Question 1

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$.
-$f(x)=\frac{2 x}{x^{2}-5 x+6}, g(x)=\frac{8}{x^{2}-6 x+8}$

Accepted Answer

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

Question 2

Determine whether the two binomials are an opposites.
-$2 x+3$ and $-2 x+3$

Accepted Answer

A) True 
 B)False

Question 3

$\frac{x}{2}-\frac{2}{x}=0$

Accepted Answer

A)  $\{-2,0,2\}$ 
B)  $\{-2,2\}$ 
C)  $\{2\}$ 
D)  $\varnothing$ 
A)  $\{-2,0,2\}$ 
B)  $\{-2,2\}$ 
C)  $\{2\}$ 
D)  $\varnothing$

Question 4

$\frac{2}{x^{2}-3 x+2}+\frac{6}{x^{2}-1}$

Accepted Answer

A)  $\frac{24 x-10}{(x-1)(x+1)(x-2)}$ 
B)  $\frac{8 x-10}{(x-1)(x-2)}$ 
C)  $\frac{8 x-10}{(x-1)(x+1)(x-2)}$ 
D)  $\frac{10 x-8}{(x-1)(x+1)(x-2)}$ 
A)  $\frac{24 x-10}{(x-1)(x+1)(x-2)}$ 
B)  $\frac{8 x-10}{(x-1)(x-2)}$ 
C)  $\frac{8 x-10}{(x-1)(x+1)(x-2)}$ 
D)  $\frac{10 x-8}{(x-1)(x+1)(x-2)}$

Question 5

The speed of a stream is $6 \mathrm{mph}$. If a boat travels 58 miles downstream in the same time that it takes to travel 29 miles upstream, what is the speed of the boat in still water?

Accepted Answer

A)  $18 \mathrm{mph}$ 
B)  $21 \mathrm{mph}$ 
C)  $12 \mathrm{mph}$ 
D)  $20 \mathrm{mph}$ 
A)  $18 \mathrm{mph}$ 
B)  $21 \mathrm{mph}$ 
C)  $12 \mathrm{mph}$ 
D)  $20 \mathrm{mph}$

Question 6

Simplify the complex fraction.
-$\frac{1-\frac{5}{x}}{1+\frac{5}{x}}$

Accepted Answer

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

Question 7

Identify the given function as a linear function, a 
-$f(x)=\frac{x^{2}}{x^{3}+3 x-28}$

Accepted Answer

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

Question 8

Evaluate the rational function.
-$r(x)=\frac{17}{x^{2}+2 x-15}, r(-9)$

Accepted Answer

A)  $-\frac{17}{48}$ 
B)  Undefined 
C)  $\frac{17}{48}$ 
D)  $\frac{1}{24}$ 
A)  $-\frac{17}{48}$ 
B)  Undefined 
C)  $\frac{17}{48}$ 
D)  $\frac{1}{24}$

Question 9

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{x^{2}-49}{(x+7)^{2}}$

Accepted Answer

A)  $\frac{x+7}{x-7}$ 
B)  $x-7$ 
C)  $\frac{x^{2}-49}{(x+7)^{2}}$ 
D)  $\frac{x-7}{x+7}$ 
A)  $\frac{x+7}{x-7}$ 
B)  $x-7$ 
C)  $\frac{x^{2}-49}{(x+7)^{2}}$ 
D)  $\frac{x-7}{x+7}$

Question 10

$\frac{8}{x}+\frac{9}{x-5}$

Accepted Answer

A)  $\frac{17 x-40}{x(5-x)}$ 
B)  $\frac{40 x-17}{x(x-5)}$ 
C)  $\frac{40 x-17}{x(5-x)}$ 
D)  $\frac{17 x-40}{x(x-5)}$ 
A)  $\frac{17 x-40}{x(5-x)}$ 
B)  $\frac{40 x-17}{x(x-5)}$ 
C)  $\frac{40 x-17}{x(5-x)}$ 
D)  $\frac{17 x-40}{x(x-5)}$

Question 11

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

Accepted Answer

A)  $\frac{-4 x^{3}+x^{2}-5 x}{\left(x^{2}-1\right)\left(x-x^{2}\right)}$ 
B)  $\frac{-4 x+5}{(x-1)(x+1)}$ 
C)  $\frac{-4 x^{2}-5 x}{x(x-1)(x+1)}$ 
D)  $\frac{-4 x-5}{(x-1)(x+1)}$ 
A)  $\frac{-4 x^{3}+x^{2}-5 x}{\left(x^{2}-1\right)\left(x-x^{2}\right)}$ 
B)  $\frac{-4 x+5}{(x-1)(x+1)}$ 
C)  $\frac{-4 x^{2}-5 x}{x(x-1)(x+1)}$ 
D)  $\frac{-4 x-5}{(x-1)(x+1)}$

Question 12

For the given functions $f(x)$ and $g(x)$, find $f(x) \div g(x)$.
-$f(x)=\frac{x^{2}-5 x+6}{x^{2}-11 x+18}, g(x)=\left(x^{2}-12 x+27\right)$

Accepted Answer

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

Question 13

$\frac{2}{x^{2}}-\frac{3}{x}$

Accepted Answer

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

Question 14

$\frac{x}{x^{2}-16}-\frac{3}{x^{2}+5 x+4}$

Accepted Answer

A)  $\frac{x^{2}-2 x+12}{(x-4)(x+4)}$ 
B)  $\frac{x^{2}+2 x+12}{(x-4)(x+4)(x+1)}$ 
C)  $\frac{x^{2}-2}{(x-4)(x+4)(x+1)}$ 
D)  $\frac{x^{2}-2 x+12}{(x-4)(x+4)(x+1)}$ 
A)  $\frac{x^{2}-2 x+12}{(x-4)(x+4)}$ 
B)  $\frac{x^{2}+2 x+12}{(x-4)(x+4)(x+1)}$ 
C)  $\frac{x^{2}-2}{(x-4)(x+4)(x+1)}$ 
D)  $\frac{x^{2}-2 x+12}{(x-4)(x+4)(x+1)}$

Question 15

$y$ varies directly as $x$. If $y=24$ when $x=8$, find $y$ when $x=16$.

Accepted Answer

A)  48 
B)  12 
C)  80 
D)  192 
A)  48 
B)  12 
C)  80 
D)  192

Question 16

Evaluate the rational function.
-$r(x)=\frac{x^{2}+5 x-14}{x^{2}+4 x-12}, r(2)$

Accepted Answer

A)  $-\frac{9}{8}$ 
B)  $\frac{5}{4}$ 
C)  $\frac{9}{8}$ 
D)  Undefined 
A)  $-\frac{9}{8}$ 
B)  $\frac{5}{4}$ 
C)  $\frac{9}{8}$ 
D)  Undefined

Question 17

The distance that a spring is stretched by a hanging object varies directly as the mass of the object. If an object with a mass of $25 \mathrm{~kg}$ stretches a spring $44 \mathrm{~cm}$, find the stretch when the object has a mass of $25 \mathrm{~kg}$.

Accepted Answer

A)  $44 \mathrm{~cm}$ 
B)  $94 \mathrm{~cm}$ 
C)  $14.2 \mathrm{~cm}$ 
D)  $1.8 \mathrm{~cm}$ 
A)  $44 \mathrm{~cm}$ 
B)  $94 \mathrm{~cm}$ 
C)  $14.2 \mathrm{~cm}$ 
D)  $1.8 \mathrm{~cm}$

Question 18

Find all values of the variable for which the rational expression is undefined.
-$\frac{2}{x-4}$

Accepted Answer

A)  0 
B)  4 
C)  -4 
D)  None 
A)  0 
B)  4 
C)  -4 
D)  None

Question 19

Find the LCD of the given rational expressions.
-$\frac{7}{15 x}, \frac{5}{9 x^{2}}$

Accepted Answer

A)  45 
B)  $45 x^{3}$ 
C)  $45 x^{2}$ 
D)  $45 x$ 
A)  45 
B)  $45 x^{3}$ 
C)  $45 x^{2}$ 
D)  $45 x$

Question 20

The sum of a number and 81 times its reciprocal is - 18. Find the number.

Accepted Answer

A)  -9 
B)  -90 
C)  9 
D)  72 
A)  -9 
B)  -90 
C)  9 
D)  72

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$ . - $f(x)=\frac{2 x}{x^{2}-5 x+6}, g(x)=\frac{8}{x^{2}-6 x+8}$

Determine whether the two binomials are an opposites. - $2 x+3$ and $-2 x+3$

$\frac{x}{2}-\frac{2}{x}=0$

$\frac{2}{x^{2}-3 x+2}+\frac{6}{x^{2}-1}$

The speed of a stream is $6 \mathrm{mph}$ . If a boat travels 58 miles downstream in the same time that it takes to travel 29 miles upstream, what is the speed of the boat in still water?

Simplify the complex fraction. - $\frac{1-\frac{5}{x}}{1+\frac{5}{x}}$

Identify the given function as a linear function, a - $f(x)=\frac{x^{2}}{x^{3}+3 x-28}$

Evaluate the rational function. - $r(x)=\frac{17}{x^{2}+2 x-15}, r(-9)$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{x^{2}-49}{(x+7)^{2}}$

$\frac{8}{x}+\frac{9}{x-5}$

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

For the given functions $f(x)$ and $g(x)$ , find $f(x) \div g(x)$ . - $f(x)=\frac{x^{2}-5 x+6}{x^{2}-11 x+18}, g(x)=\left(x^{2}-12 x+27\right)$

$\frac{2}{x^{2}}-\frac{3}{x}$

$\frac{x}{x^{2}-16}-\frac{3}{x^{2}+5 x+4}$

$y$ varies directly as $x$ . If $y=24$ when $x=8$ , find $y$ when $x=16$ .

Evaluate the rational function. - $r(x)=\frac{x^{2}+5 x-14}{x^{2}+4 x-12}, r(2)$

The distance that a spring is stretched by a hanging object varies directly as the mass of the object. If an object with a mass of $25 \mathrm{~kg}$ stretches a spring $44 \mathrm{~cm}$ , find the stretch when the object has a mass of $25 \mathrm{~kg}$ .

Find all values of the variable for which the rational expression is undefined. - $\frac{2}{x-4}$

Find the LCD of the given rational expressions. - $\frac{7}{15 x}, \frac{5}{9 x^{2}}$

The sum of a number and 81 times its reciprocal is - 18. Find the number.

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

For the given rational functions f(x)f(x)f(x) and g(x),f(x)g(x), f(x)g(x),f(x) - g(x)g(x)g(x) . - f(x)=2xx2−5x+6,g(x)=8x2−6x+8f(x)=\frac{2 x}{x^{2}-5 x+6}, g(x)=\frac{8}{x^{2}-6 x+8}f(x)=x2−5x+62x​,g(x)=x2−6x+88​

Determine whether the two binomials are an opposites. - 2x+32 x+32x+3 and −2x+3-2 x+3−2x+3

x2−2x=0\frac{x}{2}-\frac{2}{x}=02x​−x2​=0

2x2−3x+2+6x2−1\frac{2}{x^{2}-3 x+2}+\frac{6}{x^{2}-1}x2−3x+22​+x2−16​

The speed of a stream is 6mph6 \mathrm{mph}6mph . If a boat travels 58 miles downstream in the same time that it takes to travel 29 miles upstream, what is the speed of the boat in still water?

Simplify the complex fraction. - 1−5x1+5x\frac{1-\frac{5}{x}}{1+\frac{5}{x}}1+x5​1−x5​​

Identify the given function as a linear function, a - f(x)=x2x3+3x−28f(x)=\frac{x^{2}}{x^{3}+3 x-28}f(x)=x3+3x−28x2​

Evaluate the rational function. - r(x)=17x2+2x−15,r(−9)r(x)=\frac{17}{x^{2}+2 x-15}, r(-9)r(x)=x2+2x−1517​,r(−9)

Simplify the rational expression. (Assume the denominator is nonzero.) - x2−49(x+7)2\frac{x^{2}-49}{(x+7)^{2}}(x+7)2x2−49​

8x+9x−5\frac{8}{x}+\frac{9}{x-5}x8​+x−59​

xx2−1+5xx−x2\frac{x}{x^{2}-1}+\frac{5 x}{x-x^{2}}x2−1x​+x−x25x​

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)=x2−5x+6x2−11x+18,g(x)=(x2−12x+27)f(x)=\frac{x^{2}-5 x+6}{x^{2}-11 x+18}, g(x)=\left(x^{2}-12 x+27\right)f(x)=x2−11x+18x2−5x+6​,g(x)=(x2−12x+27)

2x2−3x\frac{2}{x^{2}}-\frac{3}{x}x22​−x3​

xx2−16−3x2+5x+4\frac{x}{x^{2}-16}-\frac{3}{x^{2}+5 x+4}x2−16x​−x2+5x+43​

yyy varies directly as xxx . If y=24y=24y=24 when x=8x=8x=8 , find yyy when x=16x=16x=16 .

Evaluate the rational function. - r(x)=x2+5x−14x2+4x−12,r(2)r(x)=\frac{x^{2}+5 x-14}{x^{2}+4 x-12}, r(2)r(x)=x2+4x−12x2+5x−14​,r(2)

The distance that a spring is stretched by a hanging object varies directly as the mass of the object. If an object with a mass of 25 kg25 \mathrm{~kg}25 kg stretches a spring 44 cm44 \mathrm{~cm}44 cm , find the stretch when the object has a mass of 25 kg25 \mathrm{~kg}25 kg .

Find all values of the variable for which the rational expression is undefined. - 2x−4\frac{2}{x-4}x−42​

Find the LCD of the given rational expressions. - 715x,59x2\frac{7}{15 x}, \frac{5}{9 x^{2}}15x7​,9x25​

The sum of a number and 81 times its reciprocal is - 18. Find the number.

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

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$ . - $f(x)=\frac{2 x}{x^{2}-5 x+6}, g(x)=\frac{8}{x^{2}-6 x+8}$

Determine whether the two binomials are an opposites. - $2 x+3$ and $-2 x+3$

$\frac{x}{2}-\frac{2}{x}=0$

$\frac{2}{x^{2}-3 x+2}+\frac{6}{x^{2}-1}$

The speed of a stream is $6 \mathrm{mph}$ . If a boat travels 58 miles downstream in the same time that it takes to travel 29 miles upstream, what is the speed of the boat in still water?

Simplify the complex fraction. - $\frac{1-\frac{5}{x}}{1+\frac{5}{x}}$

Identify the given function as a linear function, a - $f(x)=\frac{x^{2}}{x^{3}+3 x-28}$

Evaluate the rational function. - $r(x)=\frac{17}{x^{2}+2 x-15}, r(-9)$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{x^{2}-49}{(x+7)^{2}}$

$\frac{8}{x}+\frac{9}{x-5}$

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

For the given functions $f(x)$ and $g(x)$ , find $f(x) \div g(x)$ . - $f(x)=\frac{x^{2}-5 x+6}{x^{2}-11 x+18}, g(x)=\left(x^{2}-12 x+27\right)$

$\frac{2}{x^{2}}-\frac{3}{x}$

$\frac{x}{x^{2}-16}-\frac{3}{x^{2}+5 x+4}$

$y$ varies directly as $x$ . If $y=24$ when $x=8$ , find $y$ when $x=16$ .

Evaluate the rational function. - $r(x)=\frac{x^{2}+5 x-14}{x^{2}+4 x-12}, r(2)$

The distance that a spring is stretched by a hanging object varies directly as the mass of the object. If an object with a mass of $25 \mathrm{~kg}$ stretches a spring $44 \mathrm{~cm}$ , find the stretch when the object has a mass of $25 \mathrm{~kg}$ .

Find all values of the variable for which the rational expression is undefined. - $\frac{2}{x-4}$

Find the LCD of the given rational expressions. - $\frac{7}{15 x}, \frac{5}{9 x^{2}}$