Question 1

Add or subtract.
-$\frac { 5 x } { 4 x - 1 } - \frac { 9 } { 4 x - 1 }$

Accepted Answer

A)  $\frac { 5 x - 9 } { 2 ( 4 x - 1 ) }$ 
B)  $\frac { 5 x - 9 } { 4 x - 1 }$ 
C)  $\frac { 5 x + 9 } { 4 x - 1 }$ 
D)  $\frac { - 4 x } { 4 x - 1 }$ 
A)  $\frac { 5 x - 9 } { 2 ( 4 x - 1 ) }$ 
B)  $\frac { 5 x - 9 } { 4 x - 1 }$ 
C)  $\frac { 5 x + 9 } { 4 x - 1 }$ 
D)  $\frac { - 4 x } { 4 x - 1 }$

Question 2

Divide.
-$\frac { a ^ { 2 } - 14 a + 45 } { 9 - a } \div ( a + 5 )$

Accepted Answer

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

Question 3

Add or subtract.
-$\frac { m ^ { 2 } - 9 m } { m - 6 } + \frac { 18 } { m - 6 }$

Accepted Answer

A)  $\frac { m ^ { 2 } - 9 m + 18 } { m - 6 }$ 
B) m - 6 
C) m - 3 
D) m + 3 
A)  $\frac { m ^ { 2 } - 9 m + 18 } { m - 6 }$ 
B) m - 6 
C) m - 3 
D) m + 3

Question 4

Divide.
-$\frac { 5 x - 5 } { x } \div \frac { 8 x - 8 } { 7 x ^ { 2 } }$

Accepted Answer

A)  $\frac { 8 } { 35 x }$ 
B)  $\frac { 35 x ^ { 2 } ( x - 1 ) } { 8 x ( x - 1 ) }$ 
C)  $\frac { 40 ( x + 1 ) ^ { 2 } } { 7 x ^ { 3 } }$ 
D)  $\frac { 35 x } { 8 }$ 
A)  $\frac { 8 } { 35 x }$ 
B)  $\frac { 35 x ^ { 2 } ( x - 1 ) } { 8 x ( x - 1 ) }$ 
C)  $\frac { 40 ( x + 1 ) ^ { 2 } } { 7 x ^ { 3 } }$ 
D)  $\frac { 35 x } { 8 }$

Question 5

Perform the indicated operations.
-$\frac { 5 x ^ { 2 } + 2 x - 16 } { 2 x - 16 } \cdot \frac { x ^ { 2 } - 8 x } { 25 x ^ { 2 } - 64 } \div \frac { 5 x + 10 } { 7 x ^ { 3 } }$

Accepted Answer

A)  $\frac { 7 x ^ { 3 } } { 10 ( 5 x + 8 ) }$ 
B)  $\frac { 7 x ^ { 4 } } { 10 ( 5 x + 8 ) }$ 
C)  $\frac { 10 } { 7 x ^ { 4 } ( 5 x + 8 ) }$ 
D)  $\frac { 5 ( x + 2 ) ^ { 2 } } { 14 x ^ { 2 } ( 5 x + 8 ) }$ 
A)  $\frac { 7 x ^ { 3 } } { 10 ( 5 x + 8 ) }$ 
B)  $\frac { 7 x ^ { 4 } } { 10 ( 5 x + 8 ) }$ 
C)  $\frac { 10 } { 7 x ^ { 4 } ( 5 x + 8 ) }$ 
D)  $\frac { 5 ( x + 2 ) ^ { 2 } } { 14 x ^ { 2 } ( 5 x + 8 ) }$

Question 6

Find the least common denominator.
-$\frac { 5 } { x ^ { 2 } - 81 } + \frac { 4 } { x ( x - 9 ) }$

Accepted Answer

A) x(x + 9) 
B) x(x - 9) 
C)  $\left( x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 9 \right)$ 
D) x(x + 9)(x - 9) 
A) x(x + 9) 
B) x(x - 9) 
C)  $\left( x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 9 \right)$ 
D) x(x + 9)(x - 9)

Question 7

Multiply.
-$\frac { x ^ { 3 } + 1 } { x ^ { 3 } - x ^ { 2 } + x } \cdot \frac { 9 x } { - 27 x - 27 }$

Accepted Answer

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

Question 8

Multiply.
-$\frac { y ^ { 2 } - 25 } { y ^ { 2 } - 9 } \cdot \frac { y ^ { 2 } + 7 y + 12 } { y ^ { 2 } - 8 y + 15 }$

Accepted Answer

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

Question 9

Divide.
-$\frac { p ^ { 2 } - 10 p + p q - 10 q } { 5 p ^ { 2 } - 5 q ^ { 2 } } \div \frac { p - 10 } { 6 p - 6 q }$

Accepted Answer

A)  $\frac { 6 \left( p ^ { 2 } - 10 p + p q - 10 q \right) } { 5 ( p + q ) ( p - 10 ) }$ 
B)  $\frac { ( p - 10 ) ^ { 2 } } { 30 ( p - q ) ^ { 2 } }$ 
C) 1 
D)  $\frac { 6 } { 5 }$ 
A)  $\frac { 6 \left( p ^ { 2 } - 10 p + p q - 10 q \right) } { 5 ( p + q ) ( p - 10 ) }$ 
B)  $\frac { ( p - 10 ) ^ { 2 } } { 30 ( p - q ) ^ { 2 } }$ 
C) 1 
D)  $\frac { 6 } { 5 }$

Question 10

Find the least common denominator.
-$\frac { 5 } { x ^ { 2 } + 2 x - 3 } + \frac { 4 } { - 2 x + 2 }$

Accepted Answer

A) -2(x + 3)(x + 1) 
B) -2(x - 3)(x - 1) 
C) -2(x + 3)(x - 1) 
D) -2(x - 3)(x + 1) 
A) -2(x + 3)(x + 1) 
B) -2(x - 3)(x - 1) 
C) -2(x + 3)(x - 1) 
D) -2(x - 3)(x + 1)

Question 11

Determine the value or values that are excluded from the following expression.
-$$\frac { 7 } { x + 8 }$$

Accepted Answer

A) -8 
B) 0 
C) 8 
D) none 
A) -8 
B) 0 
C) 8 
D) none

Question 12

Divide.
-$\frac { x - y } { x y } \div \frac { x ^ { 2 } - y ^ { 2 } } { 2 x ^ { 2 } y }$

Accepted Answer

A)  $\frac { 2 x } { x - y }$ 
B)  $\frac { 2 } { y }$ 
C)  $- \frac { 2 } { y }$ 
D)  $\frac { 2 x } { x + y }$ 
A)  $\frac { 2 x } { x - y }$ 
B)  $\frac { 2 } { y }$ 
C)  $- \frac { 2 } { y }$ 
D)  $\frac { 2 x } { x + y }$

Question 13

Determine the value or values that are excluded from the following expression.
-$\frac { 11 x - 4 } { ( x + 7 ) ^ { 2 } }$

Accepted Answer

A)  $\frac { 4 } { 11 } , 7$ 
B) 7 
C) -7 
D) none 
A)  $\frac { 4 } { 11 } , 7$ 
B) 7 
C) -7 
D) none

Question 14

Determine the domain of the function.
-$f ( x ) = \frac { - 2 } { 4 x + 7 }$

Accepted Answer

A)  $\{ x \mid x \neq 2 \}$ 
B)  $\left\{ x \mid x \neq - \frac { 7 } { 4 } \right\}$ 
C)  $\{ x \mid x \neq 0 \}$ 
D)  $\left\{ x \mid x \neq \frac { 7 } { 4 } \right\}$ 
A)  $\{ x \mid x \neq 2 \}$ 
B)  $\left\{ x \mid x \neq - \frac { 7 } { 4 } \right\}$ 
C)  $\{ x \mid x \neq 0 \}$ 
D)  $\left\{ x \mid x \neq \frac { 7 } { 4 } \right\}$

Question 15

Multiply.
-$\frac { 7 y } { 14 y + 7 } \cdot \frac { 10 y + 5 } { 4 }$

Accepted Answer

A)  $\frac { 5 y } { 28 }$ 
B)  $\frac { y } { 4 }$ 
C)  $\frac { 5 y } { 4 }$ 
D)  $\frac { 5 } { 4 }$ 
A)  $\frac { 5 y } { 28 }$ 
B)  $\frac { y } { 4 }$ 
C)  $\frac { 5 y } { 4 }$ 
D)  $\frac { 5 } { 4 }$

Question 16

Divide.
-$\frac { x ^ { 2 } - 9 } { 4 x + 12 } \div \frac { 5 x ^ { 2 } - 30 x + 45 } { x ^ { 2 } + 6 x + 9 }$

Accepted Answer

A)  $\frac { 20 ( x - 3 ) } { ( x + 3 ) ^ { 2 } }$ 
B)  $\frac { ( x + 3 ) ^ { 2 } } { 20 ( x + 3 ) }$ 
C)  $\frac { ( x + 3 ) } { ( x + 3 ) ^ { 2 } }$ 
D)  $\frac { ( x + 3 ) ^ { 2 } } { 20 ( x - 3 ) }$ 
A)  $\frac { 20 ( x - 3 ) } { ( x + 3 ) ^ { 2 } }$ 
B)  $\frac { ( x + 3 ) ^ { 2 } } { 20 ( x + 3 ) }$ 
C)  $\frac { ( x + 3 ) } { ( x + 3 ) ^ { 2 } }$ 
D)  $\frac { ( x + 3 ) ^ { 2 } } { 20 ( x - 3 ) }$

Question 17

Simplify the rational expression.
-$\frac { x ^ { 2 } + 6 x - x y - 6 y } { x ^ { 2 } - x y + 5 x - 5 y }$

Accepted Answer

A)  $\frac { x + 6 } { x - 5 }$ 
B)  $\frac { x + 6 } { x + 5 }$ 
C)  $\frac { 6 x - 6 y } { 5 x + 5 y }$ 
D)  $\frac { x - 6 } { x + 5 }$ 
A)  $\frac { x + 6 } { x - 5 }$ 
B)  $\frac { x + 6 } { x + 5 }$ 
C)  $\frac { 6 x - 6 y } { 5 x + 5 y }$ 
D)  $\frac { x - 6 } { x + 5 }$

Question 18

Determine the value or values that are excluded from the following expression.
-$$f ( x ) = \frac { x } { 3 x - 8 }$$

Accepted Answer

A)  $- \frac { 3 } { 8 } , 0$ 
B)  $- \frac { 8 } { 3 }$ 
C)  $\frac { 3 } { 8 } , 0$ 
D)  $\frac { 8 } { 3 }$ 
A)  $- \frac { 3 } { 8 } , 0$ 
B)  $- \frac { 8 } { 3 }$ 
C)  $\frac { 3 } { 8 } , 0$ 
D)  $\frac { 8 } { 3 }$

Question 19

Divide.
-$\frac { x ^ { 2 } - 4 } { x ^ { 3 } - 8 } \div \frac { x ^ { 2 } + 4 x + 4 } { \left( x ^ { 2 } + 2 x + 4 \right) }$

Accepted Answer

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

Question 20

Add or subtract.
-$\frac { 5 x } { x + 8 } + \frac { 3 } { x + 8 }$

Accepted Answer

A)  $\frac { 5 x - 3 } { x + 8 }$ 
B)  $\frac { 8 x } { x + 8 }$ 
C)  $\frac { 5 x + 3 } { 2 ( x + 8 ) }$ 
D)  $\frac { 5 x + 3 } { x + 8 }$ 
A)  $\frac { 5 x - 3 } { x + 8 }$ 
B)  $\frac { 8 x } { x + 8 }$ 
C)  $\frac { 5 x + 3 } { 2 ( x + 8 ) }$ 
D)  $\frac { 5 x + 3 } { x + 8 }$

Add or subtract. - $\frac { 5 x } { 4 x - 1 } - \frac { 9 } { 4 x - 1 }$

Divide. - $\frac { a ^ { 2 } - 14 a + 45 } { 9 - a } \div ( a + 5 )$

Add or subtract. - $\frac { m ^ { 2 } - 9 m } { m - 6 } + \frac { 18 } { m - 6 }$

Divide. - $\frac { 5 x - 5 } { x } \div \frac { 8 x - 8 } { 7 x ^ { 2 } }$

Perform the indicated operations. - $\frac { 5 x ^ { 2 } + 2 x - 16 } { 2 x - 16 } \cdot \frac { x ^ { 2 } - 8 x } { 25 x ^ { 2 } - 64 } \div \frac { 5 x + 10 } { 7 x ^ { 3 } }$

Find the least common denominator. - $\frac { 5 } { x ^ { 2 } - 81 } + \frac { 4 } { x ( x - 9 ) }$

Multiply. - $\frac { x ^ { 3 } + 1 } { x ^ { 3 } - x ^ { 2 } + x } \cdot \frac { 9 x } { - 27 x - 27 }$

Multiply. - $\frac { y ^ { 2 } - 25 } { y ^ { 2 } - 9 } \cdot \frac { y ^ { 2 } + 7 y + 12 } { y ^ { 2 } - 8 y + 15 }$

Divide. - $\frac { p ^ { 2 } - 10 p + p q - 10 q } { 5 p ^ { 2 } - 5 q ^ { 2 } } \div \frac { p - 10 } { 6 p - 6 q }$

Find the least common denominator. - $\frac { 5 } { x ^ { 2 } + 2 x - 3 } + \frac { 4 } { - 2 x + 2 }$

Determine the value or values that are excluded from the following expression. - $\frac { 7 } { x + 8 }$

Divide. - $\frac { x - y } { x y } \div \frac { x ^ { 2 } - y ^ { 2 } } { 2 x ^ { 2 } y }$

Determine the value or values that are excluded from the following expression. - $\frac { 11 x - 4 } { ( x + 7 ) ^ { 2 } }$

Determine the domain of the function. - $f ( x ) = \frac { - 2 } { 4 x + 7 }$

Multiply. - $\frac { 7 y } { 14 y + 7 } \cdot \frac { 10 y + 5 } { 4 }$

Divide. - $\frac { x ^ { 2 } - 9 } { 4 x + 12 } \div \frac { 5 x ^ { 2 } - 30 x + 45 } { x ^ { 2 } + 6 x + 9 }$

Simplify the rational expression. - $\frac { x ^ { 2 } + 6 x - x y - 6 y } { x ^ { 2 } - x y + 5 x - 5 y }$

Determine the value or values that are excluded from the following expression. - $f ( x ) = \frac { x } { 3 x - 8 }$

Divide. - $\frac { x ^ { 2 } - 4 } { x ^ { 3 } - 8 } \div \frac { x ^ { 2 } + 4 x + 4 } { \left( x ^ { 2 } + 2 x + 4 \right) }$

Add or subtract. - $\frac { 5 x } { x + 8 } + \frac { 3 } { x + 8 }$

Basic Concepts

Equations and Inequalities

Graphs and Functions

Systems of Equations and Inequalities

Polynomials and Polynomial Functions

Roots, Radicals, and Complex Numbers

Quadratic Functions

Exponential and Logarithmic Functions

Characteristics of Functions and Their Graphs

Conic Sections

Sequences, Series, and Probability

Filters

Exam 6: Rational Expressions and Equations

Add or subtract. - 5x4x−1−94x−1\frac { 5 x } { 4 x - 1 } - \frac { 9 } { 4 x - 1 }4x−15x​−4x−19​

Divide. - a2−14a+459−a÷(a+5)\frac { a ^ { 2 } - 14 a + 45 } { 9 - a } \div ( a + 5 )9−aa2−14a+45​÷(a+5)

Add or subtract. - m2−9mm−6+18m−6\frac { m ^ { 2 } - 9 m } { m - 6 } + \frac { 18 } { m - 6 }m−6m2−9m​+m−618​

Divide. - 5x−5x÷8x−87x2\frac { 5 x - 5 } { x } \div \frac { 8 x - 8 } { 7 x ^ { 2 } }x5x−5​÷7x28x−8​

Perform the indicated operations. - 5x2+2x−162x−16⋅x2−8x25x2−64÷5x+107x3\frac { 5 x ^ { 2 } + 2 x - 16 } { 2 x - 16 } \cdot \frac { x ^ { 2 } - 8 x } { 25 x ^ { 2 } - 64 } \div \frac { 5 x + 10 } { 7 x ^ { 3 } }2x−165x2+2x−16​⋅25x2−64x2−8x​÷7x35x+10​

Find the least common denominator. - 5x2−81+4x(x−9)\frac { 5 } { x ^ { 2 } - 81 } + \frac { 4 } { x ( x - 9 ) }x2−815​+x(x−9)4​

Multiply. - x3+1x3−x2+x⋅9x−27x−27\frac { x ^ { 3 } + 1 } { x ^ { 3 } - x ^ { 2 } + x } \cdot \frac { 9 x } { - 27 x - 27 }x3−x2+xx3+1​⋅−27x−279x​

Multiply. - y2−25y2−9⋅y2+7y+12y2−8y+15\frac { y ^ { 2 } - 25 } { y ^ { 2 } - 9 } \cdot \frac { y ^ { 2 } + 7 y + 12 } { y ^ { 2 } - 8 y + 15 }y2−9y2−25​⋅y2−8y+15y2+7y+12​

Divide. - p2−10p+pq−10q5p2−5q2÷p−106p−6q\frac { p ^ { 2 } - 10 p + p q - 10 q } { 5 p ^ { 2 } - 5 q ^ { 2 } } \div \frac { p - 10 } { 6 p - 6 q }5p2−5q2p2−10p+pq−10q​÷6p−6qp−10​

Find the least common denominator. - 5x2+2x−3+4−2x+2\frac { 5 } { x ^ { 2 } + 2 x - 3 } + \frac { 4 } { - 2 x + 2 }x2+2x−35​+−2x+24​

Determine the value or values that are excluded from the following expression. - 7x+8\frac { 7 } { x + 8 }x+87​

Divide. - x−yxy÷x2−y22x2y\frac { x - y } { x y } \div \frac { x ^ { 2 } - y ^ { 2 } } { 2 x ^ { 2 } y }xyx−y​÷2x2yx2−y2​

Determine the value or values that are excluded from the following expression. - 11x−4(x+7)2\frac { 11 x - 4 } { ( x + 7 ) ^ { 2 } }(x+7)211x−4​

Determine the domain of the function. - f(x)=−24x+7f ( x ) = \frac { - 2 } { 4 x + 7 }f(x)=4x+7−2​

Multiply. - 7y14y+7⋅10y+54\frac { 7 y } { 14 y + 7 } \cdot \frac { 10 y + 5 } { 4 }14y+77y​⋅410y+5​

Divide. - x2−94x+12÷5x2−30x+45x2+6x+9\frac { x ^ { 2 } - 9 } { 4 x + 12 } \div \frac { 5 x ^ { 2 } - 30 x + 45 } { x ^ { 2 } + 6 x + 9 }4x+12x2−9​÷x2+6x+95x2−30x+45​

Simplify the rational expression. - x2+6x−xy−6yx2−xy+5x−5y\frac { x ^ { 2 } + 6 x - x y - 6 y } { x ^ { 2 } - x y + 5 x - 5 y }x2−xy+5x−5yx2+6x−xy−6y​

Determine the value or values that are excluded from the following expression. - f(x)=x3x−8f ( x ) = \frac { x } { 3 x - 8 }f(x)=3x−8x​

Divide. - x2−4x3−8÷x2+4x+4(x2+2x+4)\frac { x ^ { 2 } - 4 } { x ^ { 3 } - 8 } \div \frac { x ^ { 2 } + 4 x + 4 } { \left( x ^ { 2 } + 2 x + 4 \right) }x3−8x2−4​÷(x2+2x+4)x2+4x+4​

Add or subtract. - 5xx+8+3x+8\frac { 5 x } { x + 8 } + \frac { 3 } { x + 8 }x+85x​+x+83​