Question 1

Solve the proportion.
-$\frac { 1 } { 2 } = \frac { x } { 9 }$

Accepted Answer

A)  18 
B)  9 
C)  $\frac { 1 } { 18 }$ 
D)  $\frac { 9 } { 2 }$ 
A)  18 
B)  9 
C)  $\frac { 1 } { 18 }$ 
D)  $\frac { 9 } { 2 }$

Question 2

Solve.
-A boat moves 5 kilometers upstream in the same amount of time it moves 20 kilometers downstream. If the rate of the current is 9 kilometers per hour, find the rate of the boat in still water.

Accepted Answer

A)  $9 \mathrm {~km} / \mathrm { hr }$ 
B)  $6 \frac { 2 } { 3 } \mathrm {~km} / \mathrm { hr }$ 
C)  $15 \mathrm {~km} / \mathrm { hr }$ 
D)  $3 \mathrm {~km} / \mathrm { hr }$ 
A)  $9 \mathrm {~km} / \mathrm { hr }$ 
B)  $6 \frac { 2 } { 3 } \mathrm {~km} / \mathrm { hr }$ 
C)  $15 \mathrm {~km} / \mathrm { hr }$ 
D)  $3 \mathrm {~km} / \mathrm { hr }$

Question 3

Fill in the blank with one of the words or phrases listed below. Not all choices will be used. $$\begin{array} { l l l l } 
\text { least common denominator } & \text { simplifying } & \text { reciprocals } & \text { numerator } \
\text { cross products } & \text { ratio } & \text { proportion } & \text { denominator } \
\text { rational expression } & \text { domain } & \text { complex fraction } & \
\frac { - a } { b } & \frac { a } { - b } & \frac { - a } { - b } &
\end{array}$$
-The _____of the rational function $f ( x ) = \frac { 1 } { x - 3 }$ is $\{ x \mid x$ is a real number, $x \neq 3 \}$.

Accepted Answer

A)  complex fraction 
B)  denominator 
C)  domain 
D)  numerator 
A)  complex fraction 
B)  denominator 
C)  domain 
D)  numerator

Question 4

Simplify.
-$\frac { \frac { 35 } { 6 x } } { \frac { 5 } { 18 x } }$

Accepted Answer

A)  $21 \mathrm { x } ^ { 2 }$ 
B)  21 
C)  $\frac { x ^ { 2 } } { 21 }$ 
D)  $\frac { 1 } { 21 }$ 
A)  $21 \mathrm { x } ^ { 2 }$ 
B)  21 
C)  $\frac { x ^ { 2 } } { 21 }$ 
D)  $\frac { 1 } { 21 }$

Question 5

Find the product and simplify.
-$\frac { 2 z ^ { 3 } } { 5 } \cdot \frac { 10 } { z ^ { 2 } }$

Accepted Answer

A)  $\frac { 4 z ^ { 2 } } { z ^ { 3 } }$ 
B)  $\frac { \mathrm { z } } { 4 }$ 
C)  $4 \mathrm { z }$ 
D)  $\frac { 4 } { z }$ 
A)  $\frac { 4 z ^ { 2 } } { z ^ { 3 } }$ 
B)  $\frac { \mathrm { z } } { 4 }$ 
C)  $4 \mathrm { z }$ 
D)  $\frac { 4 } { z }$

Question 6

Simplify.
-$\frac { \frac { x - 1 } { 5 } } { \frac { x + 7 } { x } }$

Accepted Answer

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

Question 7

Perform the indicated operation. Simplify if possible.
-$\frac { 14 } { 5 x - 15 } + \frac { x } { x ^ { 2 } - 9 }$

Accepted Answer

A)  $\frac { 19 x } { ( x + 3 ) ( x - 3 ) }$ 
B)  $\frac { 15 x + 42 } { ( x + 3 ) ( x - 3 ) }$ 
C)  $\frac { x + 14 } { 5 ( x + 3 ) ( x - 3 ) }$ 
D)  $\frac { 19 x + 42 } { 5 ( x + 3 ) ( x - 3 ) }$ 
A)  $\frac { 19 x } { ( x + 3 ) ( x - 3 ) }$ 
B)  $\frac { 15 x + 42 } { ( x + 3 ) ( x - 3 ) }$ 
C)  $\frac { x + 14 } { 5 ( x + 3 ) ( x - 3 ) }$ 
D)  $\frac { 19 x + 42 } { 5 ( x + 3 ) ( x - 3 ) }$

Question 8

Solve.
-Five divided by the difference of a number and 3 equals the quotient of 10 and the sum of the number and 6 . Find the number.

Accepted Answer

A)  4 
B)  12 
C)  $\frac { 9 } { 5 }$ 
D)  0 
A)  4 
B)  12 
C)  $\frac { 9 } { 5 }$ 
D)  0

Question 9

Multiply or divide as indicated.
-$\frac { 4 x + 12 } { 10 } \div \frac { 5 x + 15 } { 15 }$

Accepted Answer

A)  $\frac { 9 x + 27 } { 25 }$ 
B)  $\frac { 6 } { 5 }$ 
C)  $\frac { 4 x + 12 } { 50 x }$ 
D)  $\frac { 2 } { 15 }$ 
A)  $\frac { 9 x + 27 } { 25 }$ 
B)  $\frac { 6 } { 5 }$ 
C)  $\frac { 4 x + 12 } { 50 x }$ 
D)  $\frac { 2 } { 15 }$

Question 10

Simplify the expression.
-$\frac { 20 x ^ { 2 } + 12 x } { 20 x }$

Accepted Answer

A)  $\frac { 5 x + 3 } { 5 }$ 
B)  $\frac { 5 x - 3 } { 5 }$ 
C)  $x ^ { 2 } + 12$ 
D)  $x ^ { 2 } + 3$ 
A)  $\frac { 5 x + 3 } { 5 }$ 
B)  $\frac { 5 x - 3 } { 5 }$ 
C)  $x ^ { 2 } + 12$ 
D)  $x ^ { 2 } + 3$

Question 11

Solve the equation for the indicated variable.
-$I = \frac { n E } { n r + R } \quad \text { for } n$

Accepted Answer

A)  $n = \frac { I R } { I r - E }$ 
B)  $n = \operatorname { IR } ( \operatorname { Ir } - \mathrm { E } )$ 
C)  $\mathrm { n } = \frac { \mathrm { IR } } { \mathrm { E } - \mathrm { Ir } }$ 
D)  $\mathrm { n } = \frac { \mathrm { R } } { \mathrm { E } - \mathrm { Ir } }$ 
A)  $n = \frac { I R } { I r - E }$ 
B)  $n = \operatorname { IR } ( \operatorname { Ir } - \mathrm { E } )$ 
C)  $\mathrm { n } = \frac { \mathrm { IR } } { \mathrm { E } - \mathrm { Ir } }$ 
D)  $\mathrm { n } = \frac { \mathrm { R } } { \mathrm { E } - \mathrm { Ir } }$

Question 12

Perform the indicated operation. Simplify if possible.
-$\frac { 2 } { x ^ { 2 } - 64 } - \frac { 14 } { 64 - x ^ { 2 } }$

Accepted Answer

A)  $\frac { 12 } { x ^ { 2 } - 64 }$ 
B)  $\frac { 16 } { x ^ { 2 } - 64 }$ 
C)  $- \frac { 16 } { x ^ { 2 } - 64 }$ 
D)  $- \frac { 12 } { x ^ { 2 } - 64 }$ 
A)  $\frac { 12 } { x ^ { 2 } - 64 }$ 
B)  $\frac { 16 } { x ^ { 2 } - 64 }$ 
C)  $- \frac { 16 } { x ^ { 2 } - 64 }$ 
D)  $- \frac { 12 } { x ^ { 2 } - 64 }$

Question 13

Find the domain of the rational expression.
-$f ( x ) = \frac { x ^ { 2 } - 4 } { x ^ { 2 } + 13 x + 36 }$

Accepted Answer

A)  $\{ x \mid x$ is a real number and $x \neq 0 \}$ 
B)  $\{ x \mid x$ is a real number and $x \neq - 4 , x \neq - 9 \}$ 
C)  $\{ x \mid x$ is a real number and $x \neq 4 , x \neq - 9 \}$ 
D)  $\{ x \mid x$ is a real number and $x \neq 2 , x \neq - 2 \}$ 
A)  $\{ x \mid x$ is a real number and $x \neq 0 \}$ 
B)  $\{ x \mid x$ is a real number and $x \neq - 4 , x \neq - 9 \}$ 
C)  $\{ x \mid x$ is a real number and $x \neq 4 , x \neq - 9 \}$ 
D)  $\{ x \mid x$ is a real number and $x \neq 2 , x \neq - 2 \}$

Question 14

Perform the indicated operation and simplify if possible.
-$\frac { 18 } { x ^ { 2 } - 1 } + \frac { 9 } { x + 1 }$

Accepted Answer

A)  $\frac { 9 } { x + 1 }$ 
B)  $\frac { 9 } { x - 1 }$ 
C)  $\frac { 9 } { x ^ { 2 } - 1 }$ 
D)  $\frac { 9 ( x + 3 ) } { x ^ { 2 } - 1 }$ 
A)  $\frac { 9 } { x + 1 }$ 
B)  $\frac { 9 } { x - 1 }$ 
C)  $\frac { 9 } { x ^ { 2 } - 1 }$ 
D)  $\frac { 9 ( x + 3 ) } { x ^ { 2 } - 1 }$

Question 15

Simplify.
-$\frac { \frac { x + 1 } { 7 } } { \frac { 5 x - 1 } { 14 } }$

Accepted Answer

A)  $\frac { ( x + 1 ) ( 5 x - 1 ) } { 2 }$ 
B)  $\frac { 2 ( x + 1 ) } { 5 ( x - 1 ) }$ 
C)  $\frac { 2 ( x + 1 ) } { 5 x - 1 }$ 
D)  $\frac { ( x + 1 ) ( 5 x - 1 ) } { 14 }$ 
A)  $\frac { ( x + 1 ) ( 5 x - 1 ) } { 2 }$ 
B)  $\frac { 2 ( x + 1 ) } { 5 ( x - 1 ) }$ 
C)  $\frac { 2 ( x + 1 ) } { 5 x - 1 }$ 
D)  $\frac { ( x + 1 ) ( 5 x - 1 ) } { 14 }$

Question 16

Multiply or divide as indicated.
-$\frac { ( x + 2 ) ^ { 2 } } { x - 2 } \div \frac { x ^ { 2 } - 4 } { 2 x - 4 }$

Accepted Answer

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

Question 17

Find the quotient and simplify.
-$\frac { ( s - x ) ^ { 2 } } { s + x } \div \frac { s ^ { 2 } - s x } { s }$

Accepted Answer

A)  $\frac { ( s - x ) ^ { 3 } } { s + x }$ 
B)  $\frac { x } { s + x }$ 
C)  $\frac { s - x } { s + x }$ 
D)  $\frac { s + x } { s - x }$ 
A)  $\frac { ( s - x ) ^ { 3 } } { s + x }$ 
B)  $\frac { x } { s + x }$ 
C)  $\frac { s - x } { s + x }$ 
D)  $\frac { s + x } { s - x }$

Question 18

Solve.
-Two angles are said to be complementary if their sum is $90 ^ { \circ }$. If one angle measures $\frac { 33 } { x }$ degrees, find the measure of its complement.

Accepted Answer

A)  $\left( \frac { 90 x - 33 } { x } \right) ^ { \circ }$ 
B)  $\left( \frac { 90 x - 57 } { x } \right) ^ { \circ }$ 
C)  $57 ^ { \circ }$ 
D)  $33 ^ { \circ }$ 
A)  $\left( \frac { 90 x - 33 } { x } \right) ^ { \circ }$ 
B)  $\left( \frac { 90 x - 57 } { x } \right) ^ { \circ }$ 
C)  $57 ^ { \circ }$ 
D)  $33 ^ { \circ }$

Question 19

Perform the indicated operation. Simplify if possible.
-$\frac { 8 - x } { x - 3 } - \frac { 2 x + 1 } { 3 - x }$

Accepted Answer

A)  $- \frac { x + 7 } { x - 3 }$ 
B)  $- \frac { x + 9 } { x - 3 }$ 
C)  $\frac { x + 7 } { x - 3 }$ 
D)  $\frac { x + 9 } { x - 3 }$ 
A)  $- \frac { x + 7 } { x - 3 }$ 
B)  $- \frac { x + 9 } { x - 3 }$ 
C)  $\frac { x + 7 } { x - 3 }$ 
D)  $\frac { x + 9 } { x - 3 }$

Question 20

Simplify the complex fraction.
-$\frac { \frac { x ^ { 9 } } { 7 y ^ { 5 } } } { \frac { x ^ { 2 } } { y ^ { 3 } } }$

Accepted Answer

A)  $\frac { x ^ { 11 } } { 7 y ^ { 8 } }$ 
B)  $\frac { x ^ { 7 } } { 7 y ^ { 2 } }$ 
C)  $\frac { x ^ { 7 } } { y ^ { 2 } }$ 
D)  $\frac { x ^ { 7 } } { 7 y ^ { 8 } }$ 
A)  $\frac { x ^ { 11 } } { 7 y ^ { 8 } }$ 
B)  $\frac { x ^ { 7 } } { 7 y ^ { 2 } }$ 
C)  $\frac { x ^ { 7 } } { y ^ { 2 } }$ 
D)  $\frac { x ^ { 7 } } { 7 y ^ { 8 } }$

Solve the proportion. - $\frac { 1 } { 2 } = \frac { x } { 9 }$

Solve. -A boat moves 5 kilometers upstream in the same amount of time it moves 20 kilometers downstream. If the rate of the current is 9 kilometers per hour, find the rate of the boat in still water.

Simplify. - $\frac { \frac { 35 } { 6 x } } { \frac { 5 } { 18 x } }$

Find the product and simplify. - $\frac { 2 z ^ { 3 } } { 5 } \cdot \frac { 10 } { z ^ { 2 } }$

Simplify. - $\frac { \frac { x - 1 } { 5 } } { \frac { x + 7 } { x } }$

Perform the indicated operation. Simplify if possible. - $\frac { 14 } { 5 x - 15 } + \frac { x } { x ^ { 2 } - 9 }$

Solve. -Five divided by the difference of a number and 3 equals the quotient of 10 and the sum of the number and 6 . Find the number.

Multiply or divide as indicated. - $\frac { 4 x + 12 } { 10 } \div \frac { 5 x + 15 } { 15 }$

Simplify the expression. - $\frac { 20 x ^ { 2 } + 12 x } { 20 x }$

Solve the equation for the indicated variable. - $I = \frac { n E } { n r + R } \quad \text { for } n$

Perform the indicated operation. Simplify if possible. - $\frac { 2 } { x ^ { 2 } - 64 } - \frac { 14 } { 64 - x ^ { 2 } }$

Find the domain of the rational expression. - $f ( x ) = \frac { x ^ { 2 } - 4 } { x ^ { 2 } + 13 x + 36 }$

Perform the indicated operation and simplify if possible. - $\frac { 18 } { x ^ { 2 } - 1 } + \frac { 9 } { x + 1 }$

Simplify. - $\frac { \frac { x + 1 } { 7 } } { \frac { 5 x - 1 } { 14 } }$

Multiply or divide as indicated. - $\frac { ( x + 2 ) ^ { 2 } } { x - 2 } \div \frac { x ^ { 2 } - 4 } { 2 x - 4 }$

Find the quotient and simplify. - $\frac { ( s - x ) ^ { 2 } } { s + x } \div \frac { s ^ { 2 } - s x } { s }$

Solve. -Two angles are said to be complementary if their sum is $90 ^ { \circ }$ . If one angle measures $\frac { 33 } { x }$ degrees, find the measure of its complement.

Perform the indicated operation. Simplify if possible. - $\frac { 8 - x } { x - 3 } - \frac { 2 x + 1 } { 3 - x }$

Simplify the complex fraction. - $\frac { \frac { x ^ { 9 } } { 7 y ^ { 5 } } } { \frac { x ^ { 2 } } { y ^ { 3 } } }$

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

Solve the proportion. - 12=x9\frac { 1 } { 2 } = \frac { x } { 9 }21​=9x​

Solve. -A boat moves 5 kilometers upstream in the same amount of time it moves 20 kilometers downstream. If the rate of the current is 9 kilometers per hour, find the rate of the boat in still water.

Simplify. - 356x518x\frac { \frac { 35 } { 6 x } } { \frac { 5 } { 18 x } }18x5​6x35​​

Find the product and simplify. - 2z35⋅10z2\frac { 2 z ^ { 3 } } { 5 } \cdot \frac { 10 } { z ^ { 2 } }52z3​⋅z210​

Simplify. - x−15x+7x\frac { \frac { x - 1 } { 5 } } { \frac { x + 7 } { x } }xx+7​5x−1​​

Perform the indicated operation. Simplify if possible. - 145x−15+xx2−9\frac { 14 } { 5 x - 15 } + \frac { x } { x ^ { 2 } - 9 }5x−1514​+x2−9x​