Deck 7: Linear Systems

Question 1

Let $B = \left[ \begin{array} { l l } - & 3 \\ 2 \end{array} \right] , C = \left[ \begin{array} { l l } 2 & 1 \end{array} \right]$ .
Perform the operation: $( B C ) ^ { 2 }$ .
a. $\left[ \begin{array} { c } - 16 \\ - 8 \end{array} \right]$

b. $\left[ \begin{array} { c c } 24 & 12 \\ - 16 & - 8 \end{array} \right]$

c. $\left[ \begin{array} { l l } 24 & 12 \end{array} \right]$
d. $[ 24 ]$
e. none of these

Answer 1

B

Question 2

Which fraction appears in the partial fraction decomposition of $\frac { 5 x ^ { 2 } + 17 x + 12 } { x ( x + 2 ) ^ { 2 } }$ ?
a. $\frac { 1 } { ( x + 2 ) ^ { 2 } }$
b. $\frac { 2 } { x }$
c. $\frac { 5 } { x }$
d. $\frac { 3 } { ( x + 2 ) ^ { 2 } }$
e. $\frac { 1 } { x + 2 }$

Answer 2

A

Question 3

Illustration shows three circles traced out by a figure skater during her performance.If the centers of the circles are the given distances apart, find the radius of each circle.
$$A = 10 , B = 14 , C = 20$$
a. $\quad r _ { 1 } = 2 , r _ { 2 } = 12 , r _ { 3 } = 8$
b. $\quad r _ { 1 } = 12 , r _ { 2 } = 2 , r _ { 3 } = 8$
c. $\quad r _ { 1 } = 8 , r _ { 2 } = 12 , r _ { 3 } = 2$
d. $\quad r _ { 1 } = 1 , r _ { 2 } = 8 , r _ { 3 } = 13$
e. $\quad r _ { 1 } = 2 , r _ { 2 } = 8 , r _ { 3 } = 12$

Answer 3

A

Question 4

Solve the system by the addition method, if possible. If a solution exists, give the value of y. $$\left\{ \begin{array} { l } \frac { x - 10 } { 5 } + \frac { y + 3 } { 3 } = - \frac { 37 } { 15 } \\\frac { x + 3 } { 3 } + \frac { y - 10 } { 7 } = - \frac { 17 } { 21 }\end{array} \right.$$
a. $- 5$
b. 1
$\begin{array} { l l } \text { c. } & - 3 \\ \text { d. } & 7 \end{array}$
d. 7
e. no solution

Question 5

Find $7 A$ .
$$A = \left[ \begin{array} { l l } 4 & 5 \\4 & 6\end{array} \right]$$
a. $\left[ \begin{array} { c c } 4 & 35 \\ 28 & 42 \end{array} \right]$
b. $\left[ \begin{array} { c c } 28 & 5 \\ 4 & 6 \end{array} \right]$
c. $\left[ \begin{array} { c c } 28 & 5 \\ 28 & 42 \end{array} \right]$
d. $\left[ \begin{array} { l l } 4 & 35 \\ 4 & 42 \end{array} \right]$
e. $\left[ \begin{array} { l l } 28 & 35 \\ 28 & 42 \end{array} \right]$

Question 6

What is element $a _ { 11 }$ of the inverse of the following matrix?
$$\left[ \begin{array} { c c } 8 & 1 \\1 & 16\end{array} \right]$$
a. $- \frac { 8 } { 127 }$
b. $\frac { 16 } { 127 }$
c. $- \frac { 16 } { 127 }$
d. $\quad \frac { 8 } { 127 }$
e. $\frac { 16 } { 65 }$

Question 7

Find the inverse of the matrix.
$$\left[ \begin{array} { c c } 2 & 1 \\- 1 & 16\end{array} \right]$$
a. $\left[ \begin{array} { c c } \frac { 16 } { 33 } & - \frac { 1 } { 33 } \\ - \frac { 1 } { 33 } & \frac { 2 } { 33 } \end{array} \right]$

b. $\left[ \begin{array} { c c } \frac { 16 } { 33 } & \frac { 1 } { 33 } \\ \frac { 1 } { 33 } & \frac { 2 } { 33 } \end{array} \right]$

c.
$$\left[ \begin{array} { c c } \frac { 16 } { 33 } & - \frac { 1 } { 33 } \\\frac { 1 } { 33 } & - \frac { 2 } { 33 }\end{array} \right]$$
d. $\left[ \begin{array} { c c } - \frac { 16 } { 33 } & - \frac { 1 } { 33 } \\ \frac { 1 } { 33 } & \frac { 2 } { 33 } \end{array} \right]$

e. $\left[ \begin{array} { c c } \frac { 16 } { 33 } & - \frac { 1 } { 33 } \\ \frac { 1 } { 33 } & \frac { 2 } { 33 } \end{array} \right]$

Question 8

Find the inverse of the matrix. $$\left[ \begin{array} { c c c } 4 & 1 & - 1 \\4 & 4 & - 1 \\- 1 & - 1 & 1\end{array} \right]$$
a. $\left[ \begin{array} { c c c } \frac { 1 } { 3 } & 1 & \frac { 1 } { 3 } \\ - \frac { 1 } { 3 } & \frac { 1 } { 3 } & 0 \\ - 5 & \frac { 1 } { 3 } & \frac { 4 } { 3 } \end{array} \right] \quad$

b. $\left[ \begin{array} { c c c } - \frac { 1 } { 3 } & 0 & \frac { 1 } { 3 } \\ \frac { 1 } { 3 } & \frac { 1 } { 3 } & 0 \\ 0 & \frac { 1 } { 3 } & \frac { 4 } { 3 } \end{array} \right]$


c. $\left[ \begin{array} { c c c } \frac { 1 } { 3 } & 0 & \frac { 1 } { 3 } \\ - \frac { 1 } { 3 } & \frac { 1 } { 3 } & 0 \\ 0 & \frac { 1 } { 3 } & \frac { 4 } { 3 } \end{array} \right]$

d. $\quad \left[ \begin{array} { c c c } \frac { 1 } { 3 } & 0 & \frac { 1 } { 3 } \\ \frac { 1 } { 3 } & - \frac { 1 } { 3 } & 0 \\ 0 & \frac { 1 } { 3 } & \frac { 4 } { 3 } \end{array} \right]$

e. $\left[ \begin{array} { c c c } - \frac { 1 } { 3 } & 0 & \frac { 1 } { 3 } \\ - \frac { 1 } { 3 } & \frac { 1 } { 3 } & 0 \\ 0 & \frac { 1 } { 3 } & \frac { 4 } { 3 } \end{array} \right]$

Question 9

Find the graph of the linear inequality. $$5 y \geq 2 x - 10$$
a.

b.

c.


d.

Question 10

Evaluate the determinant. $\left| \begin{array} { c c } 8 & 10 \\ - 8 & 4 \end{array} \right|$
a. $\quad D = 2,560$
b. $\quad D = 112$
c. $\quad D = - 117$
d. $\quad D = - 112$
e. $\quad D = - 48$

Question 11

Solve the system by the addition method, if possible. If a solution exists, give the value of x. $$\left\{ \begin{array} { l } 4 ( x + y ) = y + 17 \\7 ( x + 1 ) = y + 18\end{array} \right.$$
$\begin{array} { l l } \text { a. } & 2 \\ \text { b. } & 5 \end{array}$
c. 4
d. $- 1$
e. no solution

Question 12

Solve the system by Gauss-Jordan elimination, if possible. If $( x , y )$ is the solution to the system, give the value of $x$ .
$$\left\{ \begin{array} { r } x - 2 y = 35 \\y = - 15\end{array} \right.$$
a. $- 6$
b. 5
c. 6
d. $- 5$
e. no solution

Question 13

Use Cramer's rule to find the solution of the system, if possible. $$\left\{ \begin{array} { c } x - y - z = 1 \\x + y + z = 19 \\- x - y + z = - 11\end{array} \right.$$
a. $x = 10 , y = 5 , z = 4$
b. $x = 5 , y = 10 , z = 4$
c. $x = 4 , y = 5 , z = 10$
d. The system is inconsistent.
e. The equations are dependent.

Question 14

Find the graph of the linear inequality. $$2 x - 4 y > 2$$
a.

b.

c.

$\mathrm { d }$ .

Question 15

Solve the system using Gaussian elimination. If $( x , y )$ is the solution to the system, give the value of $2 x - y$ .
$$\left\{ \begin{array} { c } x + y = 24 \\x - 2 y = - 3\end{array} \right.$$
a. $\quad 19$
b. $\quad 20$
c. 21
d. 24
e. $\quad 18$

Question 16

Decompose the fraction into partial fractions. $$\frac { 7 x ^ { 2 } + 6 x + 7 } { x ^ { 3 } + x }$$
a. $\frac { 7 } { x } - \frac { 6 } { x ^ { 2 } + 1 }$
b. $\frac { 6 } { x } - \frac { 7 } { x ^ { 2 } + 1 }$
c. $\frac { 7 } { x } + \frac { 6 } { x ^ { 2 } + 1 }$
d. $\frac { 6 } { x } + \frac { 7 } { x ^ { 2 } + 1 }$
e. none of these

Question 17

Decompose the fraction into partial fractions. $\frac { 7 x - 15 } { x ^ { 2 } + 6 x - 7 }$
a. $\quad \frac { 8 } { x + 7 } + \frac { 1 } { x - 2 }$
b. $\quad \frac { 8 } { x + 7 } - \frac { 1 } { x - 1 }$
c. $\quad \frac { 6 } { x - 7 } - \frac { 1 } { x + 1 }$
d. $\quad \frac { 6 } { x - 7 } + \frac { 1 } { x + 1 }$
e. none of these

Question 18

Use a graphing calculator to approximate the solutions of the system. $$\left\{ \begin{array} { c } 3.8 x + 3.9 y = 38.03 \\- 9 x + 4 y = 14.5\end{array} \right.$$
a. $\quad ( 7.9,1.9 )$
b. $\quad ( - 6,10.8 )$
c. $\quad ( - 1.9 , - 7.9 )$
d. $\quad ( 1.9,7.9 )$
e. $\quad ( 9.8 , - 5 )$

Question 19

Find values of x and y, if any, that will make the matrices equal. $$\left[ \begin{array} { l l } x & y \\1 & 6\end{array} \right] = \left[ \begin{array} { l l } 2 & 9 \\1 & 6\end{array} \right]$$
a. $\quad x = - 2 , y = 9$
b. $\quad x = 2 , y = 9$
c. $\quad x = - 2 , y = - 9$
d. $\quad x = 9 , y = 2$
e. no solution

Question 20

Find the inverse of the matrix. $$\left[ \begin{array} { c c } 1 & 2 \\- 2 & - 1\end{array} \right]$$
a. $\left[ \begin{array} { c c } - \frac { 1 } { 3 } & - \frac { 2 } { 3 } \\ \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right]$

b.
$\left[ \begin{array} { c c } - \frac { 1 } { 3 } & \frac { 2 } { 3 } \\ \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right]$

c.
$$\left[ \begin{array} { c c } \frac { 1 } { 3 } & - \frac { 2 } { 3 } \\\frac { 2 } { 3 } & \frac { 1 } { 3 }\end{array} \right]$$

d. $\left[ \begin{array} { c c } - \frac { 1 } { 3 } & - \frac { 2 } { 3 } \\ \frac { 2 } { 3 } & - \frac { 1 } { 3 } \end{array} \right]$

e. $\left[ \begin{array} { c c } - \frac { 1 } { 3 } & - \frac { 2 } { 3 } \\ - \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right]$

Question 21

Use Cramer's rule to find the solution of the system, if possible. $$\left\{ \begin{array} { c } x - y - z = 2 \\x + y + z = 18 \\- x - y + z = - 12\end{array} \right.$$
a. $\quad x = 3 , y = 5 , z = 10$
b. $\quad x = 5 , y = 10 , z = 3$
c. $x = 10 , y = 5 , z = 3$
d. The system is inconsistent.
e. The equations are dependent.

Question 22

Let $B = \left[ \begin{array} { l l } - & 3 \\ 2 \end{array} \right] , C = \left[ \begin{array} { l l } 4 & 1 \end{array} \right]$ .
Perform the operation: $( B C ) ^ { 2 }$ .
a.
$\left[ \begin{array} { l } - 80 \\ - 20 \end{array} \right]$

b. $[ 120 ]$

c. $\left[ \begin{array} { r r } 120 & 30 \\ - 80 & - 20 \end{array} \right]$

d. $\left[ \begin{array} { l l } 120 & 30 \end{array} \right]$

e. none of these

Question 23

Use a graphing calculator to approximate the solutions of the system. $$\left\{ \begin{array} { c } 4.2 x + 7.1 y = 39.35 \\- 5 x + 4 y = 6.7\end{array} \right.$$
a. $\quad ( 4.3,2.1 )$
b. $\quad ( - 2.1 , - 4.3 )$
c. $\quad ( - 2.2,7.4 )$
d. $\quad ( 6.4 , - 1.2 )$
e. $\quad ( 2.1,4.3 )$

Question 24

Two artists make winter yard ornaments. They get $\$ 97$ for each wooden snowman they make and $\$ 77$ for each wooden Santa Claus. On average, Nina must work 4 hours and Roberta 3 hours to make a snowman. Nina must work 2 hours and Roberta 4 hours to make a Santa Claus. If neither wishes to work more than 20 hours per week, how many of each ornament should they make each week to maximize their income? The information is summarized in the table below.


a. $\quad 4$ snowmen and 3 ornaments of Santa Claus: $\$ 19$
b. $\quad 3$ snowmen and 4 ornaments of Santa Claus: $\$ 19$
c. $\quad 2$ snowmen and 5 ornaments of Santa Claus: $\$ 23$
d. $\quad 5$ snowmen and no ornaments of Santa Claus: \$485
e. 4 snowmen and 2 ornaments of Santa Claus: $\$ 542$

Question 25

Minimize $P = 16 x + 2 y$ subject to the given constraints.
$\left\{\begin{aligned}x & \geq 0 \\y & \geq 0 \\x+2 y & \geq 8 \\2 x+y & \geq 8\end{aligned}\right.$

a. $\quad P = 72$ at $( 4,4 )$
b. $\quad P = 16$ at $( 0,8 )$
c. $\quad P = 8$ at $( 0,2 )$
d. $\quad P = - 32$ at $( - 2,0 )$
e. none of these

Question 26

Find the inverse of the matrix. $$\left[ \begin{array} { c c c } 32 & 1 & - 1 \\32 & 32 & - 1 \\- 1 & - 1 & 1\end{array} \right]$$
a. $\left[ \begin{array} { c c c } - \frac { 1 } { 31 } & 0 & \frac { 1 } { 31 } \\ \frac { 1 } { 31 } & \frac { 1 } { 31 } & 0 \\ 0 & \frac { 1 } { 31 } & \frac { 32 } { 31 } \end{array} \right]$

b. $\left[ \begin{array} { c c c } \frac { 1 } { 31 } & 0 & \frac { 1 } { 31 } \\ - \frac { 1 } { 31 } & \frac { 1 } { 31 } & 0 \\ 0 & \frac { 1 } { 31 } & \frac { 32 } { 31 } \end{array} \right]$

c.
$$\left[ \begin{array} { c c c } \frac { 1 } { 31 } & 0 & \frac { 1 } { 31 } \\\frac { 1 } { 31 } & - \frac { 1 } { 31 } & 0 \\0 & \frac { 1 } { 31 } & \frac { 32 } { 31 }\end{array} \right]$$

d. $\left[ \begin{array} { c c c } \frac { 1 } { 31 } & 1 & \frac { 1 } { 31 } \\ - \frac { 1 } { 31 } & \frac { 1 } { 31 } & 0 \\ 25 & \frac { 1 } { 31 } & \frac { 32 } { 31 } \end{array} \right]$

e. $\left[ \begin{array} { c c c } - \frac { 1 } { 31 } & 0 & \frac { 1 } { 31 } \\ - \frac { 1 } { 31 } & \frac { 1 } { 31 } & 0 \\ 0 & \frac { 1 } { 31 } & \frac { 32 } { 31 } \end{array} \right]$

Question 27

Solve the system by Gauss-Jordan elimination, if possible. If (x, y) is the solution to the system, give the value of x. $$\left\{ \begin{array} { r } x - 2 y = 14 \\y = - 2\end{array} \right.$$
a. 11
b. $- 10$
c. $- 11$
d. 10
e. no solution

Question 28

Find the inverse of the matrix. $$\left[ \begin{array} { c c } 2 & 1 \\- 1 & 8\end{array} \right]$$
a. $\left[ \begin{array} { c c } \frac { 8 } { 17 } & - \frac { 1 } { 17 } \\ - \frac { 1 } { 17 } & \frac { 2 } { 17 } \end{array} \right]$

b. $\left[ \begin{array} { c c } \frac { 8 } { 17 } & \frac { 1 } { 17 } \\ \frac { 1 } { 17 } & \frac { 2 } { 17 } \end{array} \right]$

c.
$$\left[ \begin{array} { c c } - \frac { 8 } { 17 } & - \frac { 1 } { 17 } \\\frac { 1 } { 17 } & \frac { 2 } { 17 }\end{array} \right]$$

d. $\left[ \begin{array} { c c } \frac { 8 } { 17 } & - \frac { 1 } { 17 } \\ \frac { 1 } { 17 } & \frac { 2 } { 17 } \end{array} \right]$

e. $\left[ \begin{array} { c c } \frac { 8 } { 17 } & - \frac { 1 } { 17 } \\ \frac { 1 } { 17 } & - \frac { 2 } { 17 } \end{array} \right]$

Question 29

Find the solution set of the system. $$\left\{ \begin{array} { l } y \leq - 4 \\x > - 4\end{array} \right.$$
$\mathrm { a } .$

b.

c.


d.

Question 30

Find $7 A$ .
$$A = \left[ \begin{array} { l l } 8 & 8 \\3 & 2\end{array} \right]$$
a. $\left[ \begin{array} { c c } 56 & 8 \\ 3 & 2 \end{array} \right]$

b.
$\left[ \begin{array} { c c } 56 & 8 \\ 21 & 14 \end{array} \right]$

c. $\left[ \begin{array} { c c } 8 & 56 \\ 21 & 14 \end{array} \right]$

d. $\left[ \begin{array} { c c } 8 & 56 \\ 3 & 14 \end{array} \right]$

e. $\left[ \begin{array} { l l } 56 & 56 \\ 21 & 14 \end{array} \right]$

Question 31

What is element $a _ { 11 }$ of the inverse of the following matrix?
$$\left[ \begin{array} { c c } 16 & 1 \\1 & 8\end{array} \right]$$
a. $\frac { 8 } { 257 }$
b. $\frac { 8 } { 127 }$
c. $- \frac { 16 } { 127 }$
d. $- \frac { 8 } { 127 }$
e. $\frac { 16 } { 127 }$

Question 32

Illustration shows three circles traced out by a figure skater during her performance.If the centers of the circles are the given distances apart, find the radius of each circle.
$A = 10 , B = 18 , C = 20$
a. $\quad r _ { 1 } = 4 , r _ { 2 } = 6 , r _ { 3 } = 14$
b. $\quad r _ { 1 } = 3 , r _ { 2 } = 6 , r _ { 3 } = 15$
c. $\quad r _ { 1 } = 4 , r _ { 2 } = 14 , r _ { 3 } = 6$
d. $\quad r _ { 1 } = 14 , r _ { 2 } = 4 , r _ { 3 } = 6$
e. $\quad r _ { 1 } = 6 , r _ { 2 } = 14 , r _ { 3 } = 4$

Question 33

Find the inverse of the matrix. $$\left[ \begin{array} { c c } 1 & 4 \\- 4 & - 1\end{array} \right]$$
a. $\left[ \begin{array} { c c } \frac { 1 } { 15 } & - \frac { 4 } { 15 } \\ \frac { 4 } { 15 } & \frac { 1 } { 15 } \end{array} \right]$

b.
$\left[ \begin{array} { c c } - \frac { 1 } { 15 } & - \frac { 4 } { 15 } \\ - \frac { 4 } { 15 } & \frac { 1 } { 15 } \end{array} \right]$

c.
$\left[\begin{array}{cc}-\frac{1}{15} & \frac{4}{15} \\\frac{4}{15} & \frac{1}{15}\end{array}\right]$

d. $\left[ \begin{array} { c c } - \frac { 1 } { 15 } & - \frac { 4 } { 15 } \\ \frac { 4 } { 15 } & \frac { 1 } { 15 } \end{array} \right]$


e. $\left[ \begin{array} { c c } - \frac { 1 } { 15 } & - \frac { 4 } { 15 } \\ \frac { 4 } { 15 } & - \frac { 1 } { 15 } \end{array} \right]$

Question 34

Solve the system by the addition method, if possible. If a solution exists, give the value of y. $$\left\{ \begin{array} { l } \frac { x - 10 } { 5 } + \frac { y + 7 } { 3 } = \frac { 8 } { 15 } \\\frac { x + 7 } { 3 } + \frac { y - 10 } { 7 } = \frac { 52 } { 21 }\end{array} \right.$$
a. 10
b. 6
c. $- 3$
d. 4
e. no solution

Question 35

Solve the system using Gaussian elimination. If $( x , y )$ is the solution to the system, give the value of $2 x - y$ .
$$\left\{ \begin{array} { c } x + y = 23 \\x - 2 y = - 13\end{array} \right.$$
a. $\quad 10$
b. 13
c. $\quad 7$
d. $\quad 8$
e. $\quad 9$

Question 36

Evaluate the determinant. $\left| \begin{array} { c c } 8 & 10 \\ - 8 & 4 \end{array} \right|$
a. $\quad D = - 48$
b. $\quad D = 2,560$
c. $D = - 112$
d. $\quad D = - 117$
e. $\quad D = 112$

Question 37

Solve the system by the addition method, if possible. If a solution exists, give the value of x. $$\left\{ \begin{array} { c } 2 ( x + y ) = y - 19 \\7 ( x + 10 ) = y + 17\end{array} \right.$$
a. $- 5$
b. $- 11$
c. $- 2$
d. $- 8$
e. no solution

Question 38

Find the solution set of the system. $$\left\{ \begin{array} { l } y < 5 x + 4 \\y < - 4 x + 5\end{array} \right.$$
a.

b.

c.

d.

Question 39

Find values of x and y, if any, that will make the matrices equal. $$\left[ \begin{array} { l l } x & y \\3 & 9\end{array} \right] = \left[ \begin{array} { l l } 2 & 8 \\3 & 9\end{array} \right]$$
a. $x = 2 , y = 8$
b. $x = - 2 , y = 8$
c. $x = 8 , y = 2$
d. $x = - 2 , y = - 8$
e. no solution

Question 40

A diet requires at least 24 units of vitamin C and at least 26 units of vitamin B complex.Two food supplements are available that provide these nutrients, in the amounts and costs shown in the table
Below.How much of each should be used to minimize the cost?

a. $\quad 3$ grams of $A$ and 9 grams of $B : 33$ cents
b. 1 gram of $A$ and 5 grams of B: 27 cents
c. $\quad 5$ grams of $A$ and 7 grams of B: 39 cents
d. $\quad 7$ grams of $A$ and 5 grams of B: 45 cents
e. 5 grams of $A$ and 2 grams of B: 20 cents

Question 41

Two artists make winter yard ornaments.They get $83 for each wooden snowman they make and $74 for each wooden Santa Claus.On average, Nina must work 8 hours and Roberta 6 hours to make a
Snowman.Nina must work 4 hours and Roberta 8 hours to make a Santa Claus.If neither wishes to
Work more than 40 hours per week, how many of each ornament should they make each week to
Maximize their income? The information is summarized in the table below.

a. 4 snowmen and 3 ornaments of Santa Claus: $\$ 34$
b. $\quad 3$ snowmen and 4 ornaments of Santa Claus: $\$ 34$
c. 4 snowmen and 2 ornaments of Santa Claus: $\$ 480$
d. $\quad 2$ snowmen and 5 ornaments of Santa Claus: $\$ 46$
e. $\quad 5$ snowmen and no ornaments of Santa Claus: $\$ 415$

Question 42

Find the solution set of the system. $$\left\{ \begin{array} { l } y \leq - 4 \\x > - 4\end{array} \right.$$
a.

b.


C.


d.

Question 43

Find the graph of the linear inequality. $$3 y \geq 2 x - 9$$
a.

b.

C.


d.

Question 44

Find the solution set of the system. $$\left\{ \begin{array} { l } y < 7 x + 3 \\y < - 3 x + 7\end{array} \right.$$
a.

b.

c.


d.

Question 45

Decompose the fraction into partial fractions. $$\frac { 9 x ^ { 2 } + 8 x + 9 } { x ^ { 3 } + x }$$
a. $\quad \frac { 8 } { x } - \frac { 9 } { x ^ { 2 } + 1 }$
b. $\quad \frac { 9 } { x } - \frac { 8 } { x ^ { 2 } + 1 }$
c. $\frac { 8 } { x } + \frac { 9 } { x ^ { 2 } + 1 }$
d. $\frac { 9 } { x } + \frac { 8 } { x ^ { 2 } + 1 }$
e. none of these

Question 46

Decompose the fraction into partial fractions. $$\frac { 7 x - 11 } { x ^ { 2 } + 2 x - 3 }$$
a. $\quad \frac { 8 } { x + 3 } - \frac { 1 } { x - 1 }$
b. $\quad \frac { 8 } { x + 3 } + \frac { 1 } { x - 2 }$
c. $\quad \frac { 6 } { x - 3 } + \frac { 1 } { x + 1 }$
d. $\quad \frac { 6 } { x - 3 } - \frac { 1 } { x + 1 }$
e. none of these

Question 47

A diet requires at least 26 units of vitamin C and at least 31 units of vitamin B complex.Two food supplements are available that provide these nutrients, in the amounts and costs shown in the table
Below.How much of each should be used to minimize the cost?

a. $\quad 7$ grams of $A$ and 5 grams of B: 62 cents
b. 5 grams of A and 2 grams of B: 32 cents
c. 5 grams of A and 7 grams of B: 58 cents
d. $\quad 3$ grams of $A$ and 9 grams of B: 54 cents
e. 1 gram of $A$ and 5 grams of $B : 34$ cents

Question 48

Minimize $P = 18 x + 4 y$ subject to the given constraints.
$\left\{\begin{array}{rll}x & \geq & 0 \\y & \geq & 0 \\x+4 y & \geq & 9 \\4 x+y & \geq & 9\end{array}\right.$

a. $\quad P = 99$ at $( 4.5,4.5 )$
b. $\quad P = 9$ at $( 0,4 )$
c. $\quad P = - 72$ at $( - 4,0 )$
d. $\quad P = 36$ at $( 0,9 )$
e. none of these

Question 49

Which fraction appears in the partial fraction decomposition of $\frac { 5 x ^ { 2 } + 49 x + 108 } { x ( x + 6 ) ^ { 2 } }$ ?
a. $\quad \frac { 3 } { ( x + 6 ) ^ { 2 } }$
b. $\frac { 1 } { x + 6 }$
c. $\frac { 6 } { x }$
d. $\frac { 9 } { x }$
e. $\frac { 1 } { ( x + 6 ) ^ { 2 } }$

Question 50

Find the graph of the linear inequality. $$4 x - 2 y > 11$$
a.


b.

c.

d.