Question 1

Find the least common denominator (LCD).
-$\frac { 2 } { x ^ { 2 } - 10 x + 24 } , \frac { 4 } { 5 x - 30 }$

Accepted Answer

A)  $5 ( x + 4 ) ( x + 6 )$ 
B)  $5 ( x - 4 ) ( x - 6 )$ 
C)  $5 ( x + 4 ) ( x - 6 )$ 
D)  $5 ( x - 4 ) ( x + 6 )$ 
A)  $5 ( x + 4 ) ( x + 6 )$ 
B)  $5 ( x - 4 ) ( x - 6 )$ 
C)  $5 ( x + 4 ) ( x - 6 )$ 
D)  $5 ( x - 4 ) ( x + 6 )$

Question 2

Perform the indicated operation and simplify if possible.
-$\frac { x ^ { 2 } - 25 } { x ^ { 2 } + 5 x } \div \frac { x ^ { 2 } + 5 x + 1 } { 3 x - 18 }$

Accepted Answer

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

Question 3

$$\text { Find and simplify the difference quotient } \frac { f ( a + h ) - f ( a ) } { h } , h \neq 0 \text { for the given function. }$$
-$f ( x ) = \frac { 8 } { x ^ { 2 } }$

Accepted Answer

A)  $\frac { - 8 ( 2 a + h ) } { a ^ { 2 } \left( a ^ { 2 } + h \right) }$ 
B)  $\frac { 8 ( 2 a + h ) } { a ^ { 2 } ( a + h ) ^ { 2 } }$ 
C)  $\frac { 8 ( 2 a + h ) } { a ^ { 2 } \left( a ^ { 2 } + h \right) }$ 
D)  $\frac { - 8 ( 2 a + h ) } { a ^ { 2 } ( a + h ) ^ { 2 } }$ 
A)  $\frac { - 8 ( 2 a + h ) } { a ^ { 2 } \left( a ^ { 2 } + h \right) }$ 
B)  $\frac { 8 ( 2 a + h ) } { a ^ { 2 } ( a + h ) ^ { 2 } }$ 
C)  $\frac { 8 ( 2 a + h ) } { a ^ { 2 } \left( a ^ { 2 } + h \right) }$ 
D)  $\frac { - 8 ( 2 a + h ) } { a ^ { 2 } ( a + h ) ^ { 2 } }$

Question 4

Find the product and simplify.
-$\frac { x ^ { 2 } + 8 x } { 3 } \cdot \frac { 25 } { 5 x + 40 }$

Accepted Answer

A)  $\frac { x + 8 } { x + 5 }$ 
B)  $\frac { x ^ { 2 } + 8 x + 40 } { 75 }$ 
C)  $\frac { 5 x } { 3 }$ 
D)  $\frac { 25 x ^ { 2 } + 200 x } { 15 x + 120 }$ 
A)  $\frac { x + 8 } { x + 5 }$ 
B)  $\frac { x ^ { 2 } + 8 x + 40 } { 75 }$ 
C)  $\frac { 5 x } { 3 }$ 
D)  $\frac { 25 x ^ { 2 } + 200 x } { 15 x + 120 }$

Question 5

Find the domain of the rational expression.
-$f ( x ) = \frac { 1 - 8 x } { x ^ { 2 } - 4 x - 60 }$

Accepted Answer

A)  $\left\{ x \mid x \right.$ is a real number and $\left. x \neq 10 , x \neq - 6 , x \neq \frac { 1 } { 8 } , x \neq 0 \right\}$ 
B)  $\left\{ x \mid x \right.$ is a real number and $\left. x \neq 10 , x \neq - 6 , x \neq \frac { 1 } { 8 } \right\}$ 
C)  $\{ x \mid x$ is a real number and $x \neq 10 , x \neq - 6 \}$ 
D)  $\{ x \mid x$ is a real number and $x \neq - 10 , x \neq - 6 \}$ 
A)  $\left\{ x \mid x \right.$ is a real number and $\left. x \neq 10 , x \neq - 6 , x \neq \frac { 1 } { 8 } , x \neq 0 \right\}$ 
B)  $\left\{ x \mid x \right.$ is a real number and $\left. x \neq 10 , x \neq - 6 , x \neq \frac { 1 } { 8 } \right\}$ 
C)  $\{ x \mid x$ is a real number and $x \neq 10 , x \neq - 6 \}$ 
D)  $\{ x \mid x$ is a real number and $x \neq - 10 , x \neq - 6 \}$

Question 6

Solve the problem.
-A formula for the focal length of a lens is $\mathrm { f } = \frac { \mathrm { ab } } { \mathrm { b } + \mathrm { a } }$. Calculate $\mathrm { f }$ (the focal length) for $a = 10 \mathrm {~cm}$ and $\mathrm { b } = 6 \mathrm {~cm}$. Round to the nearest tenth.

Accepted Answer

A)  $3.8 \mathrm {~cm}$ 
B)  $3.0 \mathrm {~cm}$ 
C)  $6.3 \mathrm {~cm}$ 
D)  $5.0 \mathrm {~cm}$ 
A)  $3.8 \mathrm {~cm}$ 
B)  $3.0 \mathrm {~cm}$ 
C)  $6.3 \mathrm {~cm}$ 
D)  $5.0 \mathrm {~cm}$

Question 7

Perform the indicated operation. Simplify if possible.
-$\frac { 3 } { x ^ { 2 } - 3 x + 2 } + \frac { 7 } { x ^ { 2 } - 1 }$

Accepted Answer

A)  $\frac { 10 x - 11 } { ( x - 1 ) ( x - 2 ) }$ 
B)  $\frac { 42 x - 11 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
C)  $\frac { 11 x - 10 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
D)  $\frac { 10 x - 11 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
A)  $\frac { 10 x - 11 } { ( x - 1 ) ( x - 2 ) }$ 
B)  $\frac { 42 x - 11 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
C)  $\frac { 11 x - 10 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$ 
D)  $\frac { 10 x - 11 } { ( x - 1 ) ( x + 1 ) ( x - 2 ) }$

Question 8

Find the product and simplify.
-$\frac { x ^ { 2 } + 12 x + 27 } { x ^ { 2 } + 18 x + 81 } \cdot \frac { x ^ { 2 } + 17 x + 72 } { x ^ { 2 } + 11 x + 24 }$

Accepted Answer

A)  $\frac { x + 3 } { x + 9 }$ 
B)  1 
C)  $\frac { 1 } { x + 8 }$ 
D)  $\frac { x + 9 } { x + 8 }$ 
A)  $\frac { x + 3 } { x + 9 }$ 
B)  1 
C)  $\frac { 1 } { x + 8 }$ 
D)  $\frac { x + 9 } { x + 8 }$

Question 9

Simplify.
-$\frac { \frac { x + 5 } { 5 } } { \frac { x - 5 } { x } }$

Accepted Answer

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

Question 10

Simplify.
-$\frac { \frac { 5 } { x + 7 } + \frac { 10 } { x + 5 } } { \frac { 3 x + 19 } { x ^ { 2 } + 12 x + 35 } }$

Accepted Answer

A)  5 
B)  $3 x + 19$ 
C)  $\frac { 1 } { 5 }$ 
D)  15 
A)  5 
B)  $3 x + 19$ 
C)  $\frac { 1 } { 5 }$ 
D)  15

Question 11

Solve the problem.
-A company that produces inflatable rafts has costs given by the function $C ( x ) = 25 x + 30,000$, where $\mathrm { x }$ is the number of inflatable rafts manufactured and $\mathrm { C } ( \mathrm { x } )$ is measured in dollars. The average cost to manufacture each inflatable raft is given by $\bar { C } ( x ) = \frac { 25 x + 30,000 } { x }$. Find $\bar { C } ( 100 )$. (Round to the nearest dollar, if necessary.)

Accepted Answer

A)  $\$ 54$ 
B)  $\$ 335$ 
C)  $\$ 325$ 
D)  $\$ 55$ 
A)  $\$ 54$ 
B)  $\$ 335$ 
C)  $\$ 325$ 
D)  $\$ 55$

Question 12

Find the product and simplify.
-$\frac { 3 x ^ { 13 } y } { - 15 x ^ { 13 } y ^ { 6 } } \cdot y ^ { 9 }$

Accepted Answer

A)  $\frac { x ^ { 26 } y ^ { 3 } } { - 5 }$ 
B)  $\frac { y ^ { 3 } } { - 5 }$ 
C)  $\frac { x ^ { 26 } y ^ { 4 } } { - 5 }$ 
D)  $\frac { y ^ { 4 } } { - 5 }$ 
A)  $\frac { x ^ { 26 } y ^ { 3 } } { - 5 }$ 
B)  $\frac { y ^ { 3 } } { - 5 }$ 
C)  $\frac { x ^ { 26 } y ^ { 4 } } { - 5 }$ 
D)  $\frac { y ^ { 4 } } { - 5 }$

Question 13

Simplify the complex fraction.
-$\frac { \frac { 3 } { y } + \frac { 5 } { y ^ { 2 } } } { \frac { 9 } { y ^ { 2 } } - \frac { 25 } { y } }$

Accepted Answer

A)  $\frac { 1 } { 3 y - 5 }$ 
B)  $\frac { 3 y ^ { 2 } + 5 } { 9 - 25 y }$ 
C)  $\frac { 1 } { 3 - 5 y }$ 
D)  $\frac { 3 y + 5 } { 9 - 25 y }$ 
A)  $\frac { 1 } { 3 y - 5 }$ 
B)  $\frac { 3 y ^ { 2 } + 5 } { 9 - 25 y }$ 
C)  $\frac { 1 } { 3 - 5 y }$ 
D)  $\frac { 3 y + 5 } { 9 - 25 y }$

Question 14

Solve the equation.
-$1 + \frac { 1 } { x } = \frac { 42 } { x ^ { 2 } }$

Accepted Answer

A)  7,6 
B)  $- \frac { 1 } { 7 } , \frac { 1 } { 6 }$ 
C)  $- 7,6$ 
D)  $7 , - 6$ 
A)  7,6 
B)  $- \frac { 1 } { 7 } , \frac { 1 } { 6 }$ 
C)  $- 7,6$ 
D)  $7 , - 6$

Question 15

Solve.
-A square shaped pasture has a side of length $\frac { 5 } { x + 9 }$ meters.

Express the perimeter of the pasture as a rational expression.

Accepted Answer

A)  $\frac { 20 } { x + 9 } m$ 
B)  $\frac { 20 } { x + 18 } m$ 
C)  $\frac { 5 } { x + 36 } m$ 
D)  $\frac { 20 } { x + 36 } m$ 
A)  $\frac { 20 } { x + 9 } m$ 
B)  $\frac { 20 } { x + 18 } m$ 
C)  $\frac { 5 } { x + 36 } m$ 
D)  $\frac { 20 } { x + 36 } m$

Question 16

Simplify the expression.
-$\frac { 2 x - 6 } { x ^ { 2 } - 9 }$

Accepted Answer

A)  $- \frac { 2 } { x + 3 }$ 
B)  $\frac { 2 } { x - 3 }$ 
C)  $- \frac { 4 } { x - 9 }$ 
D)  $\frac { 2 } { x + 3 }$ 
A)  $- \frac { 2 } { x + 3 }$ 
B)  $\frac { 2 } { x - 3 }$ 
C)  $- \frac { 4 } { x - 9 }$ 
D)  $\frac { 2 } { x + 3 }$

Question 17

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

Question 18

Solve.
-A baker can decorate the day's cookie supply four times as fast as his new assistant. If they decorate all the cookies working together in 32 minutes, how long would it take for each of them to decorate the cookies working individually?

Accepted Answer

A)  baker: $40 \mathrm {~min}$
assistant: $160 \mathrm {~min}$ 
B)  baker: $9 \frac { 1 } { 4 } \mathrm {~min}$
assistant: $37 \mathrm {~min}$ 
C)  baker: $160 \mathrm {~min}$
assistant: $40 \mathrm {~min}$ 
D)  baker: $160 \mathrm {~min}$
assistant: $640 \mathrm {~min}$ 
A)  baker: $40 \mathrm {~min}$
assistant: $160 \mathrm {~min}$ 
B)  baker: $9 \frac { 1 } { 4 } \mathrm {~min}$
assistant: $37 \mathrm {~min}$ 
C)  baker: $160 \mathrm {~min}$
assistant: $40 \mathrm {~min}$ 
D)  baker: $160 \mathrm {~min}$
assistant: $640 \mathrm {~min}$

Question 19

Solve the equation.
-$\frac { 9 x } { 5 } + 4 = \frac { 1 } { 3 }$

Accepted Answer

A)  $- \frac { 59 } { 27 }$ 
B)  $\frac { 86 } { 27 }$ 
C)  $- \frac { 55 } { 27 }$ 
D)  $\frac { 5 } { 9 }$ 
A)  $- \frac { 59 } { 27 }$ 
B)  $\frac { 86 } { 27 }$ 
C)  $- \frac { 55 } { 27 }$ 
D)  $\frac { 5 } { 9 }$

Question 20

Find the domain of the rational expression.
-$f ( x ) = \frac { x ^ { 3 } + 8 x ^ { 4 } } { x ^ { 2 } + 36 }$

Accepted Answer

A)  $\{ x \mid x$ is a real number and $x \neq - 36 \}$ 
B)  $\left\{ x \mid x \right.$ is a real number and $\left. x \neq 0 , x \neq - \frac { 1 } { 8 } \right\}$ 
C)  $\{ x \mid x$ is a real number $\}$ 
D)  $\{ x \mid x$ is a real number and $x z - 6 \}$ 
A)  $\{ x \mid x$ is a real number and $x \neq - 36 \}$ 
B)  $\left\{ x \mid x \right.$ is a real number and $\left. x \neq 0 , x \neq - \frac { 1 } { 8 } \right\}$ 
C)  $\{ x \mid x$ is a real number $\}$ 
D)  $\{ x \mid x$ is a real number and $x z - 6 \}$

Find the least common denominator (LCD). - $\frac { 2 } { x ^ { 2 } - 10 x + 24 } , \frac { 4 } { 5 x - 30 }$

Perform the indicated operation and simplify if possible. - $\frac { x ^ { 2 } - 25 } { x ^ { 2 } + 5 x } \div \frac { x ^ { 2 } + 5 x + 1 } { 3 x - 18 }$

$\text { Find and simplify the difference quotient } \frac { f ( a + h ) - f ( a ) } { h } , h \neq 0 \text { for the given function. }$ - $f ( x ) = \frac { 8 } { x ^ { 2 } }$

Find the product and simplify. - $\frac { x ^ { 2 } + 8 x } { 3 } \cdot \frac { 25 } { 5 x + 40 }$

Find the domain of the rational expression. - $f ( x ) = \frac { 1 - 8 x } { x ^ { 2 } - 4 x - 60 }$

Solve the problem. -A formula for the focal length of a lens is $\mathrm { f } = \frac { \mathrm { ab } } { \mathrm { b } + \mathrm { a } }$ . Calculate $\mathrm { f }$ (the focal length) for $a = 10 \mathrm {~cm}$ and $\mathrm { b } = 6 \mathrm {~cm}$ . Round to the nearest tenth.

Perform the indicated operation. Simplify if possible. - $\frac { 3 } { x ^ { 2 } - 3 x + 2 } + \frac { 7 } { x ^ { 2 } - 1 }$

Find the product and simplify. - $\frac { x ^ { 2 } + 12 x + 27 } { x ^ { 2 } + 18 x + 81 } \cdot \frac { x ^ { 2 } + 17 x + 72 } { x ^ { 2 } + 11 x + 24 }$

Simplify. - $\frac { \frac { x + 5 } { 5 } } { \frac { x - 5 } { x } }$

Simplify. - $\frac { \frac { 5 } { x + 7 } + \frac { 10 } { x + 5 } } { \frac { 3 x + 19 } { x ^ { 2 } + 12 x + 35 } }$

Find the product and simplify. - $\frac { 3 x ^ { 13 } y } { - 15 x ^ { 13 } y ^ { 6 } } \cdot y ^ { 9 }$

Simplify the complex fraction. - $\frac { \frac { 3 } { y } + \frac { 5 } { y ^ { 2 } } } { \frac { 9 } { y ^ { 2 } } - \frac { 25 } { y } }$

Solve the equation. - $1 + \frac { 1 } { x } = \frac { 42 } { x ^ { 2 } }$

Solve. -A square shaped pasture has a side of length $\frac { 5 } { x + 9 }$ meters. $Solve. -A square shaped pasture has a side of length \frac { 5 } { x + 9 } meters. Express the perimeter of the pasture as a rational expression.$ Express the perimeter of the pasture as a rational expression.

Simplify the expression. - $\frac { 2 x - 6 } { x ^ { 2 } - 9 }$

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?

Solve. -A baker can decorate the day's cookie supply four times as fast as his new assistant. If they decorate all the cookies working together in 32 minutes, how long would it take for each of them to decorate the cookies working individually?

Solve the equation. - $\frac { 9 x } { 5 } + 4 = \frac { 1 } { 3 }$

Find the domain of the rational expression. - $f ( x ) = \frac { x ^ { 3 } + 8 x ^ { 4 } } { x ^ { 2 } + 36 }$

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). - 2x2−10x+24,45x−30\frac { 2 } { x ^ { 2 } - 10 x + 24 } , \frac { 4 } { 5 x - 30 }x2−10x+242​,5x−304​

Perform the indicated operation and simplify if possible. - x2−25x2+5x÷x2+5x+13x−18\frac { x ^ { 2 } - 25 } { x ^ { 2 } + 5 x } \div \frac { x ^ { 2 } + 5 x + 1 } { 3 x - 18 }x2+5xx2−25​÷3x−18x2+5x+1​

Find the product and simplify. - x2+8x3⋅255x+40\frac { x ^ { 2 } + 8 x } { 3 } \cdot \frac { 25 } { 5 x + 40 }3x2+8x​⋅5x+4025​

Find the domain of the rational expression. - f(x)=1−8xx2−4x−60f ( x ) = \frac { 1 - 8 x } { x ^ { 2 } - 4 x - 60 }f(x)=x2−4x−601−8x​

Perform the indicated operation. Simplify if possible. - 3x2−3x+2+7x2−1\frac { 3 } { x ^ { 2 } - 3 x + 2 } + \frac { 7 } { x ^ { 2 } - 1 }x2−3x+23​+x2−17​

Find the product and simplify. - x2+12x+27x2+18x+81⋅x2+17x+72x2+11x+24\frac { x ^ { 2 } + 12 x + 27 } { x ^ { 2 } + 18 x + 81 } \cdot \frac { x ^ { 2 } + 17 x + 72 } { x ^ { 2 } + 11 x + 24 }x2+18x+81x2+12x+27​⋅x2+11x+24x2+17x+72​

Simplify. - x+55x−5x\frac { \frac { x + 5 } { 5 } } { \frac { x - 5 } { x } }xx−5​5x+5​​

Simplify. - 5x+7+10x+53x+19x2+12x+35\frac { \frac { 5 } { x + 7 } + \frac { 10 } { x + 5 } } { \frac { 3 x + 19 } { x ^ { 2 } + 12 x + 35 } }x2+12x+353x+19​x+75​+x+510​​

Find the product and simplify. - 3x13y−15x13y6⋅y9\frac { 3 x ^ { 13 } y } { - 15 x ^ { 13 } y ^ { 6 } } \cdot y ^ { 9 }−15x13y63x13y​⋅y9

Simplify the complex fraction. - 3y+5y29y2−25y\frac { \frac { 3 } { y } + \frac { 5 } { y ^ { 2 } } } { \frac { 9 } { y ^ { 2 } } - \frac { 25 } { y } }y29​−y25​y3​+y25​​

Solve the equation. - 1+1x=42x21 + \frac { 1 } { x } = \frac { 42 } { x ^ { 2 } }1+x1​=x242​

Solve. -A square shaped pasture has a side of length 5x+9\frac { 5 } { x + 9 }x+95​ meters. Express the perimeter of the pasture as a rational expression.

Simplify the expression. - 2x−6x2−9\frac { 2 x - 6 } { x ^ { 2 } - 9 }x2−92x−6​