Question 1

Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$, and evaluate as $h$ approaches 0 to find the rate of change of $f(x)$.
-$f(x)=\frac{1}{3 x}$

Accepted Answer

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

Question 2

$y$ varies inversely as $x$. If $y=48$ when $x=6$, find $y$ when $x=8$.

Accepted Answer

A)  36 
B)  6 
C)  64 
D)  18 
A)  36 
B)  6 
C)  64 
D)  18

Question 3

$\frac{4 x}{x-5}+\frac{20}{5-x}$

Accepted Answer

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

Question 4

$\frac{x^{2}+8 x+15}{x^{2}+10 x+21} \cdot \frac{x^{2}+9 x+14}{x^{2}+7 x+10}$

Accepted Answer

A)  $\frac{x+5}{x+7}$ 
B)  1 
C)  $\frac{x+7}{x+2}$ 
D)  $\frac{1}{x+2}$ 
A)  $\frac{x+5}{x+7}$ 
B)  1 
C)  $\frac{x+7}{x+2}$ 
D)  $\frac{1}{x+2}$

Question 5

For the given functions $f(x)$ and $g(x)$, find $f(x) \div g(x)$.
-$f(x)=\frac{x^{2}-9 x+18}{x^{2}+12 x+36}, g(x)=\frac{x^{2}-6 x+9}{x^{2}-5 x-6}$

Accepted Answer

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

Question 6

Evaluate the rational expression for the given value of the variable.
-$\frac{x^{2}-3 x}{x-4}$ for $x=9$

Accepted Answer

A)  $\frac{54}{5}$ 
B)  $-\frac{54}{5}$ 
C)  0 
D)  Undefined 
A)  $\frac{54}{5}$ 
B)  $-\frac{54}{5}$ 
C)  0 
D)  Undefined

Question 7

Find the LCD of the given rational expressions.
-$\frac{6}{x}, \frac{9}{x-4}$

Accepted Answer

A)  $\mathrm{x}$ 
B)  54 
C)  $x(x-4)$ 
D)  $x-4$ 
A)  $\mathrm{x}$ 
B)  54 
C)  $x(x-4)$ 
D)  $x-4$

Question 8

Jeff takes 5 hours longer to build a fence than it takes Bill. When they work together, it takes them 6 hours. How long would it take Bill to do the job alone?

Accepted Answer

A)  9 hours 
B)  11 hours 
C)  12 hours 
D)  10 hours 
A)  9 hours 
B)  11 hours 
C)  12 hours 
D)  10 hours

Question 9

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{(\mathrm{x}-8)(\mathrm{x}+5)}{64-\mathrm{x}^{2}}$

Accepted Answer

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

Question 10

For the given functions $f(x)$ and $g(x)$, find $f(x) \cdot g(x)$.
-$f(x)=\frac{x^{2}+9 x+18}{x^{2}+12 x+27}, g(x)=\frac{x^{2}+9 x}{x^{2}+3 x-18}$

Accepted Answer

A)  $\frac{x}{x^{2}+12 x+27}$ 
B)  $\frac{x}{x-3}$ 
C)  $\frac{x^{2}+9 x}{x-3}$ 
D)  $\frac{1}{x-3}$ 
A)  $\frac{x}{x^{2}+12 x+27}$ 
B)  $\frac{x}{x-3}$ 
C)  $\frac{x^{2}+9 x}{x-3}$ 
D)  $\frac{1}{x-3}$

Question 11

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{(\mathrm{x}+2)(\mathrm{x}-4)}{(\mathrm{x}-4)(\mathrm{x}+7)}$

Accepted Answer

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

Question 12

Find all values of the variable for which the rational expression is undefined.
-$\frac{81 \mathrm{x}}{\mathrm{x}^{2}-3 \mathrm{x}-18}$

Accepted Answer

A)  $-9,9$ 
B)  $-6,3$ 
C)  $-3,6$ 
D)  0 
A)  $-9,9$ 
B)  $-6,3$ 
C)  $-3,6$ 
D)  0

Question 13

$\frac{3}{10 \mathrm{x}}-\frac{9}{14 \mathrm{x}^{2}}$

Accepted Answer

A)  $\frac{6}{10 x+14 x^{2}}$ 
B)  $\frac{3(7 x-15)}{70 x^{2}}$ 
C)  $-\frac{6}{140 x^{2}}$ 
D)  $\frac{21 x-45}{70 x^{2}}$ 
A)  $\frac{6}{10 x+14 x^{2}}$ 
B)  $\frac{3(7 x-15)}{70 x^{2}}$ 
C)  $-\frac{6}{140 x^{2}}$ 
D)  $\frac{21 x-45}{70 x^{2}}$

Question 14

Find $r(-2)$.

Accepted Answer

A)  1 
B)  0 
C)  -1 
D)  3 
A)  1 
B)  0 
C)  -1 
D)  3

Question 15

Charles's law states that the volume $\mathrm{V}$ of a gas varies directly as its temperature $\mathrm{T}$ in kelvin, provided that the pressure and mass of the gas remains constant. (To convert a Celsius temperature to a kelvin temperature, add 273.) A container of gas has a volume of 450 milliliters at $8^{\circ} \mathrm{C}$. If the gas is cooled and now has a volume of 390 milliliters, what is its temperature in degrees Celsius? Round your answer to the nearest hundredth of a degree as needed.

Accepted Answer

A)  $276^{\circ} \mathrm{C}$ 
B)  $-29.47^{\circ} \mathrm{C}$ 
C)  $265.13^{\circ} \mathrm{C}$ 
D)  $6.5^{\circ} \mathrm{C}$ 
A)  $276^{\circ} \mathrm{C}$ 
B)  $-29.47^{\circ} \mathrm{C}$ 
C)  $265.13^{\circ} \mathrm{C}$ 
D)  $6.5^{\circ} \mathrm{C}$

Question 16

Evaluate the rational function.
-$r(x)=\frac{x^{3}+3 x^{2}-14 x+10}{x^{2}+7 x-17}, r(2)$

Accepted Answer

A)  -2 
B)  Undefined 
C)  2 
D)  - 1 
A)  -2 
B)  Undefined 
C)  2 
D)  - 1

Question 17

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{7 x-35}{(3+x)(5-x)}$

Accepted Answer

A)  $-\frac{7}{3+x}$ 
B)  $-\frac{7 x-35}{3+x}$ 
C)  $\frac{7}{3+x}$ 
D)  $\frac{7 x-35}{3+x}$ 
A)  $-\frac{7}{3+x}$ 
B)  $-\frac{7 x-35}{3+x}$ 
C)  $\frac{7}{3+x}$ 
D)  $\frac{7 x-35}{3+x}$

Question 18

The number $G$ of gears a machine can make varies directly as the time $T$ it operates. If it can make 7665 gears in 4 hours, how many gears can it make in 2 hours?

Accepted Answer

A)  1 gears 
B)  7671 gears 
C)  1916 gears 
D)  3833 gears 
A)  1 gears 
B)  7671 gears 
C)  1916 gears 
D)  3833 gears

Question 19

$\frac{x^{2}+5 x+4}{x^{2}+10 x+24} \cdot \frac{x^{2}-11 x+30}{x^{2}-4 x-5}$

Accepted Answer

A)  -1 
B)  $\frac{x-6}{x+6}$ 
C)  $\frac{x+6}{x-6}$ 
D)  1 
A)  -1 
B)  $\frac{x-6}{x+6}$ 
C)  $\frac{x+6}{x-6}$ 
D)  1

Question 20

If $y$ varies inversely as the square of $x$. If $y=3$ when $x=24$, find $y$ when $x=6$.

Accepted Answer

A)  48 
B)  108 
C)  36 
D)  4 
A)  48 
B)  108 
C)  36 
D)  4

Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ , and evaluate as $h$ approaches 0 to find the rate of change of $f(x)$ . - $f(x)=\frac{1}{3 x}$

$y$ varies inversely as $x$ . If $y=48$ when $x=6$ , find $y$ when $x=8$ .

$\frac{4 x}{x-5}+\frac{20}{5-x}$

$\frac{x^{2}+8 x+15}{x^{2}+10 x+21} \cdot \frac{x^{2}+9 x+14}{x^{2}+7 x+10}$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \div g(x)$ . - $f(x)=\frac{x^{2}-9 x+18}{x^{2}+12 x+36}, g(x)=\frac{x^{2}-6 x+9}{x^{2}-5 x-6}$

Evaluate the rational expression for the given value of the variable. - $\frac{x^{2}-3 x}{x-4}$ for $x=9$

Find the LCD of the given rational expressions. - $\frac{6}{x}, \frac{9}{x-4}$

Jeff takes 5 hours longer to build a fence than it takes Bill. When they work together, it takes them 6 hours. How long would it take Bill to do the job alone?

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{(\mathrm{x}-8)(\mathrm{x}+5)}{64-\mathrm{x}^{2}}$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ . - $f(x)=\frac{x^{2}+9 x+18}{x^{2}+12 x+27}, g(x)=\frac{x^{2}+9 x}{x^{2}+3 x-18}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{(\mathrm{x}+2)(\mathrm{x}-4)}{(\mathrm{x}-4)(\mathrm{x}+7)}$

Find all values of the variable for which the rational expression is undefined. - $\frac{81 \mathrm{x}}{\mathrm{x}^{2}-3 \mathrm{x}-18}$

$\frac{3}{10 \mathrm{x}}-\frac{9}{14 \mathrm{x}^{2}}$

Find $r(-2)$ .

Evaluate the rational function. - $r(x)=\frac{x^{3}+3 x^{2}-14 x+10}{x^{2}+7 x-17}, r(2)$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{7 x-35}{(3+x)(5-x)}$

The number $G$ of gears a machine can make varies directly as the time $T$ it operates. If it can make 7665 gears in 4 hours, how many gears can it make in 2 hours?

$\frac{x^{2}+5 x+4}{x^{2}+10 x+24} \cdot \frac{x^{2}-11 x+30}{x^{2}-4 x-5}$

If $y$ varies inversely as the square of $x$ . If $y=3$ when $x=24$ , find $y$ when $x=6$ .

Review of Real Numbers

Linear and Absolute Value Equations and Inequalities

Graphing Linear Equations

Systems of Equations

Exponents, Polynomials, and Factoring Polynomials

Radical Expressions and Equations

Quadratic Equations

Logarithmic and Exponential Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Exam 6: Rational Expressions and Equations

Simplify the difference quotient f(x+h)−f(x)h\frac{f(x+h)-f(x)}{h}hf(x+h)−f(x)​ , and evaluate as hhh approaches 0 to find the rate of change of f(x)f(x)f(x) . - f(x)=13xf(x)=\frac{1}{3 x}f(x)=3x1​

yyy varies inversely as xxx . If y=48y=48y=48 when x=6x=6x=6 , find yyy when x=8x=8x=8 .

4xx−5+205−x\frac{4 x}{x-5}+\frac{20}{5-x}x−54x​+5−x20​

x2+8x+15x2+10x+21⋅x2+9x+14x2+7x+10\frac{x^{2}+8 x+15}{x^{2}+10 x+21} \cdot \frac{x^{2}+9 x+14}{x^{2}+7 x+10}x2+10x+21x2+8x+15​⋅x2+7x+10x2+9x+14​

For the given functions f(x)f(x)f(x) and g(x)g(x)g(x) , find f(x)÷g(x)f(x) \div g(x)f(x)÷g(x) . - f(x)=x2−9x+18x2+12x+36,g(x)=x2−6x+9x2−5x−6f(x)=\frac{x^{2}-9 x+18}{x^{2}+12 x+36}, g(x)=\frac{x^{2}-6 x+9}{x^{2}-5 x-6}f(x)=x2+12x+36x2−9x+18​,g(x)=x2−5x−6x2−6x+9​

Evaluate the rational expression for the given value of the variable. - x2−3xx−4\frac{x^{2}-3 x}{x-4}x−4x2−3x​ for x=9x=9x=9

Find the LCD of the given rational expressions. - 6x,9x−4\frac{6}{x}, \frac{9}{x-4}x6​,x−49​

Jeff takes 5 hours longer to build a fence than it takes Bill. When they work together, it takes them 6 hours. How long would it take Bill to do the job alone?

Simplify the rational expression. (Assume the denominator is nonzero.) - (x−8)(x+5)64−x2\frac{(\mathrm{x}-8)(\mathrm{x}+5)}{64-\mathrm{x}^{2}}64−x2(x−8)(x+5)​

For the given functions f(x)f(x)f(x) and g(x)g(x)g(x) , find f(x)⋅g(x)f(x) \cdot g(x)f(x)⋅g(x) . - f(x)=x2+9x+18x2+12x+27,g(x)=x2+9xx2+3x−18f(x)=\frac{x^{2}+9 x+18}{x^{2}+12 x+27}, g(x)=\frac{x^{2}+9 x}{x^{2}+3 x-18}f(x)=x2+12x+27x2+9x+18​,g(x)=x2+3x−18x2+9x​

Simplify the rational expression. (Assume the denominator is nonzero.) - (x+2)(x−4)(x−4)(x+7)\frac{(\mathrm{x}+2)(\mathrm{x}-4)}{(\mathrm{x}-4)(\mathrm{x}+7)}(x−4)(x+7)(x+2)(x−4)​

Find all values of the variable for which the rational expression is undefined. - 81xx2−3x−18\frac{81 \mathrm{x}}{\mathrm{x}^{2}-3 \mathrm{x}-18}x2−3x−1881x​

310x−914x2\frac{3}{10 \mathrm{x}}-\frac{9}{14 \mathrm{x}^{2}}10x3​−14x29​

Find r(−2)r(-2)r(−2) .

Evaluate the rational function. - r(x)=x3+3x2−14x+10x2+7x−17,r(2)r(x)=\frac{x^{3}+3 x^{2}-14 x+10}{x^{2}+7 x-17}, r(2)r(x)=x2+7x−17x3+3x2−14x+10​,r(2)

Simplify the rational expression. (Assume the denominator is nonzero.) - 7x−35(3+x)(5−x)\frac{7 x-35}{(3+x)(5-x)}(3+x)(5−x)7x−35​

The number GGG of gears a machine can make varies directly as the time TTT it operates. If it can make 7665 gears in 4 hours, how many gears can it make in 2 hours?

x2+5x+4x2+10x+24⋅x2−11x+30x2−4x−5\frac{x^{2}+5 x+4}{x^{2}+10 x+24} \cdot \frac{x^{2}-11 x+30}{x^{2}-4 x-5}x2+10x+24x2+5x+4​⋅x2−4x−5x2−11x+30​

If yyy varies inversely as the square of xxx . If y=3y=3y=3 when x=24x=24x=24 , find yyy when x=6x=6x=6 .

Review of Real Numbers

Linear and Absolute Value Equations and Inequalities

Graphing Linear Equations

Systems of Equations

Exponents, Polynomials, and Factoring Polynomials

Radical Expressions and Equations

Quadratic Equations

Logarithmic and Exponential Functions

Conic Sections

Sequences, Series, and the Binomial Theorem

Filters

Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ , and evaluate as $h$ approaches 0 to find the rate of change of $f(x)$ . - $f(x)=\frac{1}{3 x}$

$y$ varies inversely as $x$ . If $y=48$ when $x=6$ , find $y$ when $x=8$ .

$\frac{4 x}{x-5}+\frac{20}{5-x}$

$\frac{x^{2}+8 x+15}{x^{2}+10 x+21} \cdot \frac{x^{2}+9 x+14}{x^{2}+7 x+10}$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \div g(x)$ . - $f(x)=\frac{x^{2}-9 x+18}{x^{2}+12 x+36}, g(x)=\frac{x^{2}-6 x+9}{x^{2}-5 x-6}$

Evaluate the rational expression for the given value of the variable. - $\frac{x^{2}-3 x}{x-4}$ for $x=9$

Find the LCD of the given rational expressions. - $\frac{6}{x}, \frac{9}{x-4}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{(\mathrm{x}-8)(\mathrm{x}+5)}{64-\mathrm{x}^{2}}$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ . - $f(x)=\frac{x^{2}+9 x+18}{x^{2}+12 x+27}, g(x)=\frac{x^{2}+9 x}{x^{2}+3 x-18}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{(\mathrm{x}+2)(\mathrm{x}-4)}{(\mathrm{x}-4)(\mathrm{x}+7)}$

Find all values of the variable for which the rational expression is undefined. - $\frac{81 \mathrm{x}}{\mathrm{x}^{2}-3 \mathrm{x}-18}$

$\frac{3}{10 \mathrm{x}}-\frac{9}{14 \mathrm{x}^{2}}$

Find $r(-2)$ .

Evaluate the rational function. - $r(x)=\frac{x^{3}+3 x^{2}-14 x+10}{x^{2}+7 x-17}, r(2)$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{7 x-35}{(3+x)(5-x)}$

The number $G$ of gears a machine can make varies directly as the time $T$ it operates. If it can make 7665 gears in 4 hours, how many gears can it make in 2 hours?

$\frac{x^{2}+5 x+4}{x^{2}+10 x+24} \cdot \frac{x^{2}-11 x+30}{x^{2}-4 x-5}$

If $y$ varies inversely as the square of $x$ . If $y=3$ when $x=24$ , find $y$ when $x=6$ .