Question 1

Find the least common denominator (LCD).
-$\frac { 7 } { x ^ { 2 } + 10 x + 24 } , \frac { 4 } { x ^ { 2 } + 7 x + 12 }$

Accepted Answer

A)  $( x - 6 ) ( x - 4 ) ( x + 3 )$ 
B)  $( x + 4 ) ( x + 3 )$ 
C)  $( x + 6 ) ( x + 4 )$ 
D)  $( x + 6 ) ( x + 4 ) ( x + 3 )$ 
A)  $( x - 6 ) ( x - 4 ) ( x + 3 )$ 
B)  $( x + 4 ) ( x + 3 )$ 
C)  $( x + 6 ) ( x + 4 )$ 
D)  $( x + 6 ) ( x + 4 ) ( x + 3 )$

Question 2

Solve the equation.
-$\frac { 5 - a } { a } + \frac { 3 } { 4 } = \frac { 7 } { a }$

Accepted Answer

A)  4 
B)  8 
C)  $- 8$ 
D)  $- 4$ 
A)  4 
B)  8 
C)  $- 8$ 
D)  $- 4$

Question 3

Find the least common denominator (LCD).
-$\frac { 4 } { 12 x } , \frac { 6 } { 15 x ^ { 2 } }$

Accepted Answer

A)  $60 x$ 
B)  $60 \mathrm { x } ^ { 3 }$ 
C)  $60 x ^ { 2 }$ 
D)  60 
A)  $60 x$ 
B)  $60 \mathrm { x } ^ { 3 }$ 
C)  $60 x ^ { 2 }$ 
D)  60

Question 4

Simplify.
-$\frac { \frac { 3 } { a } + 3 } { \frac { 3 } { a } - 3 }$

Accepted Answer

A)  $3 - a ^ { 2 }$ 
B)  $\frac { 3 ( 1 + a ) } { 1 - a }$ 
C)  $\frac { 1 + a } { 1 - a }$ 
D)  $\frac { a ^ { 2 } } { 3 - a ^ { 2 } }$ 
A)  $3 - a ^ { 2 }$ 
B)  $\frac { 3 ( 1 + a ) } { 1 - a }$ 
C)  $\frac { 1 + a } { 1 - a }$ 
D)  $\frac { a ^ { 2 } } { 3 - a ^ { 2 } }$

Question 5

Solve.
-It is recommended that there be at least $11.1$ square feet of floor space in a classroom for every student in the class. Find the minimum floor space that 32 students require. Round to the nearest tenth if necessary.

Accepted Answer

A)  $288.3 \mathrm { sq } \mathrm { ft }$ 
B)  $355.2 \mathrm { sq } \mathrm { ft }$ 
C)  $11.1 \mathrm { sq } \mathrm { ft }$ 
D)  $34.7 \mathrm { sq } \mathrm { ft }$ 
A)  $288.3 \mathrm { sq } \mathrm { ft }$ 
B)  $355.2 \mathrm { sq } \mathrm { ft }$ 
C)  $11.1 \mathrm { sq } \mathrm { ft }$ 
D)  $34.7 \mathrm { sq } \mathrm { ft }$

Question 6

Perform the indicated operations. Simplify if possible.
-$\frac { x } { x ^ { 2 } - 25 } + \frac { 5 } { x + 5 } - \frac { 6 } { x }$

Accepted Answer

A)  $\frac { 6 x ^ { 2 } - 25 x + 150 } { x ( x + 5 ) ( x - 5 ) }$ 
B)  $\frac { 25 ( x + 6 ) } { x ( x + 5 ) ( x - 5 ) }$ 
C)  $\frac { 25 ( x - 6 ) } { ( x + 5 ) ( x - 5 ) }$ 
D)  $\frac { - 25 ( x - 6 ) } { x ( x + 5 ) ( x - 5 ) }$ 
A)  $\frac { 6 x ^ { 2 } - 25 x + 150 } { x ( x + 5 ) ( x - 5 ) }$ 
B)  $\frac { 25 ( x + 6 ) } { x ( x + 5 ) ( x - 5 ) }$ 
C)  $\frac { 25 ( x - 6 ) } { ( x + 5 ) ( x - 5 ) }$ 
D)  $\frac { - 25 ( x - 6 ) } { x ( x + 5 ) ( x - 5 ) }$

Question 7

Perform the indicated operation. Simplify if possible.
-$\frac { m + 8 } { m ^ { 2 } + 4 m - 5 } + \frac { 3 m + 2 } { m ^ { 2 } + 13 m + 40 }$

Accepted Answer

A)  $\frac { 4 m + 10 } { 2 m ^ { 2 } + 17 m + 35 }$ 
B)  $4 m + 10$ 
C)  $\frac { 4 m ^ { 2 } + 15 m + 62 } { ( m - 5 ) ( m + 1 ) ( m - 8 ) }$ 
D)  $\frac { 4 m ^ { 2 } + 15 m + 62 } { ( m + 5 ) ( m - 1 ) ( m + 8 ) }$ 
A)  $\frac { 4 m + 10 } { 2 m ^ { 2 } + 17 m + 35 }$ 
B)  $4 m + 10$ 
C)  $\frac { 4 m ^ { 2 } + 15 m + 62 } { ( m - 5 ) ( m + 1 ) ( m - 8 ) }$ 
D)  $\frac { 4 m ^ { 2 } + 15 m + 62 } { ( m + 5 ) ( m - 1 ) ( m + 8 ) }$

Question 8

Find the domain of the rational expression.
-$f ( x ) = \frac { 3 x - 5 } { x ^ { 2 } - 49 }$

Accepted Answer

A)  $\{ x \mid x$ is a real number and $x \neq 7 \}$ 
B)  $\{ x \mid x$ is a real number and $x \neq 49 \}$ 
C)  $\{ x \mid x$ is a real number and $x \neq 7 , x \neq - 7 \}$ 
D)  $\left\{ x \mid x \right.$ is a real number and $\left. x \neq \frac { 5 } { 3 } \right\}$ 
A)  $\{ x \mid x$ is a real number and $x \neq 7 \}$ 
B)  $\{ x \mid x$ is a real number and $x \neq 49 \}$ 
C)  $\{ x \mid x$ is a real number and $x \neq 7 , x \neq - 7 \}$ 
D)  $\left\{ x \mid x \right.$ is a real number and $\left. x \neq \frac { 5 } { 3 } \right\}$

Question 9

Find the product and simplify.
-$\frac { x ^ { 2 } - 16 x + 55 } { x ^ { 2 } - 16 x + 48 } \cdot \frac { x ^ { 2 } - 15 x + 36 } { x ^ { 2 } - 20 x + 99 }$

Accepted Answer

A)  $\frac { ( x - 5 ) ( x - 3 ) } { ( x - 4 ) ( x - 9 ) }$ 
B)  $\frac { ( x - 5 ) } { ( x - 9 ) }$ 
C)  $\frac { ( x + 5 ) ( x + 3 ) } { ( x + 4 ) ( x + 9 ) }$ 
D)  $\frac { \left( x ^ { 2 } - 16 x + 55 \right) \left( x ^ { 2 } - 15 x + 36 \right) } { \left( x ^ { 2 } - 16 x + 48 \right) \left( x ^ { 2 } - 20 x + 99 \right) }$ 
A)  $\frac { ( x - 5 ) ( x - 3 ) } { ( x - 4 ) ( x - 9 ) }$ 
B)  $\frac { ( x - 5 ) } { ( x - 9 ) }$ 
C)  $\frac { ( x + 5 ) ( x + 3 ) } { ( x + 4 ) ( x + 9 ) }$ 
D)  $\frac { \left( x ^ { 2 } - 16 x + 55 \right) \left( x ^ { 2 } - 15 x + 36 \right) } { \left( x ^ { 2 } - 16 x + 48 \right) \left( x ^ { 2 } - 20 x + 99 \right) }$

Question 10

Find the least common denominator (LCD).
-$\frac { 6 } { x ^ { 2 } - 4 x } , \frac { 5 } { x ^ { 2 } - 2 x - 8 }$

Accepted Answer

A)  $x ( x - 6 ) ( x + 2 )$ 
B)  $( x - 6 ) ^ { 2 }$ 
C)  $x ( x - 4 ) ( x + 2 )$ 
D)  $x ( x - 6 ) ^ { 2 }$ 
A)  $x ( x - 6 ) ( x + 2 )$ 
B)  $( x - 6 ) ^ { 2 }$ 
C)  $x ( x - 4 ) ( x + 2 )$ 
D)  $x ( x - 6 ) ^ { 2 }$

Question 11

Fill in the blank with one of the words or phrases listed below. Not all choices will be used. $$\begin{array} { l l l l } 
\text { least common denominator } & \text { simplifying } & \text { reciprocals } & \text { numerator } \
\text { cross products } & \text { ratio } & \text { proportion } & \text { denominator } \
\text { rational expression } & \text { domain } & \text { complex fraction } & \
\frac { - a } { b } & \frac { a } { - b } & \frac { - a } { - b } &
\end{array}$$
-A rational expression is undefined when the is 0.

Accepted Answer

A)  rational expression 
B)  complex fraction 
C)  denominator 
D)  numerator 
A)  rational expression 
B)  complex fraction 
C)  denominator 
D)  numerator

Question 12

Solve the equation.
-$\frac { 29 } { x } = 6 - \frac { 1 } { x }$

Accepted Answer

A)  $\frac { 29 } { 6 }$ 
B)  $\frac { 3 } { 14 }$ 
C)  5 
D)  4 
A)  $\frac { 29 } { 6 }$ 
B)  $\frac { 3 } { 14 }$ 
C)  5 
D)  4

Question 13

Solve.
-The scale on a map states that 1 centimeter corresponds to 30 kilometers. On the map, two cities are $27 \mathrm {~cm}$ apart. Find the actual distance.

Accepted Answer

A)  $81,000 \mathrm {~km}$ 
B)  $810 \mathrm {~km}$ 
C)  $81 \mathrm {~km}$ 
D)  $8100 \mathrm {~km}$ 
A)  $81,000 \mathrm {~km}$ 
B)  $810 \mathrm {~km}$ 
C)  $81 \mathrm {~km}$ 
D)  $8100 \mathrm {~km}$

Question 14

Perform the indicated operation. Simplify if possible.
-$\frac { 9 x - 8 } { x + 1 } + 1$

Accepted Answer

A)  $\frac { 8 x - 7 } { x + 1 }$ 
B)  $\frac { 9 x - 7 } { x + 1 }$ 
C)  $\frac { 10 x - 7 } { x + 1 }$ 
D)  $\frac { 9 x + 7 } { x + 1 }$ 
A)  $\frac { 8 x - 7 } { x + 1 }$ 
B)  $\frac { 9 x - 7 } { x + 1 }$ 
C)  $\frac { 10 x - 7 } { x + 1 }$ 
D)  $\frac { 9 x + 7 } { x + 1 }$

Question 15

Rewrite the rational expression as an equivalent rational expression with the given denominator.
-$\frac { - 8 ( x + 2 ) } { x ( x - 8 ) } = \frac { } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$

Accepted Answer

A)  $\frac { - 8 ( x + 2 ) ( x - 4 ) } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$ 
B)  $\frac { - 8 ( x + 2 ) ( x + 4 ) } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$ 
C)  $\frac { ( x + 2 ) ( x + 4 ) } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$ 
D)  $\frac { - 8 ( x + 4 ) } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$ 
A)  $\frac { - 8 ( x + 2 ) ( x - 4 ) } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$ 
B)  $\frac { - 8 ( x + 2 ) ( x + 4 ) } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$ 
C)  $\frac { ( x + 2 ) ( x + 4 ) } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$ 
D)  $\frac { - 8 ( x + 4 ) } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$

Question 16

Solve.
-In a race, Car A starts 1 mile behind Car B. Car A is traveling at 65 miles per hour, while Car B is traveling at 55 miles per hour. How long will it take for Car A to overtake Car B?

Accepted Answer

A)  $5 \frac { 1 } { 2 } \mathrm {~min}$ 
B)  $\frac { 1 } { 10 } \mathrm {~min}$ 
C)  $6 \mathrm {~min}$ 
D)  $33 \mathrm {~min}$ 
A)  $5 \frac { 1 } { 2 } \mathrm {~min}$ 
B)  $\frac { 1 } { 10 } \mathrm {~min}$ 
C)  $6 \mathrm {~min}$ 
D)  $33 \mathrm {~min}$

Question 17

Solve the problem.
-If $f ( x ) = \frac { x ^ { 3 } - 7 } { x ^ { 2 } - 3 }$, find $f ( 2 )$

Accepted Answer

A)  8 
B)  1 
C)  $\frac { 1 } { 4 }$ 
D)  $- 3$ 
A)  8 
B)  1 
C)  $\frac { 1 } { 4 }$ 
D)  $- 3$

Question 18

Perform the indicated operation. Simplify if possible.
-$\frac { 4 } { x } + \frac { 6 } { x - 7 }$

Accepted Answer

A)  $\frac { 28 x - 10 } { x ( x - 7 ) }$ 
B)  $\frac { 28 x - 10 } { x ( 7 - x ) }$ 
C)  $\frac { 10 x - 28 } { x ( 7 - x ) }$ 
D)  $\frac { 10 x - 28 } { x ( x - 7 ) }$ 
A)  $\frac { 28 x - 10 } { x ( x - 7 ) }$ 
B)  $\frac { 28 x - 10 } { x ( 7 - x ) }$ 
C)  $\frac { 10 x - 28 } { x ( 7 - x ) }$ 
D)  $\frac { 10 x - 28 } { x ( x - 7 ) }$

Question 19

Find the least common denominator (LCD).
-$\frac { 9 } { 6 x + 48 } , \frac { 8 } { x ^ { 2 } + 8 x }$

Accepted Answer

A)  $6 x ^ { 2 } + 48$ 
B)  $6 x + 8$ 
C)  $6 x ^ { 2 } + 8$ 
D)  $6 x ( x + 8 )$ 
A)  $6 x ^ { 2 } + 48$ 
B)  $6 x + 8$ 
C)  $6 x ^ { 2 } + 8$ 
D)  $6 x ( x + 8 )$

Question 20

Simplify the expression.
-$\frac { 8 x ^ { 2 } - 43 x + 15 } { x - 5 }$

Accepted Answer

A)  $8 x ^ { 2 } - 46$ 
B)  $\frac { 8 x ^ { 2 } - 43 x + 15 } { x - 5 }$ 
C)  $\frac { 1 } { x - 5 }$ 
D)  $8 x - 3$ 
A)  $8 x ^ { 2 } - 46$ 
B)  $\frac { 8 x ^ { 2 } - 43 x + 15 } { x - 5 }$ 
C)  $\frac { 1 } { x - 5 }$ 
D)  $8 x - 3$

Find the least common denominator (LCD). - $\frac { 7 } { x ^ { 2 } + 10 x + 24 } , \frac { 4 } { x ^ { 2 } + 7 x + 12 }$

Solve the equation. - $\frac { 5 - a } { a } + \frac { 3 } { 4 } = \frac { 7 } { a }$

Find the least common denominator (LCD). - $\frac { 4 } { 12 x } , \frac { 6 } { 15 x ^ { 2 } }$

Simplify. - $\frac { \frac { 3 } { a } + 3 } { \frac { 3 } { a } - 3 }$

Solve. -It is recommended that there be at least $11.1$ square feet of floor space in a classroom for every student in the class. Find the minimum floor space that 32 students require. Round to the nearest tenth if necessary.

Perform the indicated operations. Simplify if possible. - $\frac { x } { x ^ { 2 } - 25 } + \frac { 5 } { x + 5 } - \frac { 6 } { x }$

Perform the indicated operation. Simplify if possible. - $\frac { m + 8 } { m ^ { 2 } + 4 m - 5 } + \frac { 3 m + 2 } { m ^ { 2 } + 13 m + 40 }$

Find the domain of the rational expression. - $f ( x ) = \frac { 3 x - 5 } { x ^ { 2 } - 49 }$

Find the product and simplify. - $\frac { x ^ { 2 } - 16 x + 55 } { x ^ { 2 } - 16 x + 48 } \cdot \frac { x ^ { 2 } - 15 x + 36 } { x ^ { 2 } - 20 x + 99 }$

Find the least common denominator (LCD). - $\frac { 6 } { x ^ { 2 } - 4 x } , \frac { 5 } { x ^ { 2 } - 2 x - 8 }$

Fill in the blank with one of the words or phrases listed below. Not all choices will be used. least common denominator simplifying reciprocals numerator cross products ratio proportion denominator rational expression domain complex fraction -A rational expression is undefined when the is 0.

Solve the equation. - $\frac { 29 } { x } = 6 - \frac { 1 } { x }$

Solve. -The scale on a map states that 1 centimeter corresponds to 30 kilometers. On the map, two cities are $27 \mathrm {~cm}$ apart. Find the actual distance.

Perform the indicated operation. Simplify if possible. - $\frac { 9 x - 8 } { x + 1 } + 1$

Rewrite the rational expression as an equivalent rational expression with the given denominator. - $\frac { - 8 ( x + 2 ) } { x ( x - 8 ) } = \frac { } { x ^ { 3 } - 4 x ^ { 2 } - 32 x }$

Solve. -In a race, Car A starts 1 mile behind Car B. Car A is traveling at 65 miles per hour, while Car B is traveling at 55 miles per hour. How long will it take for Car A to overtake Car B?

Solve the problem. -If $f ( x ) = \frac { x ^ { 3 } - 7 } { x ^ { 2 } - 3 }$ , find $f ( 2 )$

Perform the indicated operation. Simplify if possible. - $\frac { 4 } { x } + \frac { 6 } { x - 7 }$

Find the least common denominator (LCD). - $\frac { 9 } { 6 x + 48 } , \frac { 8 } { x ^ { 2 } + 8 x }$

Simplify the expression. - $\frac { 8 x ^ { 2 } - 43 x + 15 } { x - 5 }$

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

Find the least common denominator (LCD). - 7x2+10x+24,4x2+7x+12\frac { 7 } { x ^ { 2 } + 10 x + 24 } , \frac { 4 } { x ^ { 2 } + 7 x + 12 }x2+10x+247​,x2+7x+124​

Solve the equation. - 5−aa+34=7a\frac { 5 - a } { a } + \frac { 3 } { 4 } = \frac { 7 } { a }a5−a​+43​=a7​

Find the least common denominator (LCD). - 412x,615x2\frac { 4 } { 12 x } , \frac { 6 } { 15 x ^ { 2 } }12x4​,15x26​

Simplify. - 3a+33a−3\frac { \frac { 3 } { a } + 3 } { \frac { 3 } { a } - 3 }a3​−3a3​+3​

Solve. -It is recommended that there be at least 11.111.111.1 square feet of floor space in a classroom for every student in the class. Find the minimum floor space that 32 students require. Round to the nearest tenth if necessary.

Perform the indicated operations. Simplify if possible. - xx2−25+5x+5−6x\frac { x } { x ^ { 2 } - 25 } + \frac { 5 } { x + 5 } - \frac { 6 } { x }x2−25x​+x+55​−x6​

Perform the indicated operation. Simplify if possible. - m+8m2+4m−5+3m+2m2+13m+40\frac { m + 8 } { m ^ { 2 } + 4 m - 5 } + \frac { 3 m + 2 } { m ^ { 2 } + 13 m + 40 }m2+4m−5m+8​+m2+13m+403m+2​

Find the domain of the rational expression. - f(x)=3x−5x2−49f ( x ) = \frac { 3 x - 5 } { x ^ { 2 } - 49 }f(x)=x2−493x−5​

Find the product and simplify. - x2−16x+55x2−16x+48⋅x2−15x+36x2−20x+99\frac { x ^ { 2 } - 16 x + 55 } { x ^ { 2 } - 16 x + 48 } \cdot \frac { x ^ { 2 } - 15 x + 36 } { x ^ { 2 } - 20 x + 99 }x2−16x+48x2−16x+55​⋅x2−20x+99x2−15x+36​

Find the least common denominator (LCD). - 6x2−4x,5x2−2x−8\frac { 6 } { x ^ { 2 } - 4 x } , \frac { 5 } { x ^ { 2 } - 2 x - 8 }x2−4x6​,x2−2x−85​