Question 1

Martha can rake the leaves in her yard in 5 hours. Her brother can do the job in 2 hours. How long will it take them to do the job working together?

Accepted Answer

A)  $\frac{1}{10}$ hours 
B)  $\frac{1}{7}$ hours 
C)  $3 \frac{1}{3}$ hours 
D)  $1 \frac{3}{7}$ hours 
A)  $\frac{1}{10}$ hours 
B)  $\frac{1}{7}$ hours 
C)  $3 \frac{1}{3}$ hours 
D)  $1 \frac{3}{7}$ hours

Question 2

Find all values of the variable for which the rational expression is undefined.
-$\frac{13}{x-6}$

Accepted Answer

A)  13 
B)  0 
C)  6 
D)  -6 
A)  13 
B)  0 
C)  6 
D)  -6

Question 3

Find the missing numerator and denominator.
-$\frac{16 x^{2}-49}{x^{2}-36} \div \frac{?}{?}=\frac{4 x-7}{x-6}$

Accepted Answer

A)  $\frac{4 x-7}{x-6}$ 
B)  $\frac{x-6}{4 x-7}$ 
C)  $\frac{4 x+7}{x+6}$ 
D)  $\frac{(4 x+7)\left(16 x^{2}-49\right)}{\left(x^{2}+6\right)(x+6)}$ 
A)  $\frac{4 x-7}{x-6}$ 
B)  $\frac{x-6}{4 x-7}$ 
C)  $\frac{4 x+7}{x+6}$ 
D)  $\frac{(4 x+7)\left(16 x^{2}-49\right)}{\left(x^{2}+6\right)(x+6)}$

Question 4

$\frac{x^{2}-4 x}{x+7}+\frac{4 x-x^{2}}{x+7}$

Accepted Answer

A)  0 
B)  1 
C)  $\frac{1}{x+7}$ 
D)  $\frac{8 x}{x+7}$ 
A)  0 
B)  1 
C)  $\frac{1}{x+7}$ 
D)  $\frac{8 x}{x+7}$

Question 5

Tom Quig traveled 220 miles east of St. Louis. For most of the trip he averaged $60 \mathrm{mph}$, but for one period of time he was slowed to $10 \mathrm{mph}$ due to a major accident. If the total travel time was 7 hours, for how many miles did he drive at the reduced speed?

Accepted Answer

A)  35 miles 
B)  50 miles 
C)  60 miles 
D)  40 miles 
A)  35 miles 
B)  50 miles 
C)  60 miles 
D)  40 miles

Question 6

Solve for the specified variable
-$\frac{1}{a}+\frac{1}{b}=\frac{1}{C}$ for $c$

Accepted Answer

A)  $c=a b(a+b)$ 
B)  $c=a+b$ 
C)  $c=\frac{a+b}{a b}$ 
D)  $c=\frac{a b}{a+b}$ 
A)  $c=a b(a+b)$ 
B)  $c=a+b$ 
C)  $c=\frac{a+b}{a b}$ 
D)  $c=\frac{a b}{a+b}$

Question 7

For the given functions $f(x)$ and $g(x)$, find $f(x) \cdot g(x)$.
-$f(x)=\frac{x^{2}+5 x+6}{x^{2}+8 x+15}, g(x)=\frac{x^{2}+5 x}{x^{2}+10 x+16}$

Accepted Answer

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

Question 8

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$.
-$f(x)=\frac{x^{2}-6 x}{x^{2}+4 x+4}, g(x)=\frac{5 x-6}{x^{2}+4 x+4}$

Accepted Answer

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

Question 9

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{2 x^{2}-50}{20-4 x}$

Accepted Answer

A)  $\frac{x+5}{2}$ 
B)  $\frac{x^{2}-25}{5-x}$ 
C)  $-\frac{x+5}{2}$ 
D)  $-2(x+5)$ 
A)  $\frac{x+5}{2}$ 
B)  $\frac{x^{2}-25}{5-x}$ 
C)  $-\frac{x+5}{2}$ 
D)  $-2(x+5)$

Question 10

Simplify the complex fraction.
-$\frac{x-6+\frac{9}{x}}{x-7+\frac{12}{x}}$

Accepted Answer

A)  $\frac{(x-3)^{2}}{x-4}$ 
B)  $\frac{x^{2}-6 x+9}{x^{2}-7 x+12}$ 
C)  $\frac{x-3}{x^{2}-4 x}$ 
D)  $\frac{x-3}{x-4}$ 
A)  $\frac{(x-3)^{2}}{x-4}$ 
B)  $\frac{x^{2}-6 x+9}{x^{2}-7 x+12}$ 
C)  $\frac{x-3}{x^{2}-4 x}$ 
D)  $\frac{x-3}{x-4}$

Question 11

$\frac{x}{7}+\frac{7}{x}=0$

Accepted Answer

A)  $\{-7,0,7\}$ 
B)  $\{-7,7\}$ 
C)  $\{7\}$ 
D)  $\varnothing$ 
A)  $\{-7,0,7\}$ 
B)  $\{-7,7\}$ 
C)  $\{7\}$ 
D)  $\varnothing$

Question 12

$\frac{2 x^{2}+14}{x^{2}+8 x-9}-\frac{11 x}{x^{2}+8 x-9}$

Accepted Answer

A)  $\frac{2 x-7}{x-1}$ 
B)  $\frac{(x+2)(2 x+7)}{(x+9)(x-1)}$ 
C)  $\frac{(x-2)(2 x+7)}{(x+9)(x-1)}$ 
D)  $\frac{(x-2)(2 x-7}{(x+9)(x-1)}$ 
A)  $\frac{2 x-7}{x-1}$ 
B)  $\frac{(x+2)(2 x+7)}{(x+9)(x-1)}$ 
C)  $\frac{(x-2)(2 x+7)}{(x+9)(x-1)}$ 
D)  $\frac{(x-2)(2 x-7}{(x+9)(x-1)}$

Question 13

Determine whether the two binomials are an opposites.
-$x+8$ and $8+x$

Accepted Answer

A) True 
 B)False

Question 14

For the given functions $f(x)$ and $g(x)$, find $f(x) \cdot g(x)$.
-$f(x)=\frac{8}{(x-8)}, g(x)=\frac{x^{2}-4 x-32}{(x+4)^{2}}$

Accepted Answer

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

Question 15

$\frac{8 x}{x^{2}-16}-\frac{x}{x-4}$

Accepted Answer

A)  $\frac{12 x-x^{2}}{(x+4)(x-4)}$ 
B)  $\frac{7 x}{(x+4)(x-4)}$ 
C)  $\frac{-x^{3}+8 x^{2}-16 x}{(x-4)\left(x^{2}-16\right)}$ 
D)  $-\frac{x}{x+4}$ 
A)  $\frac{12 x-x^{2}}{(x+4)(x-4)}$ 
B)  $\frac{7 x}{(x+4)(x-4)}$ 
C)  $\frac{-x^{3}+8 x^{2}-16 x}{(x-4)\left(x^{2}-16\right)}$ 
D)  $-\frac{x}{x+4}$

Question 16

Simplify the rational expression. (Assume the denominator is nonzero.)
-$\frac{18 x^{3}}{6 x}$

Accepted Answer

A)  12 
B)  $12 x^{2}$ 
C)  $3 x$ 
D)  $3 x^{2}$ 
A)  12 
B)  $12 x^{2}$ 
C)  $3 x$ 
D)  $3 x^{2}$

Question 17

A boat can go $90 \mathrm{mph}$ in still water. It takes as long to go 630 miles upstream as it does to go downstream 720 miles. How fast is the current?

Accepted Answer

A)  $1 \mathrm{mph}$ 
B)  $9 \mathrm{mph}$ 
C)  $5 \mathrm{mph}$ 
D)  $6 \mathrm{mph}$ 
A)  $1 \mathrm{mph}$ 
B)  $9 \mathrm{mph}$ 
C)  $5 \mathrm{mph}$ 
D)  $6 \mathrm{mph}$

Question 18

$\frac{6 \mathrm{x}+16}{\mathrm{x}-4}-\frac{5 \mathrm{x}+7}{\mathrm{x}-4}$

Accepted Answer

A)  $\frac{x+9}{x-4}$ 
B)  $\frac{5 x+9}{x-4}$ 
C)  $\frac{x+23}{x-4}$ 
D)  $\frac{x+23}{2(x-4)}$ 
A)  $\frac{x+9}{x-4}$ 
B)  $\frac{5 x+9}{x-4}$ 
C)  $\frac{x+23}{x-4}$ 
D)  $\frac{x+23}{2(x-4)}$

Question 19

$\frac{2 \mathrm{x}+8}{\mathrm{x}^{2}+6 \mathrm{x}+8}-\frac{\mathrm{x}+4}{\mathrm{x}^{2}+6 \mathrm{x}+8}$

Accepted Answer

A)  $\frac{9}{x^{2}+6 x+8}$ 
B)  $x+4$ 
C)  $\frac{x+12}{x^{2}+6 x+8}$ 
D)  $\frac{1}{x+2}$ 
A)  $\frac{9}{x^{2}+6 x+8}$ 
B)  $x+4$ 
C)  $\frac{x+12}{x^{2}+6 x+8}$ 
D)  $\frac{1}{x+2}$

Question 20

The sum of a number and its reciprocal is - 2. Find the number.

Accepted Answer

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

Martha can rake the leaves in her yard in 5 hours. Her brother can do the job in 2 hours. How long will it take them to do the job working together?

Find all values of the variable for which the rational expression is undefined. - $\frac{13}{x-6}$

Find the missing numerator and denominator. - $\frac{16 x^{2}-49}{x^{2}-36} \div \frac{?}{?}=\frac{4 x-7}{x-6}$

$\frac{x^{2}-4 x}{x+7}+\frac{4 x-x^{2}}{x+7}$

Tom Quig traveled 220 miles east of St. Louis. For most of the trip he averaged $60 \mathrm{mph}$ , but for one period of time he was slowed to $10 \mathrm{mph}$ due to a major accident. If the total travel time was 7 hours, for how many miles did he drive at the reduced speed?

Solve for the specified variable - $\frac{1}{a}+\frac{1}{b}=\frac{1}{C}$ for $c$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ . - $f(x)=\frac{x^{2}+5 x+6}{x^{2}+8 x+15}, g(x)=\frac{x^{2}+5 x}{x^{2}+10 x+16}$

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$ . - $f(x)=\frac{x^{2}-6 x}{x^{2}+4 x+4}, g(x)=\frac{5 x-6}{x^{2}+4 x+4}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{2 x^{2}-50}{20-4 x}$

Simplify the complex fraction. - $\frac{x-6+\frac{9}{x}}{x-7+\frac{12}{x}}$

$\frac{x}{7}+\frac{7}{x}=0$

$\frac{2 x^{2}+14}{x^{2}+8 x-9}-\frac{11 x}{x^{2}+8 x-9}$

Determine whether the two binomials are an opposites. - $x+8$ and $8+x$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ . - $f(x)=\frac{8}{(x-8)}, g(x)=\frac{x^{2}-4 x-32}{(x+4)^{2}}$

$\frac{8 x}{x^{2}-16}-\frac{x}{x-4}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{18 x^{3}}{6 x}$

A boat can go $90 \mathrm{mph}$ in still water. It takes as long to go 630 miles upstream as it does to go downstream 720 miles. How fast is the current?

$\frac{6 \mathrm{x}+16}{\mathrm{x}-4}-\frac{5 \mathrm{x}+7}{\mathrm{x}-4}$

$\frac{2 \mathrm{x}+8}{\mathrm{x}^{2}+6 \mathrm{x}+8}-\frac{\mathrm{x}+4}{\mathrm{x}^{2}+6 \mathrm{x}+8}$

The sum of a number and its reciprocal is - 2. 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

Martha can rake the leaves in her yard in 5 hours. Her brother can do the job in 2 hours. How long will it take them to do the job working together?

Find all values of the variable for which the rational expression is undefined. - 13x−6\frac{13}{x-6}x−613​

Find the missing numerator and denominator. - 16x2−49x2−36÷??=4x−7x−6\frac{16 x^{2}-49}{x^{2}-36} \div \frac{?}{?}=\frac{4 x-7}{x-6}x2−3616x2−49​÷??​=x−64x−7​

x2−4xx+7+4x−x2x+7\frac{x^{2}-4 x}{x+7}+\frac{4 x-x^{2}}{x+7}x+7x2−4x​+x+74x−x2​

Tom Quig traveled 220 miles east of St. Louis. For most of the trip he averaged 60mph60 \mathrm{mph}60mph , but for one period of time he was slowed to 10mph10 \mathrm{mph}10mph due to a major accident. If the total travel time was 7 hours, for how many miles did he drive at the reduced speed?

Solve for the specified variable - 1a+1b=1C\frac{1}{a}+\frac{1}{b}=\frac{1}{C}a1​+b1​=C1​ for ccc

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+5x+6x2+8x+15,g(x)=x2+5xx2+10x+16f(x)=\frac{x^{2}+5 x+6}{x^{2}+8 x+15}, g(x)=\frac{x^{2}+5 x}{x^{2}+10 x+16}f(x)=x2+8x+15x2+5x+6​,g(x)=x2+10x+16x2+5x​

For the given rational functions f(x)f(x)f(x) and g(x),f(x)+g(x)g(x), f(x)+g(x)g(x),f(x)+g(x) . - f(x)=x2−6xx2+4x+4,g(x)=5x−6x2+4x+4f(x)=\frac{x^{2}-6 x}{x^{2}+4 x+4}, g(x)=\frac{5 x-6}{x^{2}+4 x+4}f(x)=x2+4x+4x2−6x​,g(x)=x2+4x+45x−6​

Simplify the rational expression. (Assume the denominator is nonzero.) - 2x2−5020−4x\frac{2 x^{2}-50}{20-4 x}20−4x2x2−50​

Simplify the complex fraction. - x−6+9xx−7+12x\frac{x-6+\frac{9}{x}}{x-7+\frac{12}{x}}x−7+x12​x−6+x9​​

x7+7x=0\frac{x}{7}+\frac{7}{x}=07x​+x7​=0

2x2+14x2+8x−9−11xx2+8x−9\frac{2 x^{2}+14}{x^{2}+8 x-9}-\frac{11 x}{x^{2}+8 x-9}x2+8x−92x2+14​−x2+8x−911x​

Determine whether the two binomials are an opposites. - x+8x+8x+8 and 8+x8+x8+x

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)=8(x−8),g(x)=x2−4x−32(x+4)2f(x)=\frac{8}{(x-8)}, g(x)=\frac{x^{2}-4 x-32}{(x+4)^{2}}f(x)=(x−8)8​,g(x)=(x+4)2x2−4x−32​

8xx2−16−xx−4\frac{8 x}{x^{2}-16}-\frac{x}{x-4}x2−168x​−x−4x​

Simplify the rational expression. (Assume the denominator is nonzero.) - 18x36x\frac{18 x^{3}}{6 x}6x18x3​

A boat can go 90mph90 \mathrm{mph}90mph in still water. It takes as long to go 630 miles upstream as it does to go downstream 720 miles. How fast is the current?

6x+16x−4−5x+7x−4\frac{6 \mathrm{x}+16}{\mathrm{x}-4}-\frac{5 \mathrm{x}+7}{\mathrm{x}-4}x−46x+16​−x−45x+7​

2x+8x2+6x+8−x+4x2+6x+8\frac{2 \mathrm{x}+8}{\mathrm{x}^{2}+6 \mathrm{x}+8}-\frac{\mathrm{x}+4}{\mathrm{x}^{2}+6 \mathrm{x}+8}x2+6x+82x+8​−x2+6x+8x+4​

The sum of a number and its reciprocal is - 2. 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

Find all values of the variable for which the rational expression is undefined. - $\frac{13}{x-6}$

Find the missing numerator and denominator. - $\frac{16 x^{2}-49}{x^{2}-36} \div \frac{?}{?}=\frac{4 x-7}{x-6}$

$\frac{x^{2}-4 x}{x+7}+\frac{4 x-x^{2}}{x+7}$

Tom Quig traveled 220 miles east of St. Louis. For most of the trip he averaged $60 \mathrm{mph}$ , but for one period of time he was slowed to $10 \mathrm{mph}$ due to a major accident. If the total travel time was 7 hours, for how many miles did he drive at the reduced speed?

Solve for the specified variable - $\frac{1}{a}+\frac{1}{b}=\frac{1}{C}$ for $c$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ . - $f(x)=\frac{x^{2}+5 x+6}{x^{2}+8 x+15}, g(x)=\frac{x^{2}+5 x}{x^{2}+10 x+16}$

For the given rational functions $f(x)$ and $g(x), f(x)+g(x)$ . - $f(x)=\frac{x^{2}-6 x}{x^{2}+4 x+4}, g(x)=\frac{5 x-6}{x^{2}+4 x+4}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{2 x^{2}-50}{20-4 x}$

Simplify the complex fraction. - $\frac{x-6+\frac{9}{x}}{x-7+\frac{12}{x}}$

$\frac{x}{7}+\frac{7}{x}=0$

$\frac{2 x^{2}+14}{x^{2}+8 x-9}-\frac{11 x}{x^{2}+8 x-9}$

Determine whether the two binomials are an opposites. - $x+8$ and $8+x$

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ . - $f(x)=\frac{8}{(x-8)}, g(x)=\frac{x^{2}-4 x-32}{(x+4)^{2}}$

$\frac{8 x}{x^{2}-16}-\frac{x}{x-4}$

Simplify the rational expression. (Assume the denominator is nonzero.) - $\frac{18 x^{3}}{6 x}$

A boat can go $90 \mathrm{mph}$ in still water. It takes as long to go 630 miles upstream as it does to go downstream 720 miles. How fast is the current?

$\frac{6 \mathrm{x}+16}{\mathrm{x}-4}-\frac{5 \mathrm{x}+7}{\mathrm{x}-4}$

$\frac{2 \mathrm{x}+8}{\mathrm{x}^{2}+6 \mathrm{x}+8}-\frac{\mathrm{x}+4}{\mathrm{x}^{2}+6 \mathrm{x}+8}$