Question 1

Find all numbers for which the rational expression is not defined.
-$$\frac { x ^ { 2 } - 81 } { x ^ { 2 } - 7 x + 12 }$$

Accepted Answer

A)  $x = 0$ 
B)  $x = 9$ and $x = - 9$ 
C)  $x = - 3$ and $x = - 4$ 
D)  $x = 3$ and $x = 4$ 
A)  $x = 0$ 
B)  $x = 9$ and $x = - 9$ 
C)  $x = - 3$ and $x = - 4$ 
D)  $x = 3$ and $x = 4$

Question 2

Multiply. Do not simplify.
-$$\frac { x + 5 } { x + 4 } \cdot \frac { x - 5 } { x - 5 }$$

Accepted Answer

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

Question 3

Add. Simplify, if possible.
-$$\frac { 2 } { 15 x } + \frac { 5 } { 15 x }$$

Accepted Answer

A)  $\frac { 7 } { 15 x }$ 
B)  $\frac { 7 } { 30 x }$ 
C)  $\frac { 15 x } { 7 }$ 
D)  1 
A)  $\frac { 7 } { 15 x }$ 
B)  $\frac { 7 } { 30 x }$ 
C)  $\frac { 15 x } { 7 }$ 
D)  1

Question 4

Find the LCM.
-$500,900,450$

Accepted Answer

A)  1500 
B)  2250 
C)  4500 
D)  900 
A)  1500 
B)  2250 
C)  4500 
D)  900

Question 5

Subtract. Simplify, if possible.
-$$\frac { 7 } { 6 x - 5 } - \frac { 1 } { 5 - 6 x }$$

Accepted Answer

A)  $\frac { 8 } { 6 x - 5 }$ 
B)  $\frac { - 6 } { 6 x - 5 }$ 
C)  $\frac { 6 } { 6 x - 5 }$ 
D)  $\frac { - 8 } { 6 x - 5 }$ 
A)  $\frac { 8 } { 6 x - 5 }$ 
B)  $\frac { - 6 } { 6 x - 5 }$ 
C)  $\frac { 6 } { 6 x - 5 }$ 
D)  $\frac { - 8 } { 6 x - 5 }$

Question 6

Subtract. Simplify, if possible.
-$$\frac { x } { x ^ { 2 } - 16 } - \frac { 8 } { x ^ { 2 } + 5 x + 4 }$$

Accepted Answer

A)  $\frac { x ^ { 2 } - 7 x + 32 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
B)  $\frac { x ^ { 2 } - 7 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
C)  $\frac { x ^ { 2 } + 7 x + 32 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
D)  $\frac { x ^ { 2 } - 7 x + 32 } { ( x - 4 ) ( x + 4 ) }$ 
A)  $\frac { x ^ { 2 } - 7 x + 32 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
B)  $\frac { x ^ { 2 } - 7 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
C)  $\frac { x ^ { 2 } + 7 x + 32 } { ( x - 4 ) ( x + 4 ) ( x + 1 ) }$ 
D)  $\frac { x ^ { 2 } - 7 x + 32 } { ( x - 4 ) ( x + 4 ) }$

Question 7

Add. Simplify, if possible.
-$\frac { x } { x ^ { 2 } - 1 } + \frac { 3 x } { x - x ^ { 2 } }$

Accepted Answer

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

Question 8

Subtract. Simplify, if possible.
-$$\frac { 7 } { z ^ { 2 } - 6 z } - \frac { 6 } { z - 6 }$$

Accepted Answer

A)  $\frac { 7 + 6 z } { z ^ { 2 } - 6 z }$ 
B)  $\frac { 7 z + 6 } { z ^ { 2 } - 6 z }$ 
C)  $\frac { 7 - 6 z } { z ^ { 2 } - 6 z }$ 
D)  $\frac { 6 z - 7 } { z - 6 }$ 
A)  $\frac { 7 + 6 z } { z ^ { 2 } - 6 z }$ 
B)  $\frac { 7 z + 6 } { z ^ { 2 } - 6 z }$ 
C)  $\frac { 7 - 6 z } { z ^ { 2 } - 6 z }$ 
D)  $\frac { 6 z - 7 } { z - 6 }$

Question 9

Find the LCM.
-$$x ^ { 2 } + 11 x + 30,5 x + 25$$

Accepted Answer

A)  $5 ( x - 6 ) ( x - 5 )$ 
B)  $5 ( x + 6 ) ( x + 5 )$ 
C)  $5 ( x - 6 ) ( x + 5 )$ 
D)  $5 ( x + 6 ) ( x - 5 )$ 
A)  $5 ( x - 6 ) ( x - 5 )$ 
B)  $5 ( x + 6 ) ( x + 5 )$ 
C)  $5 ( x - 6 ) ( x + 5 )$ 
D)  $5 ( x + 6 ) ( x - 5 )$

Question 10

Divide and simplify.
-$$\frac { 2 p - 2 } { p } \div \frac { 3 p - 3 } { 7 p ^ { 2 } }$$

Accepted Answer

A)  $\frac { 14 p ^ { 3 } - 14 p ^ { 2 } } { 3 p ^ { 2 } - 3 p }$ 
B)  $\frac { 14 p } { 3 }$ 
C)  $\frac { 6 p ^ { 2 } + 12 p + 6 } { 7 p ^ { 3 } }$ 
D)  $\frac { 3 } { 14 \mathrm { p } }$ 
A)  $\frac { 14 p ^ { 3 } - 14 p ^ { 2 } } { 3 p ^ { 2 } - 3 p }$ 
B)  $\frac { 14 p } { 3 }$ 
C)  $\frac { 6 p ^ { 2 } + 12 p + 6 } { 7 p ^ { 3 } }$ 
D)  $\frac { 3 } { 14 \mathrm { p } }$

Question 11

Find the reciprocal.
-$\frac { x } { 2 }$

Accepted Answer

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

Question 12

Simplify.
-$\frac { 4 x + 2 } { 20 x ^ { 2 } + 18 x + 4 }$

Accepted Answer

A)  $\frac { 4 x + 5 } { 5 x + 18 }$ 
B)  $\frac { 1 } { 5 x + 2 }$ 
C)  $\frac { 4 x } { 5 x + 2 }$ 
D)  $\frac { 4 x + 2 } { 20 x ^ { 2 } + 18 x + 4 }$ 
A)  $\frac { 4 x + 5 } { 5 x + 18 }$ 
B)  $\frac { 1 } { 5 x + 2 }$ 
C)  $\frac { 4 x } { 5 x + 2 }$ 
D)  $\frac { 4 x + 2 } { 20 x ^ { 2 } + 18 x + 4 }$

Question 13

Simplify.
-$$\frac { y ^ { 2 } + 6 y - 16 } { y ^ { 2 } + 17 y + 72 }$$

Accepted Answer

A)  $\frac { 6 y + 16 } { 17 y - 17 }$ 
B)  $- \frac { y ^ { 2 } + 6 y - 16 } { y ^ { 2 } + 17 y + 72 }$ 
C)  $\frac { 6 y - 16 } { 17 y + 72 }$ 
D)  $\frac { y - 2 } { y + 9 }$ 
A)  $\frac { 6 y + 16 } { 17 y - 17 }$ 
B)  $- \frac { y ^ { 2 } + 6 y - 16 } { y ^ { 2 } + 17 y + 72 }$ 
C)  $\frac { 6 y - 16 } { 17 y + 72 }$ 
D)  $\frac { y - 2 } { y + 9 }$

Question 14

Simplify.
-$$\frac { \frac { 3 } { x + 2 } + \frac { 1 } { x } } { \frac { 2 } { x + 2 } + \frac { 4 } { x } }$$

Accepted Answer

A)  $\frac { 4 x + 6 } { 6 x + 4 }$ 
B)  $\frac { 4 x + 2 } { 6 x + 8 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 4 x + 2 } { 6 x }$ 
A)  $\frac { 4 x + 6 } { 6 x + 4 }$ 
B)  $\frac { 4 x + 2 } { 6 x + 8 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  $\frac { 4 x + 2 } { 6 x }$

Question 15

Add. Simplify, if possible.
-$$\frac { 4 } { x + y } + \frac { 3 } { y + x }$$

Accepted Answer

A)  $\frac { 1 } { x + y }$ 
B)  $\frac { 7 } { 2 x + 2 y }$ 
C)  $\frac { 4 } { x } + \frac { 3 } { y }$ 
D)  $\frac { 7 } { x + y }$ 
A)  $\frac { 1 } { x + y }$ 
B)  $\frac { 7 } { 2 x + 2 y }$ 
C)  $\frac { 4 } { x } + \frac { 3 } { y }$ 
D)  $\frac { 7 } { x + y }$

Question 16

Perform the indicated operation. Simplify, if possible.
-$$\frac { 49 x } { 9 ( 7 x + 1 ) } - \frac { 1 } { 9 x ( 7 x + 1 ) } + \frac { 4 } { x }$$

Accepted Answer

A)  $\frac { 7 ( x + 5 ) } { 63 x ^ { 2 } + 9 x }$ 
B)  $\frac { 49 x ^ { 2 } + 252 x + 35 } { 63 x ^ { 2 } + 9 x }$ 
C)  $\frac { 7 ( x + 5 ) } { 9 x }$ 
D)  $\frac { 49 x ^ { 2 } + 252 x + 35 } { 9 x }$ 
A)  $\frac { 7 ( x + 5 ) } { 63 x ^ { 2 } + 9 x }$ 
B)  $\frac { 49 x ^ { 2 } + 252 x + 35 } { 63 x ^ { 2 } + 9 x }$ 
C)  $\frac { 7 ( x + 5 ) } { 9 x }$ 
D)  $\frac { 49 x ^ { 2 } + 252 x + 35 } { 9 x }$

Question 17

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

Accepted Answer

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

Question 18

Subtract. Simplify, if possible.
-$$\frac { 5 ( y - 1 ) } { 3 y - 7 } - \frac { y - 5 } { 7 - 3 y } - \frac { 6 y } { 3 y - 7 }$$

Accepted Answer

A)  $\frac { - 10 } { 3 y - 7 }$ 
B)  $\frac { - 2 y } { 3 y - 7 }$ 
C)  $\frac { - 10 } { ( 3 y - 7 ) ( 7 - 3 y ) }$ 
D)  $\frac { 10 } { 3 y - 7 }$ 
A)  $\frac { - 10 } { 3 y - 7 }$ 
B)  $\frac { - 2 y } { 3 y - 7 }$ 
C)  $\frac { - 10 } { ( 3 y - 7 ) ( 7 - 3 y ) }$ 
D)  $\frac { 10 } { 3 y - 7 }$

Question 19

Simplify.
-$$\frac { \frac { 1 } { m } + \frac { 1 } { \mathrm { z } } } { \frac { 1 } { \mathrm {~m} } - \frac { 1 } { \mathrm { z } } }$$

Accepted Answer

A)  $\frac { z - m } { z }$ 
B)  $\frac { z + m } { z }$ 
C)  $\frac { \mathrm { z } + \mathrm { m } } { \mathrm { m } }$ 
D)  $\frac { \mathrm { z } + \mathrm { m } } { \mathrm { z } - \mathrm { m } }$ 
A)  $\frac { z - m } { z }$ 
B)  $\frac { z + m } { z }$ 
C)  $\frac { \mathrm { z } + \mathrm { m } } { \mathrm { m } }$ 
D)  $\frac { \mathrm { z } + \mathrm { m } } { \mathrm { z } - \mathrm { m } }$

Question 20

Subtract. Simplify, if possible.
-$$\frac { 2 x + 10 } { x ^ { 2 } + 8 x + 15 } - \frac { x + 5 } { x ^ { 2 } + 8 x + 15 }$$

Accepted Answer

A)  $\frac { 1 } { x + 3 }$ 
B)  $\frac { 11 } { x ^ { 2 } + 8 x + 15 }$ 
C)  $\frac { x + 15 } { x ^ { 2 } + 8 x + 15 }$ 
D)  $x + 5$ 
A)  $\frac { 1 } { x + 3 }$ 
B)  $\frac { 11 } { x ^ { 2 } + 8 x + 15 }$ 
C)  $\frac { x + 15 } { x ^ { 2 } + 8 x + 15 }$ 
D)  $x + 5$

Find all numbers for which the rational expression is not defined. - $\frac { x ^ { 2 } - 81 } { x ^ { 2 } - 7 x + 12 }$

Multiply. Do not simplify. - $\frac { x + 5 } { x + 4 } \cdot \frac { x - 5 } { x - 5 }$

Add. Simplify, if possible. - $\frac { 2 } { 15 x } + \frac { 5 } { 15 x }$

Find the LCM. - $500,900,450$

Subtract. Simplify, if possible. - $\frac { 7 } { 6 x - 5 } - \frac { 1 } { 5 - 6 x }$

Subtract. Simplify, if possible. - $\frac { x } { x ^ { 2 } - 16 } - \frac { 8 } { x ^ { 2 } + 5 x + 4 }$

Add. Simplify, if possible. - $\frac { x } { x ^ { 2 } - 1 } + \frac { 3 x } { x - x ^ { 2 } }$

Subtract. Simplify, if possible. - $\frac { 7 } { z ^ { 2 } - 6 z } - \frac { 6 } { z - 6 }$

Find the LCM. - $x ^ { 2 } + 11 x + 30,5 x + 25$

Divide and simplify. - $\frac { 2 p - 2 } { p } \div \frac { 3 p - 3 } { 7 p ^ { 2 } }$

Find the reciprocal. - $\frac { x } { 2 }$

Simplify. - $\frac { 4 x + 2 } { 20 x ^ { 2 } + 18 x + 4 }$

Simplify. - $\frac { y ^ { 2 } + 6 y - 16 } { y ^ { 2 } + 17 y + 72 }$

Simplify. - $\frac { \frac { 3 } { x + 2 } + \frac { 1 } { x } } { \frac { 2 } { x + 2 } + \frac { 4 } { x } }$

Add. Simplify, if possible. - $\frac { 4 } { x + y } + \frac { 3 } { y + x }$

Perform the indicated operation. Simplify, if possible. - $\frac { 49 x } { 9 ( 7 x + 1 ) } - \frac { 1 } { 9 x ( 7 x + 1 ) } + \frac { 4 } { x }$

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

Subtract. Simplify, if possible. - $\frac { 5 ( y - 1 ) } { 3 y - 7 } - \frac { y - 5 } { 7 - 3 y } - \frac { 6 y } { 3 y - 7 }$

Simplify. - $\frac { \frac { 1 } { m } + \frac { 1 } { \mathrm { z } } } { \frac { 1 } { \mathrm {~m} } - \frac { 1 } { \mathrm { z } } }$

Subtract. Simplify, if possible. - $\frac { 2 x + 10 } { x ^ { 2 } + 8 x + 15 } - \frac { x + 5 } { x ^ { 2 } + 8 x + 15 }$

Introduction to Real Numbers and Algebraic Expressions

Solving Equations and Inequalities

Graphs of Linear Equations

Polynomials: Operations

Polynomials: Factoring

Graphs, Functions, and Applications

Systems of Equations

More on Inequalities

Radical Expressions, Equations, and Functions

Quadratic Equations and Functions

Exponential Functions and Logarithmic Functions

Finding Measures of Central Tendency for Prices and Heights

Filters

Exam 6: Rational Expressions and Equations

Find all numbers for which the rational expression is not defined. - x2−81x2−7x+12\frac { x ^ { 2 } - 81 } { x ^ { 2 } - 7 x + 12 }x2−7x+12x2−81​

Multiply. Do not simplify. - x+5x+4⋅x−5x−5\frac { x + 5 } { x + 4 } \cdot \frac { x - 5 } { x - 5 }x+4x+5​⋅x−5x−5​

Add. Simplify, if possible. - 215x+515x\frac { 2 } { 15 x } + \frac { 5 } { 15 x }15x2​+15x5​

Find the LCM. - 500,900,450500,900,450500,900,450

Subtract. Simplify, if possible. - 76x−5−15−6x\frac { 7 } { 6 x - 5 } - \frac { 1 } { 5 - 6 x }6x−57​−5−6x1​

Subtract. Simplify, if possible. - xx2−16−8x2+5x+4\frac { x } { x ^ { 2 } - 16 } - \frac { 8 } { x ^ { 2 } + 5 x + 4 }x2−16x​−x2+5x+48​

Add. Simplify, if possible. - xx2−1+3xx−x2\frac { x } { x ^ { 2 } - 1 } + \frac { 3 x } { x - x ^ { 2 } }x2−1x​+x−x23x​

Subtract. Simplify, if possible. - 7z2−6z−6z−6\frac { 7 } { z ^ { 2 } - 6 z } - \frac { 6 } { z - 6 }z2−6z7​−z−66​

Find the LCM. - x2+11x+30,5x+25x ^ { 2 } + 11 x + 30,5 x + 25x2+11x+30,5x+25

Divide and simplify. - 2p−2p÷3p−37p2\frac { 2 p - 2 } { p } \div \frac { 3 p - 3 } { 7 p ^ { 2 } }p2p−2​÷7p23p−3​

Find the reciprocal. - x2\frac { x } { 2 }2x​

Simplify. - 4x+220x2+18x+4\frac { 4 x + 2 } { 20 x ^ { 2 } + 18 x + 4 }20x2+18x+44x+2​

Simplify. - y2+6y−16y2+17y+72\frac { y ^ { 2 } + 6 y - 16 } { y ^ { 2 } + 17 y + 72 }y2+17y+72y2+6y−16​

Simplify. - 3x+2+1x2x+2+4x\frac { \frac { 3 } { x + 2 } + \frac { 1 } { x } } { \frac { 2 } { x + 2 } + \frac { 4 } { x } }x+22​+x4​x+23​+x1​​

Add. Simplify, if possible. - 4x+y+3y+x\frac { 4 } { x + y } + \frac { 3 } { y + x }x+y4​+y+x3​

Perform the indicated operation. Simplify, if possible. - 49x9(7x+1)−19x(7x+1)+4x\frac { 49 x } { 9 ( 7 x + 1 ) } - \frac { 1 } { 9 x ( 7 x + 1 ) } + \frac { 4 } { x }9(7x+1)49x​−9x(7x+1)1​+x4​

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

Subtract. Simplify, if possible. - 5(y−1)3y−7−y−57−3y−6y3y−7\frac { 5 ( y - 1 ) } { 3 y - 7 } - \frac { y - 5 } { 7 - 3 y } - \frac { 6 y } { 3 y - 7 }3y−75(y−1)​−7−3yy−5​−3y−76y​

Simplify. - 1m+1z1 m−1z\frac { \frac { 1 } { m } + \frac { 1 } { \mathrm { z } } } { \frac { 1 } { \mathrm {~m} } - \frac { 1 } { \mathrm { z } } } m1​−z1​m1​+z1​​

Subtract. Simplify, if possible. - 2x+10x2+8x+15−x+5x2+8x+15\frac { 2 x + 10 } { x ^ { 2 } + 8 x + 15 } - \frac { x + 5 } { x ^ { 2 } + 8 x + 15 }x2+8x+152x+10​−x2+8x+15x+5​