Question 1

Simplify the rational expression.
-$$\frac { 3 x ^ { 2 } - 5 x - 12 } { 9 x ^ { 3 } + 12 x ^ { 2 } + 3 x + 4 }$$

Accepted Answer

A)  0 
B)  $\frac { x - 3 } { 3 x ^ { 2 } + 1 }$ 
C)  $\frac { x + 3 } { 3 x ^ { 2 } - 1 }$ 
D)  $- \frac { 6 } { 3 x ^ { 3 } + 6 }$ 
A)  0 
B)  $\frac { x - 3 } { 3 x ^ { 2 } + 1 }$ 
C)  $\frac { x + 3 } { 3 x ^ { 2 } - 1 }$ 
D)  $- \frac { 6 } { 3 x ^ { 3 } + 6 }$ B

Question 2

Find the domain of the rational function.$f ( x ) = \frac { 9 x ^ { 2 } - 4 } { 5 x - 30 }$

Accepted Answer

A)  $\left\{ x \mid x \text { is a real number and } x \neq \frac { 2 } { 3 } , x \neq - \frac { 2 } { 3 } \right\}$ 
B)  $\{ x \mid x \text { is a real number and } x \neq 30 \}$ 
C)  $\{ x \mid x \text { is a real number and } x \neq 6 \}$ 
D)  $\{ x \mid x \text { is a real number }$ 
A)  $\left\{ x \mid x \text { is a real number and } x \neq \frac { 2 } { 3 } , x \neq - \frac { 2 } { 3 } \right\}$ 
B)  $\{ x \mid x \text { is a real number and } x \neq 30 \}$ 
C)  $\{ x \mid x \text { is a real number and } x \neq 6 \}$ 
D)  $\{ x \mid x \text { is a real number }$ C

Question 3

Perform the indicated operation. Simplify if possible.
-$$\frac { x } { 36 - x ^ { 2 } } + \frac { 2 } { 6 x - 36 }$$

Accepted Answer

A)  $\frac { - 4 x - 12 } { 6 ( x + 6 ) ( x - 6 ) }$ 
B)  $\frac { - 4 x + 12 } { 6 ( x + 6 ) ( x - 6 ) }$ 
C)  $\frac { 4 x - 12 } { 6 ( x + 6 ) ( x - 6 ) }$ 
D)  $\frac { x + 2 } { 6 ( x + 6 ) ( x - 6 ) }$ 
A)  $\frac { - 4 x - 12 } { 6 ( x + 6 ) ( x - 6 ) }$ 
B)  $\frac { - 4 x + 12 } { 6 ( x + 6 ) ( x - 6 ) }$ 
C)  $\frac { 4 x - 12 } { 6 ( x + 6 ) ( x - 6 ) }$ 
D)  $\frac { x + 2 } { 6 ( x + 6 ) ( x - 6 ) }$ B

Question 4

Perform the indicated operation. Simplify if possible.
-$$\frac { 2 } { x + 3 } - \frac { 4 } { x - 3 }$$

Accepted Answer

A)  $\frac { - 2 } { ( x + 3 ) ( x - 3 ) }$ 
B)  $\frac { - 2 x + 18 } { ( x + 3 ) ( x - 3 ) }$ 
C)  $\frac { - 2 x - 18 } { ( x + 3 ) ( x - 3 ) }$ 
D)  $\frac { - 2 x + 6 } { ( x + 3 ) ( x - 3 ) }$ 
A)  $\frac { - 2 } { ( x + 3 ) ( x - 3 ) }$ 
B)  $\frac { - 2 x + 18 } { ( x + 3 ) ( x - 3 ) }$ 
C)  $\frac { - 2 x - 18 } { ( x + 3 ) ( x - 3 ) }$ 
D)  $\frac { - 2 x + 6 } { ( x + 3 ) ( x - 3 ) }$

Question 5

Find the domain of the rational function.
$f ( x ) = \frac { 6 x - 5 } { 7 }$

Accepted Answer

A)  $\{ x \mid x \text { is a real number and } x \neq 7 \}$ 
B)  $\{ x \mid x \text { is a real number and } x \neq 0 \}$ 
C)  $\left\{ x \mid x \text { is a real number and } x \neq \frac { 5 } { 6 } \right\}$ 
D)  $\{ x \mid x \text { is a real number } \}$ 
A)  $\{ x \mid x \text { is a real number and } x \neq 7 \}$ 
B)  $\{ x \mid x \text { is a real number and } x \neq 0 \}$ 
C)  $\left\{ x \mid x \text { is a real number and } x \neq \frac { 5 } { 6 } \right\}$ 
D)  $\{ x \mid x \text { is a real number } \}$

Question 6

Solve the problem.
-A parallelogram has area $\frac { x ^ { 2 } - 12 x + 35 } { x }$ square feet and height $\frac { x ^ { 6 } } { x - 7 }$ feet. Express the length of its base as a ration expression in $x$.

Accepted Answer

A)  $\frac { x - 5 } { x ^ { 5 } } \mathrm { ft }$ 
B)  $\frac { ( x - 5 ) ( x - 7 ) ^ { 2 } } { x ^ { 7 } } \mathrm { ft }$ 
C)  $\frac { ( x - 5 ) ( x - 7 ) ^ { 2 } } { x ^ { 5 } } \mathrm { ft }$ 
D)  $x ^ { 5 } ( x - 5 ) f t$ 
A)  $\frac { x - 5 } { x ^ { 5 } } \mathrm { ft }$ 
B)  $\frac { ( x - 5 ) ( x - 7 ) ^ { 2 } } { x ^ { 7 } } \mathrm { ft }$ 
C)  $\frac { ( x - 5 ) ( x - 7 ) ^ { 2 } } { x ^ { 5 } } \mathrm { ft }$ 
D)  $x ^ { 5 } ( x - 5 ) f t$

Question 7

Perform the indicated operation. Simplify if possible.
-$$\frac { x } { x ^ { 2 } - 16 } - \frac { 8 } { x ^ { 2 } + 5 x + 4 }$$

Accepted Answer

A)  $\frac { x ^ { 2 } - 7 x + 32 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
B)  $\frac { x ^ { 2 } - 7 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
C)  $\frac { x ^ { 2 } + 7 x + 32 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
D)  $\frac { x ^ { 2 } - 7 x + 32 } { ( x - 4 ) ( x + 4 ) }$ 
A)  $\frac { x ^ { 2 } - 7 x + 32 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
B)  $\frac { x ^ { 2 } - 7 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
C)  $\frac { x ^ { 2 } + 7 x + 32 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
D)  $\frac { x ^ { 2 } - 7 x + 32 } { ( x - 4 ) ( x + 4 ) }$

Question 8

Simplify the rational expression.
-$$\frac { x ^ { 2 } - 9 } { 3 + x }$$

Accepted Answer

A)  $x + 3$ 
B)  0 
C)  $3 - x$ 
D)  $x - 3$ 
A)  $x + 3$ 
B)  0 
C)  $3 - x$ 
D)  $x - 3$

Question 9

Find the variation equation for the variation statement.
-$y$ varies directly as the square of $x ; y = - \frac { 64 } { 5 }$ when $x = 8$

Accepted Answer

A)  $y = \left( - \frac { 64 } { 5 } x \right) ^ { 2 }$ 
B)  $y = - \frac { 64 } { 5 } x$ 
C)  $y = \left( - \frac { 1 } { 5 } x \right) ^ { 2 }$ 
D)  $y = - \frac { 1 } { 5 } x ^ { 2 }$ 
A)  $y = \left( - \frac { 64 } { 5 } x \right) ^ { 2 }$ 
B)  $y = - \frac { 64 } { 5 } x$ 
C)  $y = \left( - \frac { 1 } { 5 } x \right) ^ { 2 }$ 
D)  $y = - \frac { 1 } { 5 } x ^ { 2 }$

Question 10

Solve.
-A painter can finish painting a house in 8 hours. Her assistant takes 10 hours to finish the same job. How long would it take for them to complete the job if they were working together?

Accepted Answer

A)  $\frac { 9 } { 40 } \mathrm { hr }$ 
B)  $9 \mathrm { hr }$ 
C)  $7 \mathrm { hr }$ 
D)  $4 \frac { 4 } { 9 } \mathrm { hr }$ 
A)  $\frac { 9 } { 40 } \mathrm { hr }$ 
B)  $9 \mathrm { hr }$ 
C)  $7 \mathrm { hr }$ 
D)  $4 \frac { 4 } { 9 } \mathrm { hr }$

Question 11

Perform the indicated operations.
-$$\frac { 3 x ^ { 2 } + 8 x + 4 } { y ^ { 2 } + 2 y - 3 } \cdot \frac { y ^ { 2 } + 3 y - 4 } { 12 x ^ { 2 } + 5 x - 2 } \div \frac { 3 x ^ { 2 } + 5 x - 2 } { 8 x ^ { 2 } + 10 x - 3 }$$

Accepted Answer

A)  1 
B)  $\frac { ( y + 4 ) } { ( y + 3 ) }$ 
C)  $\frac { 3 x + 1 } { 2 x - 3 }$ 
D)  $\frac { ( y + 4 ) ( 2 x + 3 ) } { ( y + 3 ) ( 3 x - 1 ) }$ 
A)  1 
B)  $\frac { ( y + 4 ) } { ( y + 3 ) }$ 
C)  $\frac { 3 x + 1 } { 2 x - 3 }$ 
D)  $\frac { ( y + 4 ) ( 2 x + 3 ) } { ( y + 3 ) ( 3 x - 1 ) }$

Question 12

Simplify.
-$$\frac { \frac { 3 } { x ^ { 3 } } - \frac { 5 } { x ^ { 2 } } } { \frac { 2 } { x ^ { 2 } } + 5 }$$

Accepted Answer

A)  $\frac { 3 - 5 x } { 2 x + 5 x ^ { 3 } }$ 
B)  $\frac { 3 + 5 x } { 2 x + 5 x ^ { 3 } }$ 
C)  $\frac { 3 + 5 x } { 2 x - 5 x ^ { 3 } }$ 
D)  $\frac { - 2 x } { 7 }$ 
A)  $\frac { 3 - 5 x } { 2 x + 5 x ^ { 3 } }$ 
B)  $\frac { 3 + 5 x } { 2 x + 5 x ^ { 3 } }$ 
C)  $\frac { 3 + 5 x } { 2 x - 5 x ^ { 3 } }$ 
D)  $\frac { - 2 x } { 7 }$

Question 13

Simplify the rational expression.
-$\frac { 12 x - 30 x ^ { 2 } } { 6 x }$

Accepted Answer

A)  1 
B)  0 
C)  $2 + 5 x$ 
D)  $2 - 5 x$ 
A)  1 
B)  0 
C)  $2 + 5 x$ 
D)  $2 - 5 x$

Question 14

Write an equation to describe the variation. Use k for the constant of proportionality.
-w varies directly as $x$ and inversely as $y$.

Accepted Answer

A)  $w + x - y = k$ 
B)  $w x y = k$ 
C)  $w = \frac { k x } { y }$ 
D)  $w = \frac { k y } { x }$ 
A)  $w + x - y = k$ 
B)  $w x y = k$ 
C)  $w = \frac { k x } { y }$ 
D)  $w = \frac { k y } { x }$

Question 15

Solve.
-The voltage across a resistor is jointly proportional to the resistance of the resistor and the current flowing through the resistor. If the voltage across a resistor is 72 volts for a resistor whose resistance is 9 ohms and when
The current flowing through the resistor is 8 amperes, find the voltage across a resistor whose resistance is
5 ohms and when the current flowing through the resistor is 2 amperes.

Accepted Answer

A)  16 volts 
B)  10 volts 
C)  40 volts 
D)  18 volts 
A)  16 volts 
B)  10 volts 
C)  40 volts 
D)  18 volts

Question 16

Solve the problem.
-A drug is injected into a patient and the concentration of the drug is monitored. The drug's concentration, C(t), in milligrams after $t$ hours is modeled by $\mathrm { C } ( \mathrm { t } ) = \frac { 4 \mathrm { t } } { 2 \mathrm { t } ^ { 2 } + 2 }$. Estimate the drug's concentration after 2 hours. (Round to the nearest hundredth.)

Accepted Answer

A)  $0.69$ milligrams 
B)  $1.33$ milligrams 
C)  $0.80$ milligrams 
D)  $1.22$ milligrams 
A)  $0.69$ milligrams 
B)  $1.33$ milligrams 
C)  $0.80$ milligrams 
D)  $1.22$ milligrams

Question 17

If y varies inversely as x, find the inverse variation equation for the situation.
-$$y = \frac { 1 } { 8 } \text { when } x = 56$$

Accepted Answer

A)  $y = \frac { 7 } { x }$ 
B)  $y = \frac { 1 } { 7 x }$ 
C)  $y = \frac { x } { 7 }$ 
D)  $y = \frac { 1 } { 448 } x$ 
A)  $y = \frac { 7 } { x }$ 
B)  $y = \frac { 1 } { 7 x }$ 
C)  $y = \frac { x } { 7 }$ 
D)  $y = \frac { 1 } { 448 } x$

Question 18

Fill in the blank with one of the words or phrases listed below.  
-The expressions (x - 5) and (5 - x) are called .

Accepted Answer

A)  expression 
B)  rational expression 
C)  opposites 
D)  complex fraction 
A)  expression 
B)  rational expression 
C)  opposites 
D)  complex fraction

Question 19

Simplify the rational expression.
-$\frac { 25 x ^ { 2 } - 15 x + 9 } { 875 x ^ { 3 } + 189 }$

Accepted Answer

A)  $\frac { 1 } { 7 ( 5 x - 3 ) }$ 
B)  $\frac { 7 } { 5 x - 3 }$ 
C)  $\frac { 7 } { 5 x + 3 }$ 
D)  $\frac { 1 } { 7 ( 5 x + 3 ) }$ 
A)  $\frac { 1 } { 7 ( 5 x - 3 ) }$ 
B)  $\frac { 7 } { 5 x - 3 }$ 
C)  $\frac { 7 } { 5 x + 3 }$ 
D)  $\frac { 1 } { 7 ( 5 x + 3 ) }$

Question 20

Simplify.
-$$\frac { \frac { 4 } { x - 6 } } { \frac { x } { x - 6 } }$$

Accepted Answer

A)  $\frac { x } { 4 }$ 
B)  $\frac { 4 } { x }$ 
C) $ = \frac { 4 } { x }$ 
D)  $\frac { 4 } { x ( x - 6 ) }$ 
A)  $\frac { x } { 4 }$ 
B)  $\frac { 4 } { x }$ 
C) $ = \frac { 4 } { x }$ 
D)  $\frac { 4 } { x ( x - 6 ) }$

Simplify the rational expression. - $\frac { 3 x ^ { 2 } - 5 x - 12 } { 9 x ^ { 3 } + 12 x ^ { 2 } + 3 x + 4 }$

Find the domain of the rational function. $f ( x ) = \frac { 9 x ^ { 2 } - 4 } { 5 x - 30 }$

Perform the indicated operation. Simplify if possible. - $\frac { x } { 36 - x ^ { 2 } } + \frac { 2 } { 6 x - 36 }$

Perform the indicated operation. Simplify if possible. - $\frac { 2 } { x + 3 } - \frac { 4 } { x - 3 }$

Find the domain of the rational function. $f ( x ) = \frac { 6 x - 5 } { 7 }$

Perform the indicated operation. Simplify if possible. - $\frac { x } { x ^ { 2 } - 16 } - \frac { 8 } { x ^ { 2 } + 5 x + 4 }$

Simplify the rational expression. - $\frac { x ^ { 2 } - 9 } { 3 + x }$

Find the variation equation for the variation statement. - $y$ varies directly as the square of $x ; y = - \frac { 64 } { 5 }$ when $x = 8$

Solve. -A painter can finish painting a house in 8 hours. Her assistant takes 10 hours to finish the same job. How long would it take for them to complete the job if they were working together?

Perform the indicated operations. - $\frac { 3 x ^ { 2 } + 8 x + 4 } { y ^ { 2 } + 2 y - 3 } \cdot \frac { y ^ { 2 } + 3 y - 4 } { 12 x ^ { 2 } + 5 x - 2 } \div \frac { 3 x ^ { 2 } + 5 x - 2 } { 8 x ^ { 2 } + 10 x - 3 }$

Simplify. - $\frac { \frac { 3 } { x ^ { 3 } } - \frac { 5 } { x ^ { 2 } } } { \frac { 2 } { x ^ { 2 } } + 5 }$

Simplify the rational expression. - $\frac { 12 x - 30 x ^ { 2 } } { 6 x }$

Write an equation to describe the variation. Use k for the constant of proportionality. -w varies directly as $x$ and inversely as $y$ .

If y varies inversely as x, find the inverse variation equation for the situation. - $y = \frac { 1 } { 8 } \text { when } x = 56$

Fill in the blank with one of the words or phrases listed below. -The expressions (x - 5) and (5 - x) are called .

Simplify the rational expression. - $\frac { 25 x ^ { 2 } - 15 x + 9 } { 875 x ^ { 3 } + 189 }$

Simplify. - $\frac { \frac { 4 } { x - 6 } } { \frac { x } { x - 6 } }$

Real Numbers and Algebraic Expressions

Equations, Inequalities, and Problem Solving

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Exam 6: Rational Expressions

Simplify the rational expression. - 3x2−5x−129x3+12x2+3x+4\frac { 3 x ^ { 2 } - 5 x - 12 } { 9 x ^ { 3 } + 12 x ^ { 2 } + 3 x + 4 }9x3+12x2+3x+43x2−5x−12​

Find the domain of the rational function. f(x)=9x2−45x−30f ( x ) = \frac { 9 x ^ { 2 } - 4 } { 5 x - 30 }f(x)=5x−309x2−4​

Perform the indicated operation. Simplify if possible. - x36−x2+26x−36\frac { x } { 36 - x ^ { 2 } } + \frac { 2 } { 6 x - 36 }36−x2x​+6x−362​

Perform the indicated operation. Simplify if possible. - 2x+3−4x−3\frac { 2 } { x + 3 } - \frac { 4 } { x - 3 }x+32​−x−34​

Find the domain of the rational function. f(x)=6x−57f ( x ) = \frac { 6 x - 5 } { 7 }f(x)=76x−5​

Solve the problem. -A parallelogram has area x2−12x+35x\frac { x ^ { 2 } - 12 x + 35 } { x }xx2−12x+35​ square feet and height x6x−7\frac { x ^ { 6 } } { x - 7 }x−7x6​ feet. Express the length of its base as a ration expression in xxx .

Perform the indicated operation. Simplify if possible. - xx2−16−8x2+5x+4\frac { x } { x ^ { 2 } - 16 } - \frac { 8 } { x ^ { 2 } + 5 x + 4 }x2−16x​−x2+5x+48​

Simplify the rational expression. - x2−93+x\frac { x ^ { 2 } - 9 } { 3 + x }3+xx2−9​

Find the variation equation for the variation statement. - yyy varies directly as the square of x;y=−645x ; y = - \frac { 64 } { 5 }x;y=−564​ when x=8x = 8x=8