Exam 7: Rational Expressions and Equations

Solve the problem. -Amy can clean the house in 9 hours. When she works together with Tom, the job takes 4 hours. How long would it take Tom, working by himself, to clean the house?

(Multiple Choice)

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Question 21

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac { x ^ { 2 } - 4 } { x + 2 }$

(Multiple Choice)

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Question 22

Solve the problem. -A man rode a bicycle for 14 miles and hiked an additional 8 miles. The total time for the trip was 6 hours. If his bicycling rate was 5 miles per hour faster than his hiking rate, then what was his Hiking rate?

(Multiple Choice)

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Question 23

Simplify the complex fraction. - $\frac { 2 x + \frac { 4 } { x } } { \frac { 1 } { 3 x ^ { 2 } } + \frac { 1 } { 6 } }$

(Multiple Choice)

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Question 24

Add or subtract. - $\frac { 3 } { 3 x - 1 } - \frac { 4 } { 1 - 3 x }$

(Multiple Choice)

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Question 25

Find the missing numerator and denominator. - $\frac { x + 6 } { x - 6 } \cdot \frac { ? } { ? } = \frac { ( x + 6 ) ( x - 1 ) } { ( x - 6 ) ( x + 7 ) }$

(Multiple Choice)

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Question 26

Find the domain of the rational function. - $r ( x ) = \frac { x ^ { 2 } + 2 x - 8 } { x ^ { 2 } - 36 }$

(Multiple Choice)

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Question 27

Solve the problem. -Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 180 miles in the same time that Dana travels 165 miles. If Chuckʹs rate of travel is 3 mph more than Danaʹs, and they Travel the same length of time, at what speed does Chuck travel?

(Multiple Choice)

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Question 28

Subtract. - $\frac { 8 x + 7 y } { 3 x ^ { 2 } y } - \frac { 4 x - 3 y } { x y ^ { 2 } }$

(Multiple Choice)

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Question 29

Find the missing numerator and denominator. - $\frac { 36 x ^ { 2 } - 25 } { x ^ { 2 } - 4 } \div \frac { ? } { ? } = \frac { 6 x - 5 } { x - 2 }$

(Multiple Choice)

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Question 30

Find the LCD of the given rational expressions. - $\frac { 4 } { x } , \frac { 2 } { x + 5 }$

(Multiple Choice)

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Question 31

Simplify the complex fraction. - $\frac { 1 - \frac { 2 } { x } } { 1 + \frac { 2 } { x } }$

(Multiple Choice)

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Question 32

Simplify the complex fraction. - $\frac { \frac { 1 } { 8 } - \frac { 1 } { 10 } } { \frac { 1 } { 8 } + \frac { 1 } { 10 } }$

(Multiple Choice)

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Question 33

Add. - $\frac { 5 x + 9 } { x ^ { 2 } + 5 x } + \frac { x ^ { 2 } + 8 x + 31 } { x ^ { 2 } + 5 x }$

(Multiple Choice)

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Question 34

Solve. -The distance that a spring is stretched by a hanging object varies directly as the mass of the object. If an object with a mass of 27 kg stretches a spring 31 cm, find the stretch when the object has a Mass of 22 kg.

(Multiple Choice)

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Question 35

Determine whether the two binomials are or are not opposites. -3 - x and x - 3

(Multiple Choice)

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Question 36

Subtract. - $\frac { 13 } { x - 9 } - \frac { 5 } { x - 9 }$

(Multiple Choice)

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Question 37

For the given functions f(x) and g(x), find f(x) ÷ g(x). - $( x ) = \frac { x ^ { 2 } - 12 x + 35 } { x ^ { 2 } + 14 x + 49 } , g ( x ) = \frac { x ^ { 2 } - 10 x + 25 } { x ^ { 2 } - 8 x + 7 }$

(Multiple Choice)

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Question 38

Find all values of the variable for which the rational expression is undefined. - $\frac { x ^ { 2 } - 36 } { x ^ { 2 } - 14 x + 45 }$

(Multiple Choice)

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Question 39

Identify the given function as a linear function, a quadratic function, or a rational function. - $f ( x ) = \frac { 1 } { 2 } x + \frac { 1 } { 3 }$

(Multiple Choice)

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Question 40

Solve the problem. -Amy can clean the house in 9 hours. When she works together with Tom, the job takes 4 hours. How long would it take Tom, working by himself, to clean the house?

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac { x ^ { 2 } - 4 } { x + 2 }$

Solve the problem. -A man rode a bicycle for 14 miles and hiked an additional 8 miles. The total time for the trip was 6 hours. If his bicycling rate was 5 miles per hour faster than his hiking rate, then what was his Hiking rate?

Simplify the complex fraction. - $\frac { 2 x + \frac { 4 } { x } } { \frac { 1 } { 3 x ^ { 2 } } + \frac { 1 } { 6 } }$

Add or subtract. - $\frac { 3 } { 3 x - 1 } - \frac { 4 } { 1 - 3 x }$

Find the missing numerator and denominator. - $\frac { x + 6 } { x - 6 } \cdot \frac { ? } { ? } = \frac { ( x + 6 ) ( x - 1 ) } { ( x - 6 ) ( x + 7 ) }$

Find the domain of the rational function. - $r ( x ) = \frac { x ^ { 2 } + 2 x - 8 } { x ^ { 2 } - 36 }$

Solve the problem. -Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 180 miles in the same time that Dana travels 165 miles. If Chuckʹs rate of travel is 3 mph more than Danaʹs, and they Travel the same length of time, at what speed does Chuck travel?

Subtract. - $\frac { 8 x + 7 y } { 3 x ^ { 2 } y } - \frac { 4 x - 3 y } { x y ^ { 2 } }$

Find the missing numerator and denominator. - $\frac { 36 x ^ { 2 } - 25 } { x ^ { 2 } - 4 } \div \frac { ? } { ? } = \frac { 6 x - 5 } { x - 2 }$

Find the LCD of the given rational expressions. - $\frac { 4 } { x } , \frac { 2 } { x + 5 }$

Simplify the complex fraction. - $\frac { 1 - \frac { 2 } { x } } { 1 + \frac { 2 } { x } }$

Simplify the complex fraction. - $\frac { \frac { 1 } { 8 } - \frac { 1 } { 10 } } { \frac { 1 } { 8 } + \frac { 1 } { 10 } }$

Add. - $\frac { 5 x + 9 } { x ^ { 2 } + 5 x } + \frac { x ^ { 2 } + 8 x + 31 } { x ^ { 2 } + 5 x }$

Solve. -The distance that a spring is stretched by a hanging object varies directly as the mass of the object. If an object with a mass of 27 kg stretches a spring 31 cm, find the stretch when the object has a Mass of 22 kg.

Determine whether the two binomials are or are not opposites. -3 - x and x - 3

Subtract. - $\frac { 13 } { x - 9 } - \frac { 5 } { x - 9 }$

For the given functions f(x) and g(x), find f(x) ÷ g(x). - $( x ) = \frac { x ^ { 2 } - 12 x + 35 } { x ^ { 2 } + 14 x + 49 } , g ( x ) = \frac { x ^ { 2 } - 10 x + 25 } { x ^ { 2 } - 8 x + 7 }$

Find all values of the variable for which the rational expression is undefined. - $\frac { x ^ { 2 } - 36 } { x ^ { 2 } - 14 x + 45 }$

Identify the given function as a linear function, a quadratic function, or a rational function. - $f ( x ) = \frac { 1 } { 2 } x + \frac { 1 } { 3 }$

Review of Real Numbers

Linear Equations

Graphing Linear Equations

Systems of Equations

Exponents and Polynomials

Factoring and Quadratic Equations

A Transition

Radical Expressions and Equations

Quadratic Equations

Functions

Logarithmic and Exponential Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Exam 7: Rational Expressions and Equations

Solve the problem. -Amy can clean the house in 9 hours. When she works together with Tom, the job takes 4 hours. How long would it take Tom, working by himself, to clean the house?

Simplify the rational expression. (Assume the denominator is nonzero.) - x2−4x+2\frac { x ^ { 2 } - 4 } { x + 2 }x+2x2−4​

Solve the problem. -A man rode a bicycle for 14 miles and hiked an additional 8 miles. The total time for the trip was 6 hours. If his bicycling rate was 5 miles per hour faster than his hiking rate, then what was his Hiking rate?

Simplify the complex fraction. - 2x+4x13x2+16\frac { 2 x + \frac { 4 } { x } } { \frac { 1 } { 3 x ^ { 2 } } + \frac { 1 } { 6 } }3x21​+61​2x+x4​​

Add or subtract. - 33x−1−41−3x\frac { 3 } { 3 x - 1 } - \frac { 4 } { 1 - 3 x }3x−13​−1−3x4​

Find the missing numerator and denominator. - x+6x−6⋅??=(x+6)(x−1)(x−6)(x+7)\frac { x + 6 } { x - 6 } \cdot \frac { ? } { ? } = \frac { ( x + 6 ) ( x - 1 ) } { ( x - 6 ) ( x + 7 ) }x−6x+6​⋅??​=(x−6)(x+7)(x+6)(x−1)​

Find the domain of the rational function. - r(x)=x2+2x−8x2−36r ( x ) = \frac { x ^ { 2 } + 2 x - 8 } { x ^ { 2 } - 36 }r(x)=x2−36x2+2x−8​

Solve the problem. -Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 180 miles in the same time that Dana travels 165 miles. If Chuckʹs rate of travel is 3 mph more than Danaʹs, and they Travel the same length of time, at what speed does Chuck travel?

Subtract. - 8x+7y3x2y−4x−3yxy2\frac { 8 x + 7 y } { 3 x ^ { 2 } y } - \frac { 4 x - 3 y } { x y ^ { 2 } }3x2y8x+7y​−xy24x−3y​

Find the missing numerator and denominator. - 36x2−25x2−4÷??=6x−5x−2\frac { 36 x ^ { 2 } - 25 } { x ^ { 2 } - 4 } \div \frac { ? } { ? } = \frac { 6 x - 5 } { x - 2 }x2−436x2−25​÷??​=x−26x−5​

Find the LCD of the given rational expressions. - 4x,2x+5\frac { 4 } { x } , \frac { 2 } { x + 5 }x4​,x+52​

Simplify the complex fraction. - 1−2x1+2x\frac { 1 - \frac { 2 } { x } } { 1 + \frac { 2 } { x } }1+x2​1−x2​​

Simplify the complex fraction. - 18−11018+110\frac { \frac { 1 } { 8 } - \frac { 1 } { 10 } } { \frac { 1 } { 8 } + \frac { 1 } { 10 } }81​+101​81​−101​​

Add. - 5x+9x2+5x+x2+8x+31x2+5x\frac { 5 x + 9 } { x ^ { 2 } + 5 x } + \frac { x ^ { 2 } + 8 x + 31 } { x ^ { 2 } + 5 x }x2+5x5x+9​+x2+5xx2+8x+31​

Solve. -The distance that a spring is stretched by a hanging object varies directly as the mass of the object. If an object with a mass of 27 kg stretches a spring 31 cm, find the stretch when the object has a Mass of 22 kg.

Determine whether the two binomials are or are not opposites. -3 - x and x - 3

Subtract. - 13x−9−5x−9\frac { 13 } { x - 9 } - \frac { 5 } { x - 9 }x−913​−x−95​

For the given functions f(x) and g(x), find f(x) ÷ g(x). - (x)=x2−12x+35x2+14x+49,g(x)=x2−10x+25x2−8x+7( x ) = \frac { x ^ { 2 } - 12 x + 35 } { x ^ { 2 } + 14 x + 49 } , g ( x ) = \frac { x ^ { 2 } - 10 x + 25 } { x ^ { 2 } - 8 x + 7 }(x)=x2+14x+49x2−12x+35​,g(x)=x2−8x+7x2−10x+25​

Find all values of the variable for which the rational expression is undefined. - x2−36x2−14x+45\frac { x ^ { 2 } - 36 } { x ^ { 2 } - 14 x + 45 }x2−14x+45x2−36​

Identify the given function as a linear function, a quadratic function, or a rational function. - f(x)=12x+13f ( x ) = \frac { 1 } { 2 } x + \frac { 1 } { 3 }f(x)=21​x+31​

Review of Real Numbers

Linear Equations

Graphing Linear Equations

Systems of Equations

Exponents and Polynomials

Factoring and Quadratic Equations

A Transition

Radical Expressions and Equations

Quadratic Equations

Functions

Logarithmic and Exponential Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac { x ^ { 2 } - 4 } { x + 2 }$

Simplify the complex fraction. - $\frac { 2 x + \frac { 4 } { x } } { \frac { 1 } { 3 x ^ { 2 } } + \frac { 1 } { 6 } }$

Add or subtract. - $\frac { 3 } { 3 x - 1 } - \frac { 4 } { 1 - 3 x }$

Find the missing numerator and denominator. - $\frac { x + 6 } { x - 6 } \cdot \frac { ? } { ? } = \frac { ( x + 6 ) ( x - 1 ) } { ( x - 6 ) ( x + 7 ) }$

Find the domain of the rational function. - $r ( x ) = \frac { x ^ { 2 } + 2 x - 8 } { x ^ { 2 } - 36 }$

Subtract. - $\frac { 8 x + 7 y } { 3 x ^ { 2 } y } - \frac { 4 x - 3 y } { x y ^ { 2 } }$

Find the missing numerator and denominator. - $\frac { 36 x ^ { 2 } - 25 } { x ^ { 2 } - 4 } \div \frac { ? } { ? } = \frac { 6 x - 5 } { x - 2 }$

Find the LCD of the given rational expressions. - $\frac { 4 } { x } , \frac { 2 } { x + 5 }$

Simplify the complex fraction. - $\frac { 1 - \frac { 2 } { x } } { 1 + \frac { 2 } { x } }$

Simplify the complex fraction. - $\frac { \frac { 1 } { 8 } - \frac { 1 } { 10 } } { \frac { 1 } { 8 } + \frac { 1 } { 10 } }$

Add. - $\frac { 5 x + 9 } { x ^ { 2 } + 5 x } + \frac { x ^ { 2 } + 8 x + 31 } { x ^ { 2 } + 5 x }$

Subtract. - $\frac { 13 } { x - 9 } - \frac { 5 } { x - 9 }$

For the given functions f(x) and g(x), find f(x) ÷ g(x). - $( x ) = \frac { x ^ { 2 } - 12 x + 35 } { x ^ { 2 } + 14 x + 49 } , g ( x ) = \frac { x ^ { 2 } - 10 x + 25 } { x ^ { 2 } - 8 x + 7 }$

Find all values of the variable for which the rational expression is undefined. - $\frac { x ^ { 2 } - 36 } { x ^ { 2 } - 14 x + 45 }$

Identify the given function as a linear function, a quadratic function, or a rational function. - $f ( x ) = \frac { 1 } { 2 } x + \frac { 1 } { 3 }$