Chapter 5

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{11 x-10}{(x-2)(x+1)}$$

In Exercises 1–26, graph each inequality. $$x+2 y \leq 8$$

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

In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(2,-1,3)\\\ &\left\\{\begin{array}{c} x+y+z=4 \\ x-2 y-z=1 \\ 2 x-y-2 z=-1 \end{array}\right. \end{aligned} $$

Determine whether the given ordered pair is a solution of the system. \((2,3)\) \(\left\\{\begin{array}{l}x+3 y=11 \\ x-5 y=-13\end{array}\right.\)

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x+7}{(x-1)(x+3)}$$

In Exercises 1–26, graph each inequality. $$3 x-6 y \leq 12$$

In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(5,-3,-2)\\\ &\left\\{\begin{array}{cc} x+y+z= & 0 \\ x+2 y-3 z= & 5 \\ 3 x+4 y+2 z= & -1 \end{array}\right. \end{aligned} $$

Determine whether the given ordered pair is a solution of the system. \((-3,5)\) \(\left\\{\begin{array}{l}9 x+7 y=8 \\ 8 x-9 y=-69\end{array}\right.\)

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{6 x^{2}-14 x-27}{(x+2)(x-3)^{2}}$$

In Exercises 1–26, graph each inequality. $$x-2 y>10$$

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

In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(4,1,2)\\\ &\left\\{\begin{aligned} x-2 y &=2 \\ 2 x+3 y &=11 \\ y-4 z &=-7 \end{aligned}\right. \end{aligned} $$

Determine whether the given ordered pair is a solution of the system. \((2,5)\) \(\left\\{\begin{array}{l}2 x+3 y=17 \\ x+4 y=16\end{array}\right.\)

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{3 x+16}{(x+1)(x-2)^{2}}$$

In Exercises 1–26, graph each inequality. $$2 x-y>4$$

Solve each system by the substitution method. $$\left\\{\begin{array}{l} 2 x+y--5 \\ y-x^{2}+6 x+7 \end{array}\right.$$

In Exercises \(1-4,\) determine if the given ordered triple is a solution of the system. $$ \begin{aligned} &(-1,3,2)\\\ &\left\\{\begin{array}{c} x-2 z=-5 \\ y-3 z=-3 \\ 2 x-z=-4 \end{array}\right. \end{aligned} $$

Determine whether the given ordered pair is a solution of the system. \(\\{8,5)\) \(\left\\{\begin{array}{l}5 x-4 y=20 \\ 3 y=2 x+1\end{array}\right.\)

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 $$\begin{aligned}&z=3 x+2 y\\\&\left\\{\begin{array}{c}x \geq 0, y \geq 0 \\\2 x+y \leq 8 \\\x+y \geq 4\end{array}\right.\end{aligned}$$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x^{2}-6 x+7}{(x-1)\left(x^{2}+1\right)}$$

In Exercises 1–26, graph each inequality. $$y \leq \frac{1}{3} x$$

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

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{l} x+y+2 z=11 \\ x+y+3 z=14 \\ x+2 y-z=5 \end{array}\right. $$

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 $$\begin{aligned}&z=2 x+3 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+y \leq 8 \\\2 x+3 y \leq 12\end{array}\right.\end{aligned}$$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{5 x^{2}-9 x+19}{(x-4)\left(x^{2}+5\right)}$$

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

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{c} 2 x+y-2 z=-1 \\ 3 x-3 y-z=5 \\ x-2 y+3 z=6 \end{array}\right. $$

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

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 $$\begin{aligned}&z=4 x+y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+3 y \leq 12 \\\x+y \geq 3\end{array}\right.\end{aligned}$$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{x^{3}+x^{2}}{\left(x^{2}+4\right)^{2}}$$

In Exercises 1–26, graph each inequality. $$y>2 x-1$$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}x+3 y=8 \\ y=2 x-9\end{array}\right.\)

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{c} 4 x-y+2 z=11 \\ x+2 y-z=-1 \\ 2 x+2 y-3 z=-1 \end{array}\right. $$

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 $$\begin{aligned}&z=x+6 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+y \leq 10 \\\x-2 y \geq-10\end{array}\right.\end{aligned}$$

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $$\frac{7 x^{2}-9 x+3}{\left(x^{2}+7\right)^{2}}$$

In Exercises 1–26, graph each inequality. $$y>3 x+2$$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}2 x-3 y=-13 \\ y=2 x+7\end{array}\right.\)

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{l} x-y+3 z=8 \\ 3 x+y-2 z=-2 \\ 2 x+4 y+z=0 \end{array}\right. $$

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 $$\begin{aligned}&z=3 x-2 y\\\&\left\\{\begin{array}{l}1 \leq x \leq 5 \\\y \geq 2 \\\x-y \geq-3\end{array}\right.\end{aligned}$$

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

In Exercises 1–26, graph each inequality. $$x \leq 1$$

Solve each system by the substitution method. \(\left\\{\begin{array}{l}x-4 y=2 \\ x=6 y+8\end{array}\right.\)

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{l} 3 x+2 y-3 z=-2 \\ 2 x-5 y+2 z=-2 \\ 4 x-3 y+4 z=10 \end{array}\right. $$

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 $$\begin{aligned}&z=5 x-2 y\\\&\left\\{\begin{array}{l}0 \leq x \leq 5 \\\0 \leq y \leq 3 \\\x+y \geq 2\end{array}\right.\end{aligned}$$

Solve each system by the substitution method. $$\left\\{\begin{array}{l} x y--12 \\ x-2 y+14-0 \end{array}\right.$$

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

In Exercises 1–26, graph each inequality. $$x \leq-3$$

Solve each system in Exercises \(5-18\). $$ \left\\{\begin{array}{l} 2 x+3 y+7 z=13 \\ 3 x+2 y-5 z=-22 \\ 5 x+7 y-3 z=-28 \end{array}\right. $$

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 $$\begin{aligned}&z-4 x+2 y\\\&\left\\{\begin{array}{l}x \geq 0, y \geq 0 \\\2 x+3 y \leq 12\end{array}\right.\\\&\begin{array}{l}3 x+2 y \leq 12 \\\x+y \geq 2\end{array}\end{aligned}$$