Question 1

Perform the operation indicated. Simplify.
-$$\frac { \frac { x } { 8 } + \frac { 1 } { 2 } } { \frac { 1 } { 4 x } - \frac { 5 } { 8 } }$$

Accepted Answer

A)    x2+4x1−5x\frac { x ^ { 2 } + 4 x } { 1 - 5 x }1−5xx2+4x​ 
B)    x+42−5x\frac { x + 4 } { 2 - 5 x }2−5xx+4​ 
C)    x2+4x5x+2\frac { x ^ { 2 } + 4 x } { 5 x + 2 }5x+2x2+4x​ 
D)    x2+4x2−5x\frac { x ^ { 2 } + 4 x } { 2 - 5 x }2−5xx2+4x​ D

Question 2

Perform the operation indicated. Simplify.
-$$\frac { 3 } { y ^ { 2 } - 3 y + 2 } + \frac { 5 } { y - 1 }$$

Accepted Answer

A)    5y+7(y+2) (y−1) \frac { 5 y + 7 } { ( y + 2 )  ( y - 1 )  }(y+2) (y−1) 5y+7​ 
B)    8(y−2) (y−1) \frac { 8 } { ( y - 2 )  ( y - 1 )  }(y−2) (y−1) 8​ 
C)    y−7(y−2) (y−1) \frac { y - 7 } { ( y - 2 )  ( y - 1 )  }(y−2) (y−1) y−7​ 
D)    5y−7(y−2) (y−1) \frac { 5 y - 7 } { ( y - 2 )  ( y - 1 )  }(y−2) (y−1) 5y−7​ D

Question 3

Perform the operation indicated. Simplify.
-$\frac { \frac { 36 q ^ { 2 } - 25 p ^ { 2 } } { p q } } { \frac { 6 } { q } - \frac { 5 } { p } }$

Accepted Answer

A)    pq6p+5q\frac { p q } { 6 p + 5 q }6p+5qpq​ 
B)    5p+6q5 p + 6 q5p+6q 
C)    5p+6qpq\frac { 5 p + 6 q } { p q }pq5p+6q​ 
D)    6p+5q6 p + 5 q6p+5q D

Question 4

Simplify.
-$$\frac { y ^ { 2 } + 9 y + 20 } { y ^ { 2 } + 10 y + 24 }$$

Accepted Answer

A)    9y+2010y+24\frac { 9 y + 20 } { 10 y + 24 }10y+249y+20​ 
B)    −y2+9y+20y2+10y+24- \frac { y ^ { 2 } + 9 y + 20 } { y ^ { 2 } + 10 y + 24 }−y2+10y+24y2+9y+20​ 
C)    y+5y+6\frac { y + 5 } { y + 6 }y+6y+5​ 
D)    9y+510y+6\frac { 9 y + 5 } { 10 y + 6 }10y+69y+5​

Question 5

Perform the operation indicated. Simplify.
-$$\frac { 2 a b } { a ^ { 2 } - b ^ { 2 } } - \frac { b } { a - b } + \frac { 8 } { 2 }$$

Accepted Answer

A)    4a+5ba+b\frac { 4 a + 5 b } { a + b }a+b4a+5b​ 
B)    2ab−b+4a+b+1\frac { 2 a b - b + 4 } { a + b + 1 }a+b+12ab−b+4​ 
C)    (a−b) (4a+5b) a2−b2\frac { ( a - b )  ( 4 a + 5 b )  } { a ^ { 2 } - b ^ { 2 } }a2−b2(a−b) (4a+5b) ​ 
D)    4a+5ba2−b2\frac { 4 a + 5 b } { a ^ { 2 } - b ^ { 2 } }a2−b24a+5b​

Question 6

Perform the operation indicated. Simplify.
-$$\frac { 6 x } { x + 1 } + \frac { 7 } { x - 1 } - \frac { 12 } { x ^ { 2 } - 1 }$$

Accepted Answer

A)    6x−5x−1\frac { 6 x - 5 } { x - 1 }x−16x−5​ 
B)    6xx−1\frac { 6 x } { x - 1 }x−16x​ 
C)    x+1x−1\frac { x + 1 } { x - 1 }x−1x+1​ 
D)    6x−5x+1\frac { 6 x - 5 } { x + 1 }x+16x−5​

Question 7

Perform the operation indicated.
-$$\frac { 15 r ^ { 2 } - 14 r t - 8 t ^ { 2 } } { 10 r ^ { 2 } - 21 r t - 10 t ^ { 2 } } \cdot \frac { 30 r ^ { 2 } - 11 r t - 28 t ^ { 2 } } { 15 r ^ { 2 } - 8 r t - 16 t ^ { 2 } }$$

Accepted Answer

A)    6r−7t2r−5t\frac { 6 r - 7 t } { 2 r - 5 t }2r−5t6r−7t​ 
B)    (3r+4t) 2(6r+7t) (2r+5t) \frac { ( 3 r + 4 t )  ^ { 2 } } { ( 6 r + 7 t )  ( 2 r + 5 t )  }(6r+7t) (2r+5t) (3r+4t) 2​ 
C)    (3r−4t) 2(6r−7t) (2r−5t) \frac { ( 3 r - 4 t )  ^ { 2 } } { ( 6 r - 7 t )  ( 2 r - 5 t )  }(6r−7t) (2r−5t) (3r−4t) 2​ 
D)    (2r+5t) (6r+7t) \frac { ( 2 r + 5 t )  } { ( 6 r + 7 t )  }(6r+7t) (2r+5t) ​

Question 8

Perform the operation indicated. Simplify.
-$$\frac { 5 x } { x - 4 } - \frac { 20 } { x - 4 }$$

Accepted Answer

A)    5(x+4) x−4\frac { 5 ( x + 4 )  } { x - 4 }x−45(x+4) ​ 
B)    5x−4\frac { 5 } { x - 4 }x−45​ 
C)   5
D)   1

Question 9

Find the LCD. Do not combine fractions.
-$\frac { 5 } { x ^ { 2 } - 9 x + 20 } , \frac { 9 } { x ^ { 2 } - 8 x + 16 }$

Accepted Answer

A)    (x+5) (x+4) (x−4) ( x + 5 )  ( x + 4 )  ( x - 4 ) (x+5) (x+4) (x−4)  
B)    (x−4) (x−4) ( x - 4 )  ( x - 4 ) (x−4) (x−4)  
C)    (x−5) (x−4) (x−4) ( x - 5 )  ( x - 4 )  ( x - 4 ) (x−5) (x−4) (x−4)  
D)    (x−5) (x−4) ( x - 5 )  ( x - 4 ) (x−5) (x−4)

Question 10

Perform the operation indicated. Simplify.
-$$\frac { x } { x ^ { 2 } - 16 } - \frac { 6 } { x ^ { 2 } + 5 x + 4 }$$

Accepted Answer

A)    x2−5(x−4) (x+4) (x+1) \frac { x ^ { 2 } - 5 } { ( x - 4 )  ( x + 4 )  ( x + 1 )  }(x−4) (x+4) (x+1) x2−5​ 
B)    x2−5x+24(x−4) (x+4) \frac { x ^ { 2 } - 5 x + 24 } { ( x - 4 )  ( x + 4 )  }(x−4) (x+4) x2−5x+24​ 
C)   x2+5x+24(x−4) (x+4) (x+1) \frac { x ^ { 2 } + 5 x + 24 } { ( x - 4 )  ( x + 4 )  ( x + 1 )  }(x−4) (x+4) (x+1) x2+5x+24​ 
D)    x2−5x+24(x−4) (x+4) (x+1) \frac { x ^ { 2 } - 5 x + 24 } { ( x - 4 )  ( x + 4 )  ( x + 1 )  }(x−4) (x+4) (x+1) x2−5x+24​

Question 11

Simplify.
-$$\frac { ( 7 x - 5 ) ^ { 2 } } { 7 x ^ { 2 } - 61 x + 40 }$$

Accepted Answer

A)    7x−57 x - 57x−5 
B)    (7x−5) 2(x−8) (7x+5) \frac { ( 7 x - 5 )  ^ { 2 } } { ( x - 8 )  ( 7 x + 5 )  }(x−8) (7x+5) (7x−5) 2​ 
C)    1x−8\frac { 1 } { x - 8 }x−81​ 
D)    7x−5x−8\frac { 7 x - 5 } { x - 8 }x−87x−5​

Question 12

Find the LCD. Do not combine fractions.
-$$\frac { 4 } { x ^ { 2 } + 6 x } , \frac { 9 } { x ^ { 2 } + 3 x - 18 }$$

Accepted Answer

A)    x(x+9) 2x ( x + 9 )  ^ { 2 }x(x+9) 2 
B)    x(x+6) (x−3) x ( x + 6 )  ( x - 3 ) x(x+6) (x−3)  
C)    x(x+9) (x−3) x ( x + 9 )  ( x - 3 ) x(x+9) (x−3)  
D)    (x+9) 2( x + 9 )  ^ { 2 }(x+9) 2

Question 13

Perform the operation indicated. Simplify.
-$$\frac { 4 } { x ^ { 2 } - 3 x + 2 } + \frac { 5 } { x ^ { 2 } - 1 }$$

Accepted Answer

A)    9x−6(x−1) (x+1) (x−2) \frac { 9 x - 6 } { ( x - 1 )  ( x + 1 )  ( x - 2 )  }(x−1) (x+1) (x−2) 9x−6​ 
B)    40x−6(x−1) (x+1) (x−2) \frac { 40 x - 6 } { ( x - 1 )  ( x + 1 )  ( x - 2 )  }(x−1) (x+1) (x−2) 40x−6​ 
C)    9x−6(x−1) (x−2) \frac { 9 x - 6 } { ( x - 1 )  ( x - 2 )  }(x−1) (x−2) 9x−6​ 
D)    6x−9(x−1) (x+1) (x−2) \frac { 6 x - 9 } { ( x - 1 )  ( x + 1 )  ( x - 2 )  }(x−1) (x+1) (x−2) 6x−9​

Question 14

Simplify.
-$$\frac { y ^ { 2 } - 2 y - 8 } { y ^ { 2 } - 4 y - 12 }$$

Accepted Answer

A)    −y2−2y−8y2−4y−12- \frac { y ^ { 2 } - 2 y - 8 } { y ^ { 2 } - 4 y - 12 }−y2−4y−12y2−2y−8​ 
B)    −2y−8−4y−12\frac { - 2 y - 8 } { - 4 y - 12 }−4y−12−2y−8​ 
C)    −2y−2−4y−3\frac { - 2 y - 2 } { - 4 y - 3 }−4y−3−2y−2​ 
D)    y−4y−6\frac { y - 4 } { y - 6 }y−6y−4​

Question 15

Simplify.
-$$\frac { x ^ { 3 } + 10 x ^ { 2 } + 21 x } { x ^ { 2 } + 12 x + 35 }$$

Accepted Answer

A)    x+3x+5\frac { x + 3 } { x + 5 }x+5x+3​ 
B)    x3+10x2+21xx2+12x+35\frac { x ^ { 3 } + 10 x ^ { 2 } + 21 x } { x ^ { 2 } + 12 x + 35 }x2+12x+35x3+10x2+21x​ 
C)    x(x+3) x+5\frac { x ( x + 3 )  } { x + 5 }x+5x(x+3) ​ 
D)    x+3x(x+5) \frac { x + 3 } { x ( x + 5 )  }x(x+5) x+3​

Question 16

Simplify.
-$$\frac { 1 - x ^ { 2 } } { 2 x ^ { 2 } + 3 x - 5 }$$

Accepted Answer

A)    −x−12x−5\frac { - x - 1 } { 2 x - 5 }2x−5−x−1​ 
B)    x+12x+5\frac { x + 1 } { 2 x + 5 }2x+5x+1​ 
C)    −x−12x+5\frac { - x - 1 } { 2 x + 5 }2x+5−x−1​ 
D)    (1+x) (1−x) (2x+5) (x−1) \frac { ( 1 + x )  ( 1 - x )  } { ( 2 x + 5 )  ( x - 1 )  }(2x+5) (x−1) (1+x) (1−x) ​

Question 17

Find the LCD. Do not combine fractions.
-$$\frac { 8 } { 12 x ^ { 2 } y ^ { 5 } } , \frac { 8 } { 16 x ^ { 3 } y ^ { 3 } }$$

Accepted Answer

A)    48x3y348 x ^ { 3 } y ^ { 3 }48x3y3 
B)    48xy48 x y48xy 
C)    48x2y548 x ^ { 2 } y ^ { 5 }48x2y5 
D)    48x3y548 x ^ { 3 } y ^ { 5 }48x3y5

Question 18

Perform the operation indicated.
-$$\frac { x ^ { 2 } + 9 x + 20 } { x ^ { 2 } + 11 x + 28 } \cdot \frac { x ^ { 2 } + 7 x } { x ^ { 2 } - 2 x - 35 }$$

Accepted Answer

A)    x2+7xx−7\frac { x ^ { 2 } + 7 x } { x - 7 }x−7x2+7x​ 
B)    xx−7\frac { x } { x - 7 }x−7x​ 
C)    xx2+11x+28\frac { x } { x ^ { 2 } + 11 x + 28 }x2+11x+28x​ 
D)    1x−7\frac { 1 } { x - 7 }x−71​

Question 19

Perform the operation indicated. Simplify.
-$$\frac { 6 } { 4 x - 11 } - \frac { 19 } { 11 - 4 x }$$

Accepted Answer

A)    =−134x−11=- \frac { 13 } { 4 x - 11 }=−4x−1113​ 
B)    134x−11\frac { 13 } { 4 x - 11 }4x−1113​ 
C)    −254x−11- \frac { 25 } { 4 x - 11 }−4x−1125​ 
D)    254x−11\frac { 25 } { 4 x - 11 }4x−1125​

Question 20

Simplify.
-$$\frac { 5 x - 25 } { x ^ { 2 } - 5 x }$$

Accepted Answer

A)   5
B)    5x\frac { 5 } { x }x5​ 
C)    5x5 x5x 
D)    x5\frac { x } { 5 }5x​

Exam 7: Rational Expressions and Equations

Perform the operation indicated. Simplify. - $\frac { \frac { x } { 8 } + \frac { 1 } { 2 } } { \frac { 1 } { 4 x } - \frac { 5 } { 8 } }$

Perform the operation indicated. Simplify. - $\frac { 3 } { y ^ { 2 } - 3 y + 2 } + \frac { 5 } { y - 1 }$

Perform the operation indicated. Simplify. - $\frac { \frac { 36 q ^ { 2 } - 25 p ^ { 2 } } { p q } } { \frac { 6 } { q } - \frac { 5 } { p } }$

Simplify. - $\frac { y ^ { 2 } + 9 y + 20 } { y ^ { 2 } + 10 y + 24 }$

Perform the operation indicated. Simplify. - $\frac { 2 a b } { a ^ { 2 } - b ^ { 2 } } - \frac { b } { a - b } + \frac { 8 } { 2 }$

Perform the operation indicated. Simplify. - $\frac { 6 x } { x + 1 } + \frac { 7 } { x - 1 } - \frac { 12 } { x ^ { 2 } - 1 }$

Perform the operation indicated. - $\frac { 15 r ^ { 2 } - 14 r t - 8 t ^ { 2 } } { 10 r ^ { 2 } - 21 r t - 10 t ^ { 2 } } \cdot \frac { 30 r ^ { 2 } - 11 r t - 28 t ^ { 2 } } { 15 r ^ { 2 } - 8 r t - 16 t ^ { 2 } }$

Perform the operation indicated. Simplify. - $\frac { 5 x } { x - 4 } - \frac { 20 } { x - 4 }$

Find the LCD. Do not combine fractions. - $\frac { 5 } { x ^ { 2 } - 9 x + 20 } , \frac { 9 } { x ^ { 2 } - 8 x + 16 }$

Perform the operation indicated. Simplify. - $\frac { x } { x ^ { 2 } - 16 } - \frac { 6 } { x ^ { 2 } + 5 x + 4 }$

Simplify. - $\frac { ( 7 x - 5 ) ^ { 2 } } { 7 x ^ { 2 } - 61 x + 40 }$

Find the LCD. Do not combine fractions. - $\frac { 4 } { x ^ { 2 } + 6 x } , \frac { 9 } { x ^ { 2 } + 3 x - 18 }$

Perform the operation indicated. Simplify. - $\frac { 4 } { x ^ { 2 } - 3 x + 2 } + \frac { 5 } { x ^ { 2 } - 1 }$

Simplify. - $\frac { y ^ { 2 } - 2 y - 8 } { y ^ { 2 } - 4 y - 12 }$

Simplify. - $\frac { x ^ { 3 } + 10 x ^ { 2 } + 21 x } { x ^ { 2 } + 12 x + 35 }$

Simplify. - $\frac { 1 - x ^ { 2 } } { 2 x ^ { 2 } + 3 x - 5 }$

Find the LCD. Do not combine fractions. - $\frac { 8 } { 12 x ^ { 2 } y ^ { 5 } } , \frac { 8 } { 16 x ^ { 3 } y ^ { 3 } }$

Perform the operation indicated. - $\frac { x ^ { 2 } + 9 x + 20 } { x ^ { 2 } + 11 x + 28 } \cdot \frac { x ^ { 2 } + 7 x } { x ^ { 2 } - 2 x - 35 }$

Perform the operation indicated. Simplify. - $\frac { 6 } { 4 x - 11 } - \frac { 19 } { 11 - 4 x }$

Simplify. - $\frac { 5 x - 25 } { x ^ { 2 } - 5 x }$

Exam 7: Rational Expressions and Equations

Perform the operation indicated. Simplify. - x8+1214x−58\frac { \frac { x } { 8 } + \frac { 1 } { 2 } } { \frac { 1 } { 4 x } - \frac { 5 } { 8 } }4x1​−85​8x​+21​​

Perform the operation indicated. Simplify. - 3y2−3y+2+5y−1\frac { 3 } { y ^ { 2 } - 3 y + 2 } + \frac { 5 } { y - 1 }y2−3y+23​+y−15​

Perform the operation indicated. Simplify. - 36q2−25p2pq6q−5p\frac { \frac { 36 q ^ { 2 } - 25 p ^ { 2 } } { p q } } { \frac { 6 } { q } - \frac { 5 } { p } }q6​−p5​pq36q2−25p2​​

Simplify. - y2+9y+20y2+10y+24\frac { y ^ { 2 } + 9 y + 20 } { y ^ { 2 } + 10 y + 24 }y2+10y+24y2+9y+20​

Perform the operation indicated. Simplify. - 2aba2−b2−ba−b+82\frac { 2 a b } { a ^ { 2 } - b ^ { 2 } } - \frac { b } { a - b } + \frac { 8 } { 2 }a2−b22ab​−a−bb​+28​

Perform the operation indicated. Simplify. - 6xx+1+7x−1−12x2−1\frac { 6 x } { x + 1 } + \frac { 7 } { x - 1 } - \frac { 12 } { x ^ { 2 } - 1 }x+16x​+x−17​−x2−112​

Perform the operation indicated. Simplify. - 5xx−4−20x−4\frac { 5 x } { x - 4 } - \frac { 20 } { x - 4 }x−45x​−x−420​

Find the LCD. Do not combine fractions. - 5x2−9x+20,9x2−8x+16\frac { 5 } { x ^ { 2 } - 9 x + 20 } , \frac { 9 } { x ^ { 2 } - 8 x + 16 }x2−9x+205​,x2−8x+169​

Perform the operation indicated. Simplify. - xx2−16−6x2+5x+4\frac { x } { x ^ { 2 } - 16 } - \frac { 6 } { x ^ { 2 } + 5 x + 4 }x2−16x​−x2+5x+46​

Simplify. - (7x−5) 27x2−61x+40\frac { ( 7 x - 5 ) ^ { 2 } } { 7 x ^ { 2 } - 61 x + 40 }7x2−61x+40(7x−5) 2​

Find the LCD. Do not combine fractions. - 4x2+6x,9x2+3x−18\frac { 4 } { x ^ { 2 } + 6 x } , \frac { 9 } { x ^ { 2 } + 3 x - 18 }x2+6x4​,x2+3x−189​

Perform the operation indicated. Simplify. - 4x2−3x+2+5x2−1\frac { 4 } { x ^ { 2 } - 3 x + 2 } + \frac { 5 } { x ^ { 2 } - 1 }x2−3x+24​+x2−15​

Simplify. - y2−2y−8y2−4y−12\frac { y ^ { 2 } - 2 y - 8 } { y ^ { 2 } - 4 y - 12 }y2−4y−12y2−2y−8​

Simplify. - x3+10x2+21xx2+12x+35\frac { x ^ { 3 } + 10 x ^ { 2 } + 21 x } { x ^ { 2 } + 12 x + 35 }x2+12x+35x3+10x2+21x​

Simplify. - 1−x22x2+3x−5\frac { 1 - x ^ { 2 } } { 2 x ^ { 2 } + 3 x - 5 }2x2+3x−51−x2​

Find the LCD. Do not combine fractions. - 812x2y5,816x3y3\frac { 8 } { 12 x ^ { 2 } y ^ { 5 } } , \frac { 8 } { 16 x ^ { 3 } y ^ { 3 } }12x2y58​,16x3y38​

Perform the operation indicated. - x2+9x+20x2+11x+28⋅x2+7xx2−2x−35\frac { x ^ { 2 } + 9 x + 20 } { x ^ { 2 } + 11 x + 28 } \cdot \frac { x ^ { 2 } + 7 x } { x ^ { 2 } - 2 x - 35 }x2+11x+28x2+9x+20​⋅x2−2x−35x2+7x​

Perform the operation indicated. Simplify. - 64x−11−1911−4x\frac { 6 } { 4 x - 11 } - \frac { 19 } { 11 - 4 x }4x−116​−11−4x19​

Simplify. - 5x−25x2−5x\frac { 5 x - 25 } { x ^ { 2 } - 5 x }x2−5x5x−25​