Question 1

$\frac{8 x}{x^{2}-1}-\frac{5}{1-x}$

Accepted Answer

A)  $\frac{8 x-5}{x^{2}-1}$ 
B)  $\frac{13 x-5}{x^{2}-1}$ 
C)  $\frac{13 x+5}{x^{2}-1}$ 
D)  $\frac{3 x-5}{x^{2}-1}$ 
A)  $\frac{8 x-5}{x^{2}-1}$ 
B)  $\frac{13 x-5}{x^{2}-1}$ 
C)  $\frac{13 x+5}{x^{2}-1}$ 
D)  $\frac{3 x-5}{x^{2}-1}$

Question 2

A man rode a bicycle for 12 miles and hiked an additional 8 miles. The total time for the trip was 5 hours. If his bicycling rate was 10 miles per hour faster than his hiking rate, then what was his hiking rate?

Accepted Answer

A)  $2 \mathrm{mph}$ 
B)  $4 \mathrm{mph}$ 
C)  $1 \mathrm{mph}$ 
D)  $3 \mathrm{mph}$ 
A)  $2 \mathrm{mph}$ 
B)  $4 \mathrm{mph}$ 
C)  $1 \mathrm{mph}$ 
D)  $3 \mathrm{mph}$

Question 3

$\frac{x^{2}-6 x}{x^{2}+8 x+16}+\frac{5 x-20}{x^{2}+8 x+16}$

Accepted Answer

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

Question 4

$y$ varies inversely as $x$. If $y=6$ when $x=9$, find $y$ when $x=27$.

Accepted Answer

A)  81 
B)  18 
C)  2 
D)  3 
A)  81 
B)  18 
C)  2 
D)  3

Question 5

Evaluate the rational expression for the given value of the variable.
-$\frac{2 x-2}{2 x^{2}+5 x-3}$ for $x=-4$

Accepted Answer

A)  $-\frac{2}{3}$ 
B)  $-\frac{10}{9}$ 
C)  $-\frac{2}{11}$ 
D)  $\frac{2}{3}$ 
A)  $-\frac{2}{3}$ 
B)  $-\frac{10}{9}$ 
C)  $-\frac{2}{11}$ 
D)  $\frac{2}{3}$

Question 6

Charles's law states that the volume $V$ of a gas varies directly as its temperature $T$ 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 2.2 liters at $24^{\circ} \mathrm{C}$. If the gas is heated to $49^{\circ} \mathrm{C}$, what will its volume be? Round your answer to the nearest hundredth of a liter as needed.

Accepted Answer

A)  2.58 liters 
B)  4.02 liters 
C)  2.39 liters 
D)  4.49 liters 
A)  2.58 liters 
B)  4.02 liters 
C)  2.39 liters 
D)  4.49 liters

Question 7

Find the LCD of the given rational expressions.
-$\frac{1}{x^{2}+6 x+9}, \frac{x-7}{x^{2}+3 x}$

Accepted Answer

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

Question 8

$\frac{x^{2}+18 x+81}{x^{2}-49} \cdot \frac{x+7}{x+9}$

Accepted Answer

A)  $\frac{x+9}{x+7}$ 
B)  $\frac{x+9}{x-7}$ 
C)  $\frac{x+9}{(x-7)^{2}}$ 
D)  $\frac{x-9}{x-7}$ 
A)  $\frac{x+9}{x+7}$ 
B)  $\frac{x+9}{x-7}$ 
C)  $\frac{x+9}{(x-7)^{2}}$ 
D)  $\frac{x-9}{x-7}$

Question 9

Find any vertical asymptotes as well as the location of any holes in the graph of the given rational function.
-$f(x)=\frac{x^{2}-49}{x^{2}+7 x+10}$

Accepted Answer

A)  vertical asymptotes: $x=-7, x=7$; holes: none 
B)  vertical asymptotes: $x=-5, x=-2$; holes: none 
C)  vertical asymptotes: $x=-5, x=-7$; hole: $(7,0)$ 
D)  vertical asymptotes: $x=5, x=2$; hole: $(7,0)$ 
A)  vertical asymptotes: $x=-7, x=7$; holes: none 
B)  vertical asymptotes: $x=-5, x=-2$; holes: none 
C)  vertical asymptotes: $x=-5, x=-7$; hole: $(7,0)$ 
D)  vertical asymptotes: $x=5, x=2$; hole: $(7,0)$

Question 10

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{x^{2}+2 x-15}{x^{2}+6 x-27}$

Accepted Answer

A)  $\frac{2 x-5}{6 x-9}$ 
B)  $\frac{x+5}{x+9}$ 
C)  $-\frac{x^{2}+2 x-15}{x^{2}+6 x-27}$ 
D)  $\frac{2 x-15}{6 x-27}$ 
A)  $\frac{2 x-5}{6 x-9}$ 
B)  $\frac{x+5}{x+9}$ 
C)  $-\frac{x^{2}+2 x-15}{x^{2}+6 x-27}$ 
D)  $\frac{2 x-15}{6 x-27}$

Question 11

$\frac{x^{2}-1}{x} \div \frac{x-1}{x+6}$

Accepted Answer

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

Question 12

The time $T$ necessary to make an enlargement of a photo negative varies directly as the area $A$ of the enlargement. If 245 seconds are required to make a 5 - by- 7 in. enlargement, find the time required for a 7 - by- 10 in. enlargement.

Accepted Answer

A)  560 seconds 
B)  630 seconds 
C)  420 seconds 
D)  490 seconds 
A)  560 seconds 
B)  630 seconds 
C)  420 seconds 
D)  490 seconds

Question 13

Simplify the complex fraction.
-$\frac{4+\frac{1}{3}}{3-\frac{2}{27}}$

Accepted Answer

A)  $\frac{117}{79}$ 
B)  $\frac{351}{158}$ 
C)  $\frac{234}{79}$ 
D)  $\frac{351}{79}$ 
A)  $\frac{117}{79}$ 
B)  $\frac{351}{158}$ 
C)  $\frac{234}{79}$ 
D)  $\frac{351}{79}$

Question 14

$y$ varies directly as the square of $x$. If $y=72$ when $x=3$, find $y$ when $x$ is 9 .

Accepted Answer

A)  648 
B)  24 
C)  216 
D)  0 
E)  27 
A)  648 
B)  24 
C)  216 
D)  0 
E)  27

Question 15

$\frac{x^{2}+12 x+27}{x^{2}+18 x+81} \cdot \frac{x^{2}+9 x}{x^{2}-3 x-18}$

Accepted Answer

A)  $\frac{x}{x-6}$ 
B)  $\frac{1}{x-6}$ 
C)  $\frac{x}{x^{2}+18 x+81}$ 
D)  $\frac{x^{2}+9 x}{x-6}$ 
A)  $\frac{x}{x-6}$ 
B)  $\frac{1}{x-6}$ 
C)  $\frac{x}{x^{2}+18 x+81}$ 
D)  $\frac{x^{2}+9 x}{x-6}$

Question 16

$\frac{23}{x-9}-\frac{11}{x-9}$

Accepted Answer

A)  $\frac{12}{x}$ 
B)  $\frac{23(\mathrm{x}-9)}{11(\mathrm{x}-9)}$ 
C)  $\frac{34}{x-9}$ 
D)  $\frac{12}{x-9}$ 
A)  $\frac{12}{x}$ 
B)  $\frac{23(\mathrm{x}-9)}{11(\mathrm{x}-9)}$ 
C)  $\frac{34}{x-9}$ 
D)  $\frac{12}{x-9}$

Question 17

Evaluate the rational expression for the given value of the variable.
-$\frac{x^{2}-5 x-15}{x^{2}-10 x+14}$ for $x=2$

Accepted Answer

A)  -5 
B)  $\frac{23}{4}$ 
C)  -2 
D)  $\frac{21}{2}$ 
A)  -5 
B)  $\frac{23}{4}$ 
C)  -2 
D)  $\frac{21}{2}$

Question 18

$\frac{72}{5 x-45}+\frac{8 x}{45-5 x}$

Accepted Answer

A)  $-\frac{8(x+9)}{5(x-9)}$ 
B)  $\frac{8(x+9)}{5(x-9)}$ 
C)  $-\frac{8}{5}$ 
D)  $\frac{8}{5}$ 
A)  $-\frac{8(x+9)}{5(x-9)}$ 
B)  $\frac{8(x+9)}{5(x-9)}$ 
C)  $-\frac{8}{5}$ 
D)  $\frac{8}{5}$

Question 19

A contractor finds that it takes Julie 11 hours to construct a wall of a certain size. It takes Leann 8 hours to construct the same wall. How long would it take if they worked together?

Accepted Answer

A)  $\frac{19}{88}$ hours 
B)  19 hours 
C)  1 hour 
D)  $4 \frac{12}{19}$ hours 
A)  $\frac{19}{88}$ hours 
B)  19 hours 
C)  1 hour 
D)  $4 \frac{12}{19}$ hours

Question 20

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

Accepted Answer

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

$\frac{8 x}{x^{2}-1}-\frac{5}{1-x}$

A man rode a bicycle for 12 miles and hiked an additional 8 miles. The total time for the trip was 5 hours. If his bicycling rate was 10 miles per hour faster than his hiking rate, then what was his hiking rate?

$\frac{x^{2}-6 x}{x^{2}+8 x+16}+\frac{5 x-20}{x^{2}+8 x+16}$

$y$ varies inversely as $x$ . If $y=6$ when $x=9$ , find $y$ when $x=27$ .

Evaluate the rational expression for the given value of the variable. - $\frac{2 x-2}{2 x^{2}+5 x-3}$ for $x=-4$

Find the LCD of the given rational expressions. - $\frac{1}{x^{2}+6 x+9}, \frac{x-7}{x^{2}+3 x}$

$\frac{x^{2}+18 x+81}{x^{2}-49} \cdot \frac{x+7}{x+9}$

Find any vertical asymptotes as well as the location of any holes in the graph of the given rational function. - $f(x)=\frac{x^{2}-49}{x^{2}+7 x+10}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{x^{2}+2 x-15}{x^{2}+6 x-27}$

$\frac{x^{2}-1}{x} \div \frac{x-1}{x+6}$

The time $T$ necessary to make an enlargement of a photo negative varies directly as the area $A$ of the enlargement. If 245 seconds are required to make a 5 - by- 7 in. enlargement, find the time required for a 7 - by- 10 in. enlargement.

Simplify the complex fraction. - $\frac{4+\frac{1}{3}}{3-\frac{2}{27}}$

$y$ varies directly as the square of $x$ . If $y=72$ when $x=3$ , find $y$ when $x$ is 9 .

$\frac{x^{2}+12 x+27}{x^{2}+18 x+81} \cdot \frac{x^{2}+9 x}{x^{2}-3 x-18}$

$\frac{23}{x-9}-\frac{11}{x-9}$

Evaluate the rational expression for the given value of the variable. - $\frac{x^{2}-5 x-15}{x^{2}-10 x+14}$ for $x=2$

$\frac{72}{5 x-45}+\frac{8 x}{45-5 x}$

A contractor finds that it takes Julie 11 hours to construct a wall of a certain size. It takes Leann 8 hours to construct the same wall. How long would it take if they worked together?

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

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

8xx2−1−51−x\frac{8 x}{x^{2}-1}-\frac{5}{1-x}x2−18x​−1−x5​

A man rode a bicycle for 12 miles and hiked an additional 8 miles. The total time for the trip was 5 hours. If his bicycling rate was 10 miles per hour faster than his hiking rate, then what was his hiking rate?

x2−6xx2+8x+16+5x−20x2+8x+16\frac{x^{2}-6 x}{x^{2}+8 x+16}+\frac{5 x-20}{x^{2}+8 x+16}x2+8x+16x2−6x​+x2+8x+165x−20​

yyy varies inversely as xxx . If y=6y=6y=6 when x=9x=9x=9 , find yyy when x=27x=27x=27 .

Evaluate the rational expression for the given value of the variable. - 2x−22x2+5x−3\frac{2 x-2}{2 x^{2}+5 x-3}2x2+5x−32x−2​ for x=−4x=-4x=−4

Find the LCD of the given rational expressions. - 1x2+6x+9,x−7x2+3x\frac{1}{x^{2}+6 x+9}, \frac{x-7}{x^{2}+3 x}x2+6x+91​,x2+3xx−7​

x2+18x+81x2−49⋅x+7x+9\frac{x^{2}+18 x+81}{x^{2}-49} \cdot \frac{x+7}{x+9}x2−49x2+18x+81​⋅x+9x+7​

Find any vertical asymptotes as well as the location of any holes in the graph of the given rational function. - f(x)=x2−49x2+7x+10f(x)=\frac{x^{2}-49}{x^{2}+7 x+10}f(x)=x2+7x+10x2−49​

Simplify the rational expression. (Assume the denominator is nonzero.) - x2+2x−15x2+6x−27\frac{x^{2}+2 x-15}{x^{2}+6 x-27}x2+6x−27x2+2x−15​

x2−1x÷x−1x+6\frac{x^{2}-1}{x} \div \frac{x-1}{x+6}xx2−1​÷x+6x−1​

The time TTT necessary to make an enlargement of a photo negative varies directly as the area AAA of the enlargement. If 245 seconds are required to make a 5 - by- 7 in. enlargement, find the time required for a 7 - by- 10 in. enlargement.

Simplify the complex fraction. - 4+133−227\frac{4+\frac{1}{3}}{3-\frac{2}{27}}3−272​4+31​​

yyy varies directly as the square of xxx . If y=72y=72y=72 when x=3x=3x=3 , find yyy when xxx is 9 .

x2+12x+27x2+18x+81⋅x2+9xx2−3x−18\frac{x^{2}+12 x+27}{x^{2}+18 x+81} \cdot \frac{x^{2}+9 x}{x^{2}-3 x-18}x2+18x+81x2+12x+27​⋅x2−3x−18x2+9x​

23x−9−11x−9\frac{23}{x-9}-\frac{11}{x-9}x−923​−x−911​

Evaluate the rational expression for the given value of the variable. - x2−5x−15x2−10x+14\frac{x^{2}-5 x-15}{x^{2}-10 x+14}x2−10x+14x2−5x−15​ for x=2x=2x=2

725x−45+8x45−5x\frac{72}{5 x-45}+\frac{8 x}{45-5 x}5x−4572​+45−5x8x​

A contractor finds that it takes Julie 11 hours to construct a wall of a certain size. It takes Leann 8 hours to construct the same wall. How long would it take if they worked together?

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

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

$\frac{8 x}{x^{2}-1}-\frac{5}{1-x}$

$\frac{x^{2}-6 x}{x^{2}+8 x+16}+\frac{5 x-20}{x^{2}+8 x+16}$

$y$ varies inversely as $x$ . If $y=6$ when $x=9$ , find $y$ when $x=27$ .

Evaluate the rational expression for the given value of the variable. - $\frac{2 x-2}{2 x^{2}+5 x-3}$ for $x=-4$

Find the LCD of the given rational expressions. - $\frac{1}{x^{2}+6 x+9}, \frac{x-7}{x^{2}+3 x}$

$\frac{x^{2}+18 x+81}{x^{2}-49} \cdot \frac{x+7}{x+9}$

Find any vertical asymptotes as well as the location of any holes in the graph of the given rational function. - $f(x)=\frac{x^{2}-49}{x^{2}+7 x+10}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{x^{2}+2 x-15}{x^{2}+6 x-27}$

$\frac{x^{2}-1}{x} \div \frac{x-1}{x+6}$

The time $T$ necessary to make an enlargement of a photo negative varies directly as the area $A$ of the enlargement. If 245 seconds are required to make a 5 - by- 7 in. enlargement, find the time required for a 7 - by- 10 in. enlargement.

Simplify the complex fraction. - $\frac{4+\frac{1}{3}}{3-\frac{2}{27}}$

$y$ varies directly as the square of $x$ . If $y=72$ when $x=3$ , find $y$ when $x$ is 9 .

$\frac{x^{2}+12 x+27}{x^{2}+18 x+81} \cdot \frac{x^{2}+9 x}{x^{2}-3 x-18}$

$\frac{23}{x-9}-\frac{11}{x-9}$

Evaluate the rational expression for the given value of the variable. - $\frac{x^{2}-5 x-15}{x^{2}-10 x+14}$ for $x=2$

$\frac{72}{5 x-45}+\frac{8 x}{45-5 x}$

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