Question 1

List four equivalent forms for the rational expression.
-$- \frac { 7 y - 2 } { y - 15 }$

Accepted Answer

A)  $\frac { - ( 7 y - 2 ) } { y - 15 } , \frac { - 7 y - 2 } { y - 15 } , \frac { 7 y - 2 } { - ( y - 15 ) } , \frac { 7 y - 2 } { - y - 15 }$ 
B)  $\frac { - ( 7 y - 2 ) } { y - 15 } , \frac { - 7 y + 2 } { y - 15 } , \frac { 7 y - 2 } { - ( y - 15 ) } , \frac { 7 y - 2 } { - y + 15 }$ 
C)  $\frac { - ( 7 y - 2 ) } { y - 15 } , \frac { - 7 y + 2 } { y - 15 } , \frac { 7 y - 2 } { - ( y - 15 ) } , \frac { 7 y - 2 } { - y - 15 }$ 
D)  $\frac { - ( 7 y - 2 ) } { y - 15 } , \frac { - 7 y - 2 } { y - 15 } , \frac { 7 y - 2 } { - ( y - 15 ) } , \frac { 7 y - 2 } { - y + 15 }$ 
A)  $\frac { - ( 7 y - 2 ) } { y - 15 } , \frac { - 7 y - 2 } { y - 15 } , \frac { 7 y - 2 } { - ( y - 15 ) } , \frac { 7 y - 2 } { - y - 15 }$ 
B)  $\frac { - ( 7 y - 2 ) } { y - 15 } , \frac { - 7 y + 2 } { y - 15 } , \frac { 7 y - 2 } { - ( y - 15 ) } , \frac { 7 y - 2 } { - y + 15 }$ 
C)  $\frac { - ( 7 y - 2 ) } { y - 15 } , \frac { - 7 y + 2 } { y - 15 } , \frac { 7 y - 2 } { - ( y - 15 ) } , \frac { 7 y - 2 } { - y - 15 }$ 
D)  $\frac { - ( 7 y - 2 ) } { y - 15 } , \frac { - 7 y - 2 } { y - 15 } , \frac { 7 y - 2 } { - ( y - 15 ) } , \frac { 7 y - 2 } { - y + 15 }$

Question 2

Find the least common denominator (LCD).
-$\frac { 3 x } { x - 20 } , \frac { 4 x } { 20 - x }$

Accepted Answer

A)  $12 x$ 
B)  $( x - 20 ) ( 20 - x )$ 
C)  $- 1$ 
D)  $x - 20$ or $20 - x$ 
A)  $12 x$ 
B)  $( x - 20 ) ( 20 - x )$ 
C)  $- 1$ 
D)  $x - 20$ or $20 - x$

Question 3

Solve the equation.
-$x - \frac { 10 } { x - 1 } = 6 - \frac { 4 x } { x - 1 }$

Accepted Answer

A)  1,4 
B)  $- 1,4$ 
C)  $- 4,1$ 
D)  $- 4 , - 1$ 
A)  1,4 
B)  $- 1,4$ 
C)  $- 4,1$ 
D)  $- 4 , - 1$

Question 4

Perform the indicated operation. Simplify if possible.
-$\frac { 6 } { x ^ { 2 } } - \frac { x } { 5 x + 1 }$

Accepted Answer

A)  $\frac { 6 + 30 x - x ^ { 3 } } { x ^ { 2 } ( 5 x + 1 ) }$ 
B)  $\frac { 30 x ^ { 3 } + 6 x ^ { 2 } } { x ^ { 2 } ( 5 x + 1 ) }$ 
C)  $\frac { 6 + 30 x - x ^ { 2 } } { x ( 5 x + 1 ) }$ 
D)  $\frac { 6 + 30 x - x ^ { 2 } } { x ^ { 2 } ( 5 x + 1 ) }$ 
A)  $\frac { 6 + 30 x - x ^ { 3 } } { x ^ { 2 } ( 5 x + 1 ) }$ 
B)  $\frac { 30 x ^ { 3 } + 6 x ^ { 2 } } { x ^ { 2 } ( 5 x + 1 ) }$ 
C)  $\frac { 6 + 30 x - x ^ { 2 } } { x ( 5 x + 1 ) }$ 
D)  $\frac { 6 + 30 x - x ^ { 2 } } { x ^ { 2 } ( 5 x + 1 ) }$

Question 5

Simplify.
-$\frac { \frac { x } { 3 } } { \frac { 4 } { x + 8 } }$

Accepted Answer

A)  $12 x ( x + 8 )$ 
B)  $\frac { 3 x } { 4 ( x + 8 ) }$ 
C)  $\frac { x ( x + 8 ) } { 12 }$ 
D)  $\frac { x + 8 } { 12 x }$ 
A)  $12 x ( x + 8 )$ 
B)  $\frac { 3 x } { 4 ( x + 8 ) }$ 
C)  $\frac { x ( x + 8 ) } { 12 }$ 
D)  $\frac { x + 8 } { 12 x }$

Question 6

Fill in the blank with one of the words or phrases listed below. Not all choices will be used. $$\begin{array} { l l l l } 
\text { least common denominator } & \text { simplifying } & \text { reciprocals } & \text { numerator } \
\text { cross products } & \text { ratio } & \text { proportion } & \text { denominator } \
\text { rational expression } & \text { domain } & \text { complex fraction } & \
\frac { - a } { b } & \frac { a } { - b } & \frac { - a } { - b } &
\end{array}$$
-For a rational expression, $ -\frac{\mathrm{a}}{\mathrm{b}}$= ____= ____

Accepted Answer

A)  $\frac { - \mathrm { a } } { \mathrm { b } } , \frac { - \mathrm { a } } { - \mathrm { b } }$ 
B)  numerator, denominator 
C)  $\frac { - a } { b } , \frac { a } { - b }$ 
D)  $\frac { - a } { - b } , \frac { a } { - b }$ 
A)  $\frac { - \mathrm { a } } { \mathrm { b } } , \frac { - \mathrm { a } } { - \mathrm { b } }$ 
B)  numerator, denominator 
C)  $\frac { - a } { b } , \frac { a } { - b }$ 
D)  $\frac { - a } { - b } , \frac { a } { - b }$

Question 7

Simplify.
-$\frac { \frac { 3 } { x } + 4 } { \frac { 9 } { x ^ { 2 } } - 16 }$

Accepted Answer

A)  $\frac { x } { 3 x - 4 }$ 
B)  $\frac { 1 } { 3 x - 4 }$ 
C)  $\frac { x } { 3 - 4 x }$ 
D)  $\frac { 1 } { 3 - 4 x }$ 
A)  $\frac { x } { 3 x - 4 }$ 
B)  $\frac { 1 } { 3 x - 4 }$ 
C)  $\frac { x } { 3 - 4 x }$ 
D)  $\frac { 1 } { 3 - 4 x }$

Question 8

Simplify.
-$\frac { \frac { 4 } { x } - \frac { 5 } { y } - \frac { 3 } { x y } } { 5 y ^ { - 1 } + 4 x ^ { - 3 } }$

Accepted Answer

A)  $- \frac { 4 x ^ { 2 } } { 5 x ^ { 3 } + 4 y }$ 
B)  $- \frac { 4 \left( x ^ { 3 } + y \right) } { 9 ( x - y - x y ) }$ 
C)  $\frac { 4 x ^ { 2 } y - 5 x ^ { 3 } - 3 x ^ { 2 } } { 5 x ^ { 3 } + 4 y }$ 
D)  $\frac { 80 x y - 100 x ^ { 2 } - 60 x } { 4 x ^ { 3 } + 5 y }$ 
A)  $- \frac { 4 x ^ { 2 } } { 5 x ^ { 3 } + 4 y }$ 
B)  $- \frac { 4 \left( x ^ { 3 } + y \right) } { 9 ( x - y - x y ) }$ 
C)  $\frac { 4 x ^ { 2 } y - 5 x ^ { 3 } - 3 x ^ { 2 } } { 5 x ^ { 3 } + 4 y }$ 
D)  $\frac { 80 x y - 100 x ^ { 2 } - 60 x } { 4 x ^ { 3 } + 5 y }$

Question 9

Perform the indicated operation. Simplify if possible.
-$\frac { 7 } { 8 x } + \frac { 1 } { 3 x }$

Accepted Answer

A)  $\frac { 8 } { 11 x }$ 
B)  $\frac { 29 } { 24 x ^ { 2 } }$ 
C)  $\frac { 29 } { 24 x }$ 
D)  $\frac { 59 } { 24 x }$ 
A)  $\frac { 8 } { 11 x }$ 
B)  $\frac { 29 } { 24 x ^ { 2 } }$ 
C)  $\frac { 29 } { 24 x }$ 
D)  $\frac { 59 } { 24 x }$

Question 10

Solve.
-A cyclist bikes at a constant speed for 24 miles. He then returns home at the same speed but takes a different route. His return trip takes one hour longer and is 29 miles. Find his speed.

Accepted Answer

A)  $6 \mathrm { mph }$ 
B)  $7 \mathrm { mph }$ 
C)  $4 \mathrm { mph }$ 
D)  $5 \mathrm { mph }$ 
A)  $6 \mathrm { mph }$ 
B)  $7 \mathrm { mph }$ 
C)  $4 \mathrm { mph }$ 
D)  $5 \mathrm { mph }$

Question 11

Solve the equation for the indicated variable.
-$A = \frac { b h } { 2 } \quad$ for $b$

Accepted Answer

A)  $b = \frac { h } { 2 A }$ 
B)  $b = \frac { A h } { 2 }$ 
C)  $b = \frac { 2 \mathrm {~A} } { \mathrm {~h} }$ 
D)  $b = \frac { A } { 2 h }$ 
A)  $b = \frac { h } { 2 A }$ 
B)  $b = \frac { A h } { 2 }$ 
C)  $b = \frac { 2 \mathrm {~A} } { \mathrm {~h} }$ 
D)  $b = \frac { A } { 2 h }$

Question 12

Find the quotient and simplify.
-$\frac { ( x + 4 ) } { ( x + 4 ) ( 4 x + 1 ) } \div \frac { 4 } { 4 x + 1 }$

Accepted Answer

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

Question 13

Solve.
-On a map of Nature's Wonder Hiking Trails, 1 centimeter corresponds to 3 miles. Find the length of a trail represented by a line that is $4 \frac { 1 } { 2 }$ centimeters long on the map.

Accepted Answer

A)  $85 \frac { 1 } { 2 } \mathrm { mi }$ 
B)  $\frac { 2 } { 3 } \mathrm { mi }$ 
C)  $13 \frac { 1 } { 2 } \mathrm { mi }$ 
D)  $1 \frac { 1 } { 2 } \mathrm { mi }$ 
A)  $85 \frac { 1 } { 2 } \mathrm { mi }$ 
B)  $\frac { 2 } { 3 } \mathrm { mi }$ 
C)  $13 \frac { 1 } { 2 } \mathrm { mi }$ 
D)  $1 \frac { 1 } { 2 } \mathrm { mi }$

Question 14

Rewrite the rational expression as an equivalent rational expression with the given denominator.
-$\frac { x + 14 } { 9 x ^ { 2 } + 67 x + 28 } = \frac { } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$

Accepted Answer

A)  $\frac { x ( x + 14 ) } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$ 
B)  $\frac { x ( x + 14 ) ( x + 10 ) } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$ 
C)  $\frac { x ( x + 14 ) ( 9 x + 4 ) ( x + 7 ) ( x + 10 ) } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$ 
D)  $\frac { ( x + 14 ) ( x + 10 ) } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$ 
A)  $\frac { x ( x + 14 ) } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$ 
B)  $\frac { x ( x + 14 ) ( x + 10 ) } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$ 
C)  $\frac { x ( x + 14 ) ( 9 x + 4 ) ( x + 7 ) ( x + 10 ) } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$ 
D)  $\frac { ( x + 14 ) ( x + 10 ) } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$

Question 15

Simplify.
-$\frac { \frac { 15 } { 4 x } } { \frac { 3 } { 16 x } }$

Accepted Answer

A)  $\frac { 1 } { 20 }$ 
B)  $\frac { x ^ { 2 } } { 20 }$ 
C)  $20 x ^ { 2 }$ 
D)  20 
A)  $\frac { 1 } { 20 }$ 
B)  $\frac { x ^ { 2 } } { 20 }$ 
C)  $20 x ^ { 2 }$ 
D)  20

Question 16

Find the product and simplify.
-$\frac { 2 y } { 4 y + 2 } \cdot \frac { 10 y + 5 } { 7 }$

Accepted Answer

A)  $\frac { y } { 7 }$ 
B)  $\frac { 5 y } { 14 }$ 
C)  $\frac { 5 } { 7 }$ 
D)  $\frac { 5 y } { 7 }$ 
A)  $\frac { y } { 7 }$ 
B)  $\frac { 5 y } { 14 }$ 
C)  $\frac { 5 } { 7 }$ 
D)  $\frac { 5 y } { 7 }$

Question 17

Solve the problem.
-A publishing company estimates the revenue from the sale of a popular book by the rational function $R ( x ) = \frac { 950 x ^ { 2 } } { x ^ { 2 } + 4 }$ where $x$ is the number of years since publication and $R ( x )$ is the total revenue in millions of dollars. Find the total revenue at the end of the first year.

Accepted Answer

A)  $\$ 230$ million 
B)  $\$ 150$ million 
C)  $250 million 
D)  $\$ 190$ million 
A)  $\$ 230$ million 
B)  $\$ 150$ million 
C)  $250 million 
D)  $\$ 190$ million

Question 18

Find the domain of the rational expression.
-$f ( x ) = \frac { x ^ { 2 } - 49 } { x ^ { 2 } - 11 x + 30 }$

Accepted Answer

A)  $\{ x \mid x$ is a real number and $x \neq 6 , x \neq 5 \}$ 
B)  $\{ x \mid x$ is a real number and $x \neq 0 \}$ 
C)  $\{ x \mid x$ is a real number and $x \neq 7 , x \neq - 7 \}$ 
D)  $\{ x \mid x$ is a real number and $x \neq - 6 , x \neq - 5 \}$ 
A)  $\{ x \mid x$ is a real number and $x \neq 6 , x \neq 5 \}$ 
B)  $\{ x \mid x$ is a real number and $x \neq 0 \}$ 
C)  $\{ x \mid x$ is a real number and $x \neq 7 , x \neq - 7 \}$ 
D)  $\{ x \mid x$ is a real number and $x \neq - 6 , x \neq - 5 \}$

Question 19

Simplify.
-$\frac { \frac { 7 } { y } } { \frac { 9 } { y + 6 } }$

Accepted Answer

A)  $\frac { 9 y } { 7 ( y + 6 ) }$ 
B)  $\frac { 7 ( \mathrm { y } + 6 ) } { 9 \mathrm { y } }$ 
C)  $\frac { y + 6 } { 63 y }$ 
D)  $63 y ( y + 6 )$ 
A)  $\frac { 9 y } { 7 ( y + 6 ) }$ 
B)  $\frac { 7 ( \mathrm { y } + 6 ) } { 9 \mathrm { y } }$ 
C)  $\frac { y + 6 } { 63 y }$ 
D)  $63 y ( y + 6 )$

Question 20

Find the product and simplify.
-$\frac { 7 x ^ { 8 } y ^ { 5 } } { 56 x y ^ { 11 } } \cdot y ^ { 6 }$

Accepted Answer

A)  $\frac { x ^ { 8 } } { 8 y }$ 
B)  $\frac { x ^ { 7 } y } { 8 }$ 
C)  $\frac { x ^ { 8 } } { 8 }$ 
D)  $\frac { x ^ { 7 } } { 8 }$ 
A)  $\frac { x ^ { 8 } } { 8 y }$ 
B)  $\frac { x ^ { 7 } y } { 8 }$ 
C)  $\frac { x ^ { 8 } } { 8 }$ 
D)  $\frac { x ^ { 7 } } { 8 }$

List four equivalent forms for the rational expression. - $- \frac { 7 y - 2 } { y - 15 }$

Find the least common denominator (LCD). - $\frac { 3 x } { x - 20 } , \frac { 4 x } { 20 - x }$

Solve the equation. - $x - \frac { 10 } { x - 1 } = 6 - \frac { 4 x } { x - 1 }$

Perform the indicated operation. Simplify if possible. - $\frac { 6 } { x ^ { 2 } } - \frac { x } { 5 x + 1 }$

Simplify. - $\frac { \frac { x } { 3 } } { \frac { 4 } { x + 8 } }$

Simplify. - $\frac { \frac { 3 } { x } + 4 } { \frac { 9 } { x ^ { 2 } } - 16 }$

Simplify. - $\frac { \frac { 4 } { x } - \frac { 5 } { y } - \frac { 3 } { x y } } { 5 y ^ { - 1 } + 4 x ^ { - 3 } }$

Perform the indicated operation. Simplify if possible. - $\frac { 7 } { 8 x } + \frac { 1 } { 3 x }$

Solve. -A cyclist bikes at a constant speed for 24 miles. He then returns home at the same speed but takes a different route. His return trip takes one hour longer and is 29 miles. Find his speed.

Solve the equation for the indicated variable. - $A = \frac { b h } { 2 } \quad$ for $b$

Find the quotient and simplify. - $\frac { ( x + 4 ) } { ( x + 4 ) ( 4 x + 1 ) } \div \frac { 4 } { 4 x + 1 }$

Solve. -On a map of Nature's Wonder Hiking Trails, 1 centimeter corresponds to 3 miles. Find the length of a trail represented by a line that is $4 \frac { 1 } { 2 }$ centimeters long on the map.

Rewrite the rational expression as an equivalent rational expression with the given denominator. - $\frac { x + 14 } { 9 x ^ { 2 } + 67 x + 28 } = \frac { } { x ( 9 x + 4 ) ( x + 10 ) ( x + 7 ) }$

Simplify. - $\frac { \frac { 15 } { 4 x } } { \frac { 3 } { 16 x } }$

Find the product and simplify. - $\frac { 2 y } { 4 y + 2 } \cdot \frac { 10 y + 5 } { 7 }$

Find the domain of the rational expression. - $f ( x ) = \frac { x ^ { 2 } - 49 } { x ^ { 2 } - 11 x + 30 }$

Simplify. - $\frac { \frac { 7 } { y } } { \frac { 9 } { y + 6 } }$

Find the product and simplify. - $\frac { 7 x ^ { 8 } y ^ { 5 } } { 56 x y ^ { 11 } } \cdot y ^ { 6 }$

Review of Real Numbers

Equations, Inequalities, and Problem Solving

Graphing

Solving Systems of Linear Equations

Exponents and Polynomials

Factoring Polynomials

More on Functions and Graphs

Inequalities and Absolute Value

Rational Exponents, Radicals, and Complex Numbers

Quadratic Equations and Functions

Exponential and Logarithmic Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Exam 7: Rational Expressions

List four equivalent forms for the rational expression. - −7y−2y−15- \frac { 7 y - 2 } { y - 15 }−y−157y−2​

Find the least common denominator (LCD). - 3xx−20,4x20−x\frac { 3 x } { x - 20 } , \frac { 4 x } { 20 - x }x−203x​,20−x4x​

Solve the equation. - x−10x−1=6−4xx−1x - \frac { 10 } { x - 1 } = 6 - \frac { 4 x } { x - 1 }x−x−110​=6−x−14x​

Perform the indicated operation. Simplify if possible. - 6x2−x5x+1\frac { 6 } { x ^ { 2 } } - \frac { x } { 5 x + 1 }x26​−5x+1x​

Simplify. - x34x+8\frac { \frac { x } { 3 } } { \frac { 4 } { x + 8 } }x+84​3x​​

Simplify. - 3x+49x2−16\frac { \frac { 3 } { x } + 4 } { \frac { 9 } { x ^ { 2 } } - 16 }x29​−16x3​+4​

Simplify. - 4x−5y−3xy5y−1+4x−3\frac { \frac { 4 } { x } - \frac { 5 } { y } - \frac { 3 } { x y } } { 5 y ^ { - 1 } + 4 x ^ { - 3 } }5y−1+4x−3x4​−y5​−xy3​​

Perform the indicated operation. Simplify if possible. - 78x+13x\frac { 7 } { 8 x } + \frac { 1 } { 3 x }8x7​+3x1​