Question 1

Simplify the complex fraction.
-$$\frac { \frac { x } { 5 } - \frac { 3 } { x } } { \frac { 3 } { x } + \frac { 3 } { 5 } }$$

Accepted Answer

A)  $\frac { x ^ { 2 } - 15 } { x + 15 }$ 
B)  $\frac { x ^ { 2 } - 15 } { 3 x + 15 }$ 
C)  $\frac { x ^ { 2 } - 15 } { 3 ( x + 5 ) }$ 
D)  $\frac { x ^ { 2 } - 3 } { 3 x + 15 }$ 
A)  $\frac { x ^ { 2 } - 15 } { x + 15 }$ 
B)  $\frac { x ^ { 2 } - 15 } { 3 x + 15 }$ 
C)  $\frac { x ^ { 2 } - 15 } { 3 ( x + 5 ) }$ 
D)  $\frac { x ^ { 2 } - 3 } { 3 x + 15 }$ C

Question 2

Solve.
-y varies directly as x. If y = 36 when x = 6 $$\text { find } y \text { when } x$$

Accepted Answer

A)  216 
B)  144 
C)  12 
D)  108 
A)  216 
B)  144 
C)  12 
D)  108 D

Question 3

Simplify the rational expression. (Assume the denominator is nonzero.)
-$$\frac { x ^ { 2 } + 14 x + 45 } { x ^ { 2 } + 13 x + 40 } \cdot \frac { x ^ { 2 } + 17 x + 72 } { x ^ { 2 } + 18 x + 81 }$$

Accepted Answer

A)  $\frac { x + 8 } { x + 9 }$ 
B)  $\frac { x + 9 } { x + 8 }$ 
C)  $\frac { 1 } { x + 9 }$ 
D)  1 
A)  $\frac { x + 8 } { x + 9 }$ 
B)  $\frac { x + 9 } { x + 8 }$ 
C)  $\frac { 1 } { x + 9 }$ 
D)  1 D

Question 4

Find all values of the variable for which the rational expression is undefined.
-$$\frac { x ^ { 2 } - 49 } { x ^ { 2 } + 2 x - 24 }$$

Accepted Answer

A)  $- 7,7$ 
B)  0 
C)  $- 4,6$ 
D)  $- 6,4$ 
A)  $- 7,7$ 
B)  0 
C)  $- 4,6$ 
D)  $- 6,4$

Question 5

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac { 2 x + 4 } { x - 3 } \cdot \frac { 4 x ^ { 2 } - 24 x + 36 } { x ^ { 2 } - 4 }$

Accepted Answer

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

Question 6

For the given rational functions f(x) and g(x), f(x) -g(x).
-$$f ( x ) = \frac { 2 x + 6 } { x ^ { 2 } + 4 x + 3 } , g ( x ) = \frac { x + 3 } { x ^ { 2 } + 4 x + 3 }$$

Accepted Answer

A)  $\frac { 1 } { x + 1 }$ 
B)  $\frac { 7 } { x ^ { 2 } + 4 x + 3 }$ 
C)  $x + 3$ 
D)  $\frac { x + 9 } { x ^ { 2 } + 4 x + 3 }$ 
A)  $\frac { 1 } { x + 1 }$ 
B)  $\frac { 7 } { x ^ { 2 } + 4 x + 3 }$ 
C)  $x + 3$ 
D)  $\frac { x + 9 } { x ^ { 2 } + 4 x + 3 }$

Question 7

Identify the given function as a linear function, a quadratic function, or a rational function.
-$$f ( x ) = x ^ { 2 } - 3 x - 18$$

Accepted Answer

A)  Rational 
B)  Linear 
C)  Quadratic 
A)  Rational 
B)  Linear 
C)  Quadratic

Question 8

Identify the given function as a linear function, a quadratic function, or a rational function.
-$$f ( x ) = \frac { x ^ { 2 } - 16 } { x ^ { 2 } + 9 x + 20 }$$

Accepted Answer

A)  Quadratic 
B)  Rational 
C)  Linear 
A)  Quadratic 
B)  Rational 
C)  Linear

Question 9

Simplify the complex fraction.
-$I = \frac { n E } { n r + R }$ for $n$

Accepted Answer

A)  $n = \frac { I R } { E - I r }$ 
B)  $\mathrm { n } = \frac { \mathrm { I } ( \mathrm { nr } + \mathrm { R } ) } { \mathrm { E } }$ 
C)  $n = I R - E + I r$ 
D)  $\mathrm { n } = \frac { \mathrm { IR } } { \mathrm { Ir } - \mathrm { E } }$ 
A)  $n = \frac { I R } { E - I r }$ 
B)  $\mathrm { n } = \frac { \mathrm { I } ( \mathrm { nr } + \mathrm { R } ) } { \mathrm { E } }$ 
C)  $n = I R - E + I r$ 
D)  $\mathrm { n } = \frac { \mathrm { IR } } { \mathrm { Ir } - \mathrm { E } }$

Question 10

Subtract.
-$$\frac { 8 x } { x ^ { 2 } - 49 } - \frac { 3 } { 7 - x }$$

Accepted Answer

A)  $\frac { 11 x + 21 } { x ^ { 2 } - 49 }$ 
B)  $\frac { 11 x - 21 } { x ^ { 2 } - 49 }$ 
C)  $\frac { 8 x - 3 } { x ^ { 2 } - 49 }$ 
D)  $\frac { 5 x - 21 } { x ^ { 2 } - 49 }$ 
A)  $\frac { 11 x + 21 } { x ^ { 2 } - 49 }$ 
B)  $\frac { 11 x - 21 } { x ^ { 2 } - 49 }$ 
C)  $\frac { 8 x - 3 } { x ^ { 2 } - 49 }$ 
D)  $\frac { 5 x - 21 } { x ^ { 2 } - 49 }$

Question 11

Find the missing numerator and denominator.
-$$\frac { x ^ { 2 } - 64 } { x ^ { 2 } - 10 x + 25 } \div \frac { ? } { ? } = \frac { ( x - 8 ) ( x + 10 ) } { 8 ( x - 5 ) }$$

Accepted Answer

A)  $\frac { 8 x + 64 } { x ^ { 2 } + 5 x - 50 }$ 
B)  $\frac { x - 5 } { ( x - 8 ) ( x + 10 ) }$ 
C)  $\frac { ( x - 8 ) ( x + 10 ) } { x - 5 }$ 
D)  $\frac { 8 x - 64 } { x ^ { 2 } + 5 x - 50 }$ 
A)  $\frac { 8 x + 64 } { x ^ { 2 } + 5 x - 50 }$ 
B)  $\frac { x - 5 } { ( x - 8 ) ( x + 10 ) }$ 
C)  $\frac { ( x - 8 ) ( x + 10 ) } { x - 5 }$ 
D)  $\frac { 8 x - 64 } { x ^ { 2 } + 5 x - 50 }$

Question 12

Determine whether the two binomials are or are not opposites.
-2x - 5 and 5x - 2

Accepted Answer

A)  Opposites 
B)  Not opposites 
A)  Opposites 
B)  Not opposites

Question 13

For the given functions f(x) and g(x), find f(x) ÷ g(x).
-$$f ( x ) = \frac { x ^ { 2 } - 11 x + 30 } { x ^ { 2 } - 8 x + 15 } , g ( x ) = \left( x ^ { 2 } - 9 x + 18 \right)$$

Accepted Answer

A)  $( x + 3 ) ^ { 2 }$ 
B)  $\frac { 1 } { ( x + 3 ) ^ { 2 } }$ 
C)  $( x - 6 ) ^ { 2 }$ 
D)  $\frac { 1 } { ( x - 3 ) ^ { 2 } }$ 
A)  $( x + 3 ) ^ { 2 }$ 
B)  $\frac { 1 } { ( x + 3 ) ^ { 2 } }$ 
C)  $( x - 6 ) ^ { 2 }$ 
D)  $\frac { 1 } { ( x - 3 ) ^ { 2 } }$

Question 14

Simplify the rational expression. (Assume the denominator is nonzero.)
-$$\frac { x ^ { 2 } + 4 x - 45 } { x ^ { 2 } + 2 x - 63 }$$

Accepted Answer

A)  $\frac { x - 5 } { x - 7 }$ 
B)  $\frac { 4 x - 5 } { 2 x - 7 }$ 
C)  $\frac { 4 x - 45 } { 2 x - 63 }$ 
D)  $- \frac { x ^ { 2 } + 4 x - 45 } { x ^ { 2 } + 2 x - 63 }$ 
A)  $\frac { x - 5 } { x - 7 }$ 
B)  $\frac { 4 x - 5 } { 2 x - 7 }$ 
C)  $\frac { 4 x - 45 } { 2 x - 63 }$ 
D)  $- \frac { x ^ { 2 } + 4 x - 45 } { x ^ { 2 } + 2 x - 63 }$

Question 15

Find the domain of the rational function.
-$$r ( x ) = \frac { x ^ { 2 } - 11 x + 28 } { x ^ { 2 } - 7 x }$$

Accepted Answer

A)  All real numbers except 0 and 7 . 
B)  All real numbers except 0 and $- 7$. 
C)  All real numbers except $- 7$ and 7 . 
D)  All real numbers except 0 . 
A)  All real numbers except 0 and 7 . 
B)  All real numbers except 0 and $- 7$. 
C)  All real numbers except $- 7$ and 7 . 
D)  All real numbers except 0 .

Question 16

Find the missing numerator.
-$$\frac { 4 x } { x ^ { 2 } - 7 x + 12 } - \frac { ? } { x ^ { 2 } - 7 x + 12 } = \frac { 4 } { x - 3 }$$

Accepted Answer

A)  $\frac { 16 } { x - 4 }$ 
B)  $\frac { 4 ( x + 4 ) } { ( x - 4 ) ( x - 3 ) }$ 
C)  16 
D)  $4 ( x - 4 )$ 
A)  $\frac { 16 } { x - 4 }$ 
B)  $\frac { 4 ( x + 4 ) } { ( x - 4 ) ( x - 3 ) }$ 
C)  16 
D)  $4 ( x - 4 )$

Question 17

Simplify the complex fraction.
-$\frac { \frac { 1 } { 7 } + \frac { 1 } { 8 } } { \frac { 1 } { 7 } - \frac { 1 } { 8 } }$

Accepted Answer

A)  0 
B)  $- \frac { 1 } { 15 }$ 
C)  $- 1$ 
D)  15 
A)  0 
B)  $- \frac { 1 } { 15 }$ 
C)  $- 1$ 
D)  15

Question 18

Simplify the rational expression. (Assume the denominator is nonzero.)
-$$\frac { x ^ { 2 } - 5 x - 24 } { x ^ { 2 } + 12 x + 27 }$$

Accepted Answer

A)  $\frac { - 5 x - 24 } { 12 x + 27 }$ 
B)  $\frac { - 5 x + 24 } { 12 x - 12 }$ 
C)  $\frac { x - 8 } { x + 9 }$ 
D)  $- \frac { x ^ { 2 } - 5 x - 24 } { x ^ { 2 } + 12 x + 27 }$ 
A)  $\frac { - 5 x - 24 } { 12 x + 27 }$ 
B)  $\frac { - 5 x + 24 } { 12 x - 12 }$ 
C)  $\frac { x - 8 } { x + 9 }$ 
D)  $- \frac { x ^ { 2 } - 5 x - 24 } { x ^ { 2 } + 12 x + 27 }$

Question 19

Solve the problem.
-A plane flies 400 miles with the wind and 320 miles against the wind in the same length of time. If the speed of the wind is 22 mph, what is the speed of the plane in calm air?

Accepted Answer

A)  198 mph 
B)  223 mph 
C)  188 mph 
D)  203 mph 
A)  198 mph 
B)  223 mph 
C)  188 mph 
D)  203 mph

Question 20

Solve.
-A number minus 20 times its reciprocal is 1. Find the number.

Accepted Answer

A)  5 
B)  5 or 4 
C) -5 or 4 
D) 5 or -4 
A)  5 
B)  5 or 4 
C) -5 or 4 
D) 5 or -4

Simplify the complex fraction. - $\frac { \frac { x } { 5 } - \frac { 3 } { x } } { \frac { 3 } { x } + \frac { 3 } { 5 } }$

Solve. -y varies directly as x. If y = 36 when x = 6 $\text { find } y \text { when } x$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac { x ^ { 2 } + 14 x + 45 } { x ^ { 2 } + 13 x + 40 } \cdot \frac { x ^ { 2 } + 17 x + 72 } { x ^ { 2 } + 18 x + 81 }$

Find all values of the variable for which the rational expression is undefined. - $\frac { x ^ { 2 } - 49 } { x ^ { 2 } + 2 x - 24 }$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac { 2 x + 4 } { x - 3 } \cdot \frac { 4 x ^ { 2 } - 24 x + 36 } { x ^ { 2 } - 4 }$

For the given rational functions f(x) and g(x), f(x) -g(x). - $f ( x ) = \frac { 2 x + 6 } { x ^ { 2 } + 4 x + 3 } , g ( x ) = \frac { x + 3 } { x ^ { 2 } + 4 x + 3 }$

Identify the given function as a linear function, a quadratic function, or a rational function. - $f ( x ) = x ^ { 2 } - 3 x - 18$

Identify the given function as a linear function, a quadratic function, or a rational function. - $f ( x ) = \frac { x ^ { 2 } - 16 } { x ^ { 2 } + 9 x + 20 }$

Simplify the complex fraction. - $I = \frac { n E } { n r + R }$ for $n$

Subtract. - $\frac { 8 x } { x ^ { 2 } - 49 } - \frac { 3 } { 7 - x }$

Find the missing numerator and denominator. - $\frac { x ^ { 2 } - 64 } { x ^ { 2 } - 10 x + 25 } \div \frac { ? } { ? } = \frac { ( x - 8 ) ( x + 10 ) } { 8 ( x - 5 ) }$

Determine whether the two binomials are or are not opposites. -2x - 5 and 5x - 2

For the given functions f(x) and g(x), find f(x) ÷ g(x). - $f ( x ) = \frac { x ^ { 2 } - 11 x + 30 } { x ^ { 2 } - 8 x + 15 } , g ( x ) = \left( x ^ { 2 } - 9 x + 18 \right)$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac { x ^ { 2 } + 4 x - 45 } { x ^ { 2 } + 2 x - 63 }$

Find the domain of the rational function. - $r ( x ) = \frac { x ^ { 2 } - 11 x + 28 } { x ^ { 2 } - 7 x }$

Find the missing numerator. - $\frac { 4 x } { x ^ { 2 } - 7 x + 12 } - \frac { ? } { x ^ { 2 } - 7 x + 12 } = \frac { 4 } { x - 3 }$

Simplify the complex fraction. - $\frac { \frac { 1 } { 7 } + \frac { 1 } { 8 } } { \frac { 1 } { 7 } - \frac { 1 } { 8 } }$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac { x ^ { 2 } - 5 x - 24 } { x ^ { 2 } + 12 x + 27 }$

Solve the problem. -A plane flies 400 miles with the wind and 320 miles against the wind in the same length of time. If the speed of the wind is 22 mph, what is the speed of the plane in calm air?

Solve. -A number minus 20 times its reciprocal is 1. Find the number.

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

Simplify the complex fraction. - x5−3x3x+35\frac { \frac { x } { 5 } - \frac { 3 } { x } } { \frac { 3 } { x } + \frac { 3 } { 5 } }x3​+53​5x​−x3​​

Solve. -y varies directly as x. If y = 36 when x = 6 find y when x\text { find } y \text { when } x find y when x

Simplify the rational expression. (Assume the denominator is nonzero.) - x2+14x+45x2+13x+40⋅x2+17x+72x2+18x+81\frac { x ^ { 2 } + 14 x + 45 } { x ^ { 2 } + 13 x + 40 } \cdot \frac { x ^ { 2 } + 17 x + 72 } { x ^ { 2 } + 18 x + 81 }x2+13x+40x2+14x+45​⋅x2+18x+81x2+17x+72​

Find all values of the variable for which the rational expression is undefined. - x2−49x2+2x−24\frac { x ^ { 2 } - 49 } { x ^ { 2 } + 2 x - 24 }x2+2x−24x2−49​

Simplify the rational expression. (Assume the denominator is nonzero.) - 2x+4x−3⋅4x2−24x+36x2−4\frac { 2 x + 4 } { x - 3 } \cdot \frac { 4 x ^ { 2 } - 24 x + 36 } { x ^ { 2 } - 4 }x−32x+4​⋅x2−44x2−24x+36​

For the given rational functions f(x) and g(x), f(x) -g(x). - f(x)=2x+6x2+4x+3,g(x)=x+3x2+4x+3f ( x ) = \frac { 2 x + 6 } { x ^ { 2 } + 4 x + 3 } , g ( x ) = \frac { x + 3 } { x ^ { 2 } + 4 x + 3 }f(x)=x2+4x+32x+6​,g(x)=x2+4x+3x+3​

Identify the given function as a linear function, a quadratic function, or a rational function. - f(x)=x2−3x−18f ( x ) = x ^ { 2 } - 3 x - 18f(x)=x2−3x−18

Identify the given function as a linear function, a quadratic function, or a rational function. - f(x)=x2−16x2+9x+20f ( x ) = \frac { x ^ { 2 } - 16 } { x ^ { 2 } + 9 x + 20 }f(x)=x2+9x+20x2−16​

Simplify the complex fraction. - I=nEnr+RI = \frac { n E } { n r + R }I=nr+RnE​ for nnn

Subtract. - 8xx2−49−37−x\frac { 8 x } { x ^ { 2 } - 49 } - \frac { 3 } { 7 - x }x2−498x​−7−x3​

Find the missing numerator and denominator. - x2−64x2−10x+25÷??=(x−8)(x+10)8(x−5)\frac { x ^ { 2 } - 64 } { x ^ { 2 } - 10 x + 25 } \div \frac { ? } { ? } = \frac { ( x - 8 ) ( x + 10 ) } { 8 ( x - 5 ) }x2−10x+25x2−64​÷??​=8(x−5)(x−8)(x+10)​