize cargo capacity.

Write the partial fraction decomposition of each rational expression. $$\frac{2 x^{2}-18 x-12}{x^{3}-4 x}$$

In Exercises 1–26, graph each inequality. $$y < x^{2}-9$$

In Exercises \(19-22,\) find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. $$ (-2,7),(1,-2),(2,3) $$

Solve each system by the addition method. \(\left\\{\begin{array}{l}x+y-6 \\ x-y--2\end{array}\right.\)

A theater is presenting a program for students and their parents on drinking and driving. The proceeds will be donated to a local alcohol information center. Admission is \(\$ 2.00\) for parents and \(\$ 1.00\) for students. However, the situation has two constraints: The theater can hold no more than 150 people and every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?

Write the partial fraction decomposition of each rational expression. $$\frac{6 x-11}{(x-1)^{2}}$$

In Exercises 1–26, graph each inequality. $$y \geq x^{2}-9$$