Deck 6: Rational Expression and Equations

Question 1

Perform the indicated operation. Simplify, if possible. $\frac { 6 x ^ { 2 } + 7 x - 10 } { 7 x ^ { 2 } + 8 x - 12 } \cdot \frac { 7 x ^ { 2 } + x - 6 } { 36 x ^ { 2 } - 25 }$

A) $\frac { x + 1 } { 6 x + 5 }$
B) $\frac { x - 1 } { 6 x - 5 }$
C) $\frac { 2 x + 9 } { 290 }$
D) $\frac { ( x + 2 ) ( 7 x + 6 ) } { ( x - 2 ) ( 6 x + 5 ) }$

Answer 1

$\frac { x + 1 } { 6 x + 5 }$

Question 2

Simplify:

A)
B)
C)
D)

Answer 2

Question 3

Perform the indicated operation. Simplify, if possible. $\frac { x ^ { 2 } - 8 x - 9 } { x ^ { 2 } + x - 6 } \div \frac { x ^ { 2 } - 12 x + 27 } { x ^ { 2 } - 4 x - 21 } \cdot \frac { x ^ { 2 } - 5 x + 6 } { x ^ { 2 } - 6 x - 7 }$

A) $\frac { ( x + 1 ) ^ { 2 } ( x - 7 ) ^ { 2 } } { ( x - 2 ) ^ { 2 } ( x - 3 ) ^ { 2 } }$
B) 1
C) $\frac { ( x + 9 ) ( x - 2 ) } { ( x + 2 ) ( x - 9 ) }$
D) 0

Answer 3

1

Question 4

Simplify the rational expression.

A)
B)
C)
D)
E)

Answer 4

Question 5

Perform the indicated operation. Simplify, if possible. $\frac { x ^ { 2 } - 3 } { x ^ { 3 } + 4 x ^ { 2 } - 3 x - 12 } \cdot \frac { 2 x ^ { 2 } y + 8 x y } { 4 x y }$

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

Answer 5

$\frac { 1 } { 2 }$

Question 6

Find the domain of the rational function. Express your answer in interval notation.

A)
B)
C)
D)
E)

Answer 6

Question 7

Perform the indicated operation. Simplify, if possible. $\frac { 2 x ^ { 2 } - 8 } { 10 x ^ { 2 } - 15 x - 10 } \div \frac { x ^ { 2 } + 4 x + 4 } { 4 x + 2 }$

A) $\frac { ( x + 2 ) ^ { 3 } } { 5 ( 2 x + 1 ) ^ { 2 } }$
B) $\frac { 2 ( 2 x - 4 ) ( 2 x + 1 ) } { ( 10 x + 5 ) ( x + 2 ) ^ { 2 } }$
C) $\frac { 4 ( x - 2 ) ( 2 x + 1 ) } { 5 ( 2 x - 1 ) ( x + 2 ) ^ { 2 } }$
D) $\frac { 4 } { 5 ( x + 2 ) }$

Question 8

Simplify the rational expression.

A)
B)
C)
D)
E)

Question 9

Perform the indicated operation. Simplify, if possible. $\frac { 2 x ^ { 2 } + 7 x - 4 } { 3 x ^ { 2 } + 8 x - 16 } \div \frac { 2 x ^ { 2 } + 3 x - 2 } { 3 x + 6 }$

A) $\frac { ( 2 x - 1 ) ^ { 2 } } { 3 ( 3 x - 4 ) }$
B) $\frac { 3 } { 3 x - 4 }$
C) $\frac { 2 x - 1 } { ( 3 x - 4 ) ( x + 1 ) }$
D) none of these

Question 10

Simplify the rational expression if possible.

A)
B)
C)
D)
E) Does not simplify

Question 11

Perform the indicated operation. Simplify, if possible. $\frac { 2 y ^ { 2 } - 18 } { 6 y - 12 } \cdot \frac { y ^ { 2 } - 5 y + 6 } { y ^ { 2 } - 6 y + 9 }$

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

Question 12

Simplify:

A)
B)
C)
D)

Question 13

To multiply rational expressions, multiply their numerators and multiply their _________.

Question 14

The graph of rational function (for ) is shown in the illustration. Find the value(s) of for which .

A)
B)
C)
D)
E)

Question 15

Perform the indicated operation. Simplify, if possible. $\frac { 8 x ^ { 2 } + 2 x - 3 } { 8 x ^ { 2 } - 2 } \div \frac { 12 x + 9 } { 16 x ^ { 2 } + 12 x + 2 }$

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

Question 16

Find the domain of .

A)
B)
C)
D)

Question 17

A room should have at least 108 square feet of windows for natural light. Convert this to square yards. __________

Question 18

Simplify the rational expression if possible.

A)
B)
C)
D)
E) Does not simplify

Question 19

Perform the indicated operation. Simplify, if possible. $\left( 4 x ^ { 2 } - 25 \right) \div \frac { 2 x ^ { 2 } - 3 x - 5 } { x + 4 } \div ( 2 x + 5 )$

A) $\frac { ( 2 x + 5 ) ^ { 2 } ( x + 4 ) } { x + 1 }$
B) $\frac { x + 1 } { x + 4 }$
C) $\frac { x + 4 } { x + 1 }$
D) $\frac { ( x + 4 ) } { ( 2 x - 5 ) ( 2 x + 5 ) ( x - 5 ) }$

Question 20

Find the domain of each rational function. Express your answer in interval notation.

A)
B)
C)
D)
E)

Question 21

According to the Sacramento Bee newspaper, the number of words an average man speaks a day is about 12,000. How many words does an average man speak in 6 years?
( Hint: 365 days = 1 year.) __________ words

Question 22

Simplify the rational expression.

A)
B)
C)
D)
E)

Question 23

The __________ of .

Question 24

An electric company charges per month plus cents for each kilowatt hour (kwh) of electricity used. Find the average cost per kwh when kwh are used.

A) cents
B) cents
C) cents
D) cents
E) cents

Question 25

Perform the indicated operation. Simplify, if possible. $\frac { 25 y ^ { 2 } + 40 y + 16 } { 12 y + 18 } \div \frac { 10 y ^ { 2 } - 7 y - 12 } { 4 y ^ { 2 } - 9 }$

A) $\frac { ( 5 y - 4 ) ( 2 y - 3 ) } { 6 ( 2 y + 3 ) }$
B) $\frac { 5 y + 4 } { 6 }$
C) $6 ( 5 y + 4 )$
D) $5 y + 4$

Question 26

Simplify the rational expression if possible.

A)
B)
C)
D)
E) Does not simplify

Question 27

Multiply, then simplify if possible.

Question 28

Simplify the rational expression.

A)
B)
C)
D)
E)

Question 29

Perform the operations and simplify the result when possible. $\frac { s + 6 } { s - 6 } - \frac { s - 6 } { s + 6 }$

A) $\frac { s ^ { 2 } - 24 s - 36 } { ( s + 6 ) ( s - 6 ) }$
B) 0
C) $\frac { 24 \mathrm {~s} } { ( s + 6 ) ( s - 6 ) }$
D) 2s
E) $\frac { s ^ { 2 } + 24 s + 36 } { ( s + 6 ) ( s - 6 ) }$

Question 30

The graph of the function approaches the positive -axis. When a graph approaches a line, we call the line a(n) __________.

A) curve
B) asymptote
C) number line
D) axis
E) quotient

Question 31

Insert either a multiplication symbol or a division symbol in each blank to make a true statement.

Question 32

Simplify the rational expression if possible.

A)
B)
C)
D)
E)

Question 33

The __________ of $\frac { x - 8 } { x + 7 }$ and $\frac { 6 - 5 x } { x }$ is $x ( x + 7 )$ .

A) greatest common denominator
B) greatest common numerator
C) least common denominator
D) least common numerator
E) greatest common factor

Question 34

Simplify the rational expression.

A)
B)
C)
D)
E)

Question 35

An aluminum brace used to support a wooden shelf has a length that is 3 inches less than twice the width of the shelf. The brace is anchored to the wall 3 inches below the shelf, as shown. Find the width of the shelf and the length of the brace. w = __________ in. l = __________ in.

Question 36

Simplify the rational expression if possible.

A)
B)
C)
D)
E) Does not simplify

Question 37

Multiply, then simplify if possible.

Question 38

A quotient of two polynomials, such as , is called a(n) __________ expression.

A) regular
B) rational
C) undefined
D) algebraic
E) arithmetic

Question 39

Multiply, then simplify if possible.

Question 40

Find the domain of .

A)
B)
C)
D)

Question 41

Match each function with the correct graph shown below.

A)
B)
C)
D)
E)
F)

Question 42

Add, and then simplify, if possible. $\frac { 5 x } { 2 x + 2 } + \frac { 7 x + 12 } { 2 x + 2 }$

A) 6
B) 7
C) 3
D) 4 x
E) 6 x

Question 43

Find the domain of .

A)
B)
C)
D)

Question 44

Multiply, then simplify if possible.

Question 45

Simplify:

A)
B)
C)
D)

Question 46

Divide, then simplify if possible.

Question 47

Match each function with the correct graph shown below.

A)
B)
C)
D)
E)
F)

Question 48

Perform the operations and simplify the result when possible. $\frac { 4 } { a b ^ { 2 } } - \frac { 7 } { a ^ { 2 } b }$

A) $- \frac { 3 ( a - b ) } { a ^ { 2 } b ^ { 2 } }$
B) $\frac { 4 a - 7 b } { a b ^ { 2 } }$
C) $- \frac { 3 a b } { a ^ { 2 } b ^ { 2 } }$
D) $\frac { 4 a - 7 b } { a ^ { 2 } b ^ { 2 } }$
E) $\frac { 4 a - 7 b } { a b }$

Question 49

To simplify a rational expression, we remove factors ________ to the numerator and denominator.

Question 50

Divide, then simplify if possible.

Question 51

The ________ of a function is the set of all permissible input values for the variable.

Question 52

Complete the unit conversion. _____ __________ ( ft , years , square ft , ft per year , or square ft per year )

Question 53

Because of division by 0, the expression is ________.

Question 54

Perform the operations and simplify the result when possible. $\frac { 8 x } { x + 1 } + \frac { 5 } { x + 1 } - \frac { 3 x } { x + 1 }$

A) 9
B) 5 x
C) 8 x
D) 7
E) 5

Question 55

Match each function with the correct graph shown below.

A)
B)
C)
D)
E)
F)

Question 56

Insert either a multiplication symbol or a division symbol in each blank to make a true statement.

Question 57

Simplify:

A)
B)
C)
D)

Question 58

The table contains algebraic expressions for the rate an object travels, and the time traveled at that rate in terms of a constant. Complete the table by finding a . Rate (mph) Time (hr) Distance (mi) = __________

Question 59

Match each function with the correct graph shown below.

A)
B)
C)
D)
E)
F)

Question 60

Find the power:

Question 61

The graph of the function approaches the positive -axis. When a graph approaches a line, we call the line a(n) __________.

Question 62

Perform the operations and simplify the result when possible. $\frac { a + b } { 2 } + \frac { a - b } { 7 }$

A) $\frac { 2 ( 9 a + 5 b ) } { 9 }$
B) $\frac { 5 a - 9 b } { 7 }$
C) $\frac { 9 a + 5 b } { 14 }$
D) $\frac { 9 a + 5 b } { 9 }$
E) $\frac { 9 a + b } { 7 }$

Question 63

Match each function with the correct graph shown below.

A)
B)
C)
D)
E)
F)

Question 64

Divide, then simplify if possible.

Question 65

A quotient of two polynomials, such as , is called a(n) __________ expression.

Question 66

Multiply, then simplify if possible.

Question 67

Simplify:

A)
B)
C)
D)

Question 68

Complete the unit conversion. _____ __________ ( meters per min , meters per sec , meters , sec , or min )

Question 69

Simplify:

A)
B)
C)
D)

Question 70

Perform the indicated operation. Simplify, if possible. $\frac { 4 x - 1 } { x ^ { 2 } - 3 x - 10 } - \frac { x } { x ^ { 2 } - 4 }$

A) $\frac { 3 x - 1 } { ( x - 5 ) ( x + 2 ) ( x - 2 ) }$
B) $\frac { 3 x ^ { 2 } - 14 x + 2 } { ( x - 5 ) ( x + 2 ) ( x - 2 ) }$
C) $\frac { 3 x ^ { 2 } - 4 x + 2 } { ( x - 5 ) ( x + 2 ) ( x - 2 ) }$
D) $\frac { 3 x - 1 } { ( x - 5 ) ( x + 2 ) }$

Question 71

Simplify:

A) Does not simplify
B)
C)
D)

Question 72

Perform the operations and simplify the result when possible. $\frac { 5 } { x ^ { 2 } - 3 x - 4 } - \frac { x } { 3 x ^ { 2 } - 10 x - 8 }$

A) $\frac { - x ^ { 2 } + 14 x + 10 } { ( x - 4 ) ( x + 1 ) ( 3 x + 2 ) }$
B) $\frac { x ^ { 2 } + 14 x + 10 } { ( x - 3 ) ( x + 1 ) ( 3 x + 2 ) }$
C) $\frac { - x ^ { 2 } + 14 x + 10 } { ( x - 4 ) ( x + 1 ) \left( 3 x ^ { 2 } - 7 x - 6 \right) }$
D) $\frac { - x ^ { 2 } + 14 x + 10 } { ( x + 1 ) ( 3 x + 2 ) }$
E) $\frac { - x ^ { 2 } + 10 x + 10 } { ( x - 4 ) ( x + 1 ) ( 3 x + 2 ) }$

Question 73

Find the domain of the rational function. Express your answer in interval notation.

Question 74

Perform the indicated operation. Simplify, if possible. $\frac { 2 x } { x ^ { 2 } - x - 6 } + \frac { x } { x ^ { 2 } + 8 x + 12 }$

A) $\frac { 3 x ^ { 2 } + 9 x } { ( x + 2 ) ( x - 3 ) ( x + 6 ) }$
B) $\frac { 3 x } { ( x + 2 ) ( x - 3 ) ( x + 6 ) }$
C) $\frac { 3 x ^ { 2 } + 15 x } { ( x + 2 ) ( x - 3 ) ( x + 6 ) }$
D) $\frac { 3 x ^ { 2 } + 15 } { ( x + 2 ) ( x - 3 ) ( x + 6 ) }$

Question 75

Simplify:

A)
B)
C)
D) simplified

Question 76

Perform the operations and simplify the result when possible. $\frac { 3 x } { 2 x - 1 } + \frac { x + 3 } { 3 x + 2 } - \frac { 9 x } { 6 x ^ { 3 } + x ^ { 2 } - 2 x }$

A) $\frac { 11 x ^ { 2 } + 11 x - 12 } { x ( 3 x + 2 ) ( 2 x - 1 ) }$
B) $\frac { 11 x ^ { 2 } + 11 x - 12 } { ( 3 x + 2 ) ( 2 x - 1 ) }$
C) $\frac { 11 x ^ { 2 } + x - 12 } { ( 3 x + 2 ) ( 2 x - 1 ) }$
D) $\frac { 11 x ^ { 2 } + 11 x - 9 } { ( 3 x + 2 ) ( 2 x - 1 ) }$
E) $\frac { 11 x ^ { 2 } - 11 x + 9 } { ( 3 x + 2 ) ( 2 x - 1 ) }$

Question 77

Match each function with the correct graph shown below.

A)
B)
C)
D)
E)
F)

Question 78

Perform the indicated operation. Simplify, if possible. $\frac { 5 } { x + 7 } + \frac { 3 } { x - 4 }$

A) $\frac { 8 x + 1 } { ( x + 7 ) ( x - 4 ) }$
B) $\frac { 8 } { ( x + 7 ) ( x - 4 ) }$
C) $\frac { 8 x ^ { 2 } + 1 } { ( x + 7 ) ( x - 4 ) }$
D) $\frac { 8 x + 7 } { ( x + 7 ) ( x - 4 ) }$

Question 79

The simplest of all rational functions, , is called the ________ function.

Question 80

Perform the indicated operation. Simplify, if possible. $\frac { 6 } { 4 x - 28 } + \frac { 4 } { 7 - x }$

A) $\frac { 11 } { 2 ( x - 7 ) }$
B) $\frac { 5 } { 2 ( x - 7 ) }$
C) $\frac { - 4 } { x - 7 }$
D) $\frac { - 5 } { 2 ( x - 7 ) }$