Question 1

Determine whether the two binomials are an opposites.
-$3 x-6$ and $3 x+6$

Accepted Answer

A) True 
 B)False

Question 2

Find all values of the variable for which the rational expression is undefined.
-$\frac{7}{x+8}$

Accepted Answer

A)  8 
B)  _ 8 
C)  0 
D)  Nono 
A)  8 
B)  _ 8 
C)  0 
D)  Nono

Question 3

Simplify the complex fraction.
-$\frac{\frac{x}{6}-\frac{3}{x}}{\frac{3}{x}+\frac{1}{2}}$

Accepted Answer

A)  $\frac{x^{2}-18}{3 x+18}$ 
B)  $\frac{x^{2}-3}{3 x+18}$ 
C)  $\frac{x^{2}-18}{x+18}$ 
D)  $\frac{x^{2}-18}{3(x+6)}$ 
A)  $\frac{x^{2}-18}{3 x+18}$ 
B)  $\frac{x^{2}-3}{3 x+18}$ 
C)  $\frac{x^{2}-18}{x+18}$ 
D)  $\frac{x^{2}-18}{3(x+6)}$

Question 4

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$.
-$f(x)=\frac{24}{x-6}, g(x)=\frac{4 x}{x-6}$

Accepted Answer

A)  $\frac{4}{x-6}$ 
B)  -4 
C)  4 
D)  $-\frac{4}{x-6}$ 
A)  $\frac{4}{x-6}$ 
B)  -4 
C)  4 
D)  $-\frac{4}{x-6}$

Question 5

$y$ varies directly as $x$. If $y=\frac{17}{2}$ when $x=17$, find $y$ when $x=16$.

Accepted Answer

A)  $\frac{289}{32}$ 
B)  8 
C)  $\frac{17}{16}$ 
D)  $\frac{1}{2}$ 
A)  $\frac{289}{32}$ 
B)  8 
C)  $\frac{17}{16}$ 
D)  $\frac{1}{2}$

Question 6

$\frac{4}{\mathrm{x}}+\frac{7}{3 \mathrm{x}}$

Accepted Answer

A)  $-\frac{19}{x}$ 
B)  $\frac{19}{3 x}$ 
C)  $\frac{11}{x^{2}}$ 
D)  $\frac{19 x}{3}$ 
A)  $-\frac{19}{x}$ 
B)  $\frac{19}{3 x}$ 
C)  $\frac{11}{x^{2}}$ 
D)  $\frac{19 x}{3}$

Question 7

Simplify the complex fraction.
-$\frac{\frac{1}{x}-2}{\frac{1}{x}-6}$

Accepted Answer

A)  $\frac{1-2 x}{1-6 x}$ 
B)  $\frac{1-6 x}{1-2 x}$ 
C)  $\frac{x-6}{x-2}$ 
D)  $\frac{x-2}{x-6}$ 
A)  $\frac{1-2 x}{1-6 x}$ 
B)  $\frac{1-6 x}{1-2 x}$ 
C)  $\frac{x-6}{x-2}$ 
D)  $\frac{x-2}{x-6}$

Question 8

Find the LCD of the given rational expressions.
-$\frac{7}{x^{2}-2 x-24}, \frac{7}{6 x+24}$

Accepted Answer

A)  $6(x-6)(x+4)$ 
B)  $6(x+6)(x-4)$ 
C)  $6(x+6)(x+4)$ 
D)  $6(x-6)(x-4)$ 
A)  $6(x-6)(x+4)$ 
B)  $6(x+6)(x-4)$ 
C)  $6(x+6)(x+4)$ 
D)  $6(x-6)(x-4)$

Question 9

$\frac{18}{x-2}=1+\frac{20}{x+2}$

Accepted Answer

A)  $\{-20,10\}$ 
B)  $\{8,-10\}$ 
C)  $\{-8,10\}$ 
D)  $\varnothing$ 
A)  $\{-20,10\}$ 
B)  $\{8,-10\}$ 
C)  $\{-8,10\}$ 
D)  $\varnothing$

Question 10

Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 245 miles in the same time that Dana travels 215 miles. If Chuck's rate of travel is $6 \mathrm{mph}$ more than Dana's, and they travel the same length of time, at what speed does Chuck travel?

Accepted Answer

A)  $56 \mathrm{mph}$ 
B)  $42 \mathrm{mph}$ 
C)  $49 \mathrm{mph}$ 
D)  $43 \mathrm{mph}$ 
A)  $56 \mathrm{mph}$ 
B)  $42 \mathrm{mph}$ 
C)  $49 \mathrm{mph}$ 
D)  $43 \mathrm{mph}$

Question 11

$\frac{x^{2}-9}{4 x^{2}-25} \div \frac{x-3}{5-2 x}$

Accepted Answer

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

Question 12

Find the domain of the rational function
-$r(x)=\frac{x^{2}+4 x-32}{x^{2}-9}$

Accepted Answer

A)  All real numbers except - 8 and 4 . 
B)  All real numbers except - 3 and 3. 
C)  All real numbers except -4 and 8. 
D)  All real numbers. 
A)  All real numbers except - 8 and 4 . 
B)  All real numbers except - 3 and 3. 
C)  All real numbers except -4 and 8. 
D)  All real numbers.

Question 13

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

Accepted Answer

A)  Undefined 
B)  0 
C)  $-\frac{1}{9}$ 
D)  $\frac{1}{2}$ 
A)  Undefined 
B)  0 
C)  $-\frac{1}{9}$ 
D)  $\frac{1}{2}$

Question 14

For the given functions $f(x)$ and $g(x)$, find $f(x) \cdot g(x)$.
-$f(x)=x+8, g(x)=\frac{x^{2}+5 x-24}{4 x^{2}+24 x-64}$

Accepted Answer

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

Question 15

A boat can go 180 miles downstream in the same time it can go 120 miles upstream. The speed of the current is 4 miles per hour. Find the speed of the boat in still water.

Accepted Answer

A)  $16 \mathrm{mph}$ 
B)  $24 \mathrm{mph}$ 
C)  $30 \mathrm{mph}$ 
D)  $20 \mathrm{mph}$ 
A)  $16 \mathrm{mph}$ 
B)  $24 \mathrm{mph}$ 
C)  $30 \mathrm{mph}$ 
D)  $20 \mathrm{mph}$

Question 16

$\frac{25}{x-5}+\frac{9}{x-5}$

Accepted Answer

A)  $\frac{16}{x-5}$ 
B)  $\frac{25(x-5)}{9(x-5)}$ 
C)  $\frac{34}{x-5}$ 
D)  $\frac{16}{x}$ 
A)  $\frac{16}{x-5}$ 
B)  $\frac{25(x-5)}{9(x-5)}$ 
C)  $\frac{34}{x-5}$ 
D)  $\frac{16}{x}$

Question 17

Determine whether the two binomials are an opposites.
-$x-5$ and $5-x$

Accepted Answer

A) True 
 B)False

Question 18

Solve for the specified variable
-$S=\frac{H}{m\left(t_{1}-t_{2}\right)}$ for $t_{2}$

Accepted Answer

A)  $t_{2}=\frac{\text { Smt }_{1}-H}{m}$ 
B)  $t_{2}=\frac{m t_{1}-\mathrm{SH}}{\mathrm{Sm}}$ 
C)  $t_{2}=\frac{H-S_{t} 1}{S m}$ 
D)  $\mathrm{t}_{2}=\frac{\mathrm{Smt}_{1}-\mathrm{H}}{\mathrm{Sm}}$ 
A)  $t_{2}=\frac{\text { Smt }_{1}-H}{m}$ 
B)  $t_{2}=\frac{m t_{1}-\mathrm{SH}}{\mathrm{Sm}}$ 
C)  $t_{2}=\frac{H-S_{t} 1}{S m}$ 
D)  $\mathrm{t}_{2}=\frac{\mathrm{Smt}_{1}-\mathrm{H}}{\mathrm{Sm}}$

Question 19

Solve for the specified variable
-$I=\frac{E}{R+r}$ for $r$

Accepted Answer

A)  $r=\frac{E-I R}{I}$ 
B)  $r=\frac{I R}{E}$ 
C)  $r=\frac{E-R}{I}$ 
D)  $r=\frac{E}{I}-I R$ 
A)  $r=\frac{E-I R}{I}$ 
B)  $r=\frac{I R}{E}$ 
C)  $r=\frac{E-R}{I}$ 
D)  $r=\frac{E}{I}-I R$

Question 20

Find the domain of the rational function
-$r(x)=\frac{x^{2}+x-36}{x^{2}-17 x+72}$

Accepted Answer

A)  All real numbers except - 9 and - 8 . 
B)  All real numbers except - 9 and 8. 
C)  All real numbers except - 8 and 9 . 
D)  All real numbers except 9 and 8. 
A)  All real numbers except - 9 and - 8 . 
B)  All real numbers except - 9 and 8. 
C)  All real numbers except - 8 and 9 . 
D)  All real numbers except 9 and 8.

Determine whether the two binomials are an opposites. - $3 x-6$ and $3 x+6$

Find all values of the variable for which the rational expression is undefined. - $\frac{7}{x+8}$

Simplify the complex fraction. - $\frac{\frac{x}{6}-\frac{3}{x}}{\frac{3}{x}+\frac{1}{2}}$

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$ . - $f(x)=\frac{24}{x-6}, g(x)=\frac{4 x}{x-6}$

$y$ varies directly as $x$ . If $y=\frac{17}{2}$ when $x=17$ , find $y$ when $x=16$ .

$\frac{4}{\mathrm{x}}+\frac{7}{3 \mathrm{x}}$

Simplify the complex fraction. - $\frac{\frac{1}{x}-2}{\frac{1}{x}-6}$

Find the LCD of the given rational expressions. - $\frac{7}{x^{2}-2 x-24}, \frac{7}{6 x+24}$

$\frac{18}{x-2}=1+\frac{20}{x+2}$

Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 245 miles in the same time that Dana travels 215 miles. If Chuck's rate of travel is $6 \mathrm{mph}$ more than Dana's, and they travel the same length of time, at what speed does Chuck travel?

$\frac{x^{2}-9}{4 x^{2}-25} \div \frac{x-3}{5-2 x}$

Find the domain of the rational function - $r(x)=\frac{x^{2}+4 x-32}{x^{2}-9}$

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

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ . - $f(x)=x+8, g(x)=\frac{x^{2}+5 x-24}{4 x^{2}+24 x-64}$

A boat can go 180 miles downstream in the same time it can go 120 miles upstream. The speed of the current is 4 miles per hour. Find the speed of the boat in still water.

$\frac{25}{x-5}+\frac{9}{x-5}$

Determine whether the two binomials are an opposites. - $x-5$ and $5-x$

Solve for the specified variable - $S=\frac{H}{m\left(t_{1}-t_{2}\right)}$ for $t_{2}$

Solve for the specified variable - $I=\frac{E}{R+r}$ for $r$

Find the domain of the rational function - $r(x)=\frac{x^{2}+x-36}{x^{2}-17 x+72}$

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

Determine whether the two binomials are an opposites. - 3x−63 x-63x−6 and 3x+63 x+63x+6

Find all values of the variable for which the rational expression is undefined. - 7x+8\frac{7}{x+8}x+87​

Simplify the complex fraction. - x6−3x3x+12\frac{\frac{x}{6}-\frac{3}{x}}{\frac{3}{x}+\frac{1}{2}}x3​+21​6x​−x3​​

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)=24x−6,g(x)=4xx−6f(x)=\frac{24}{x-6}, g(x)=\frac{4 x}{x-6}f(x)=x−624​,g(x)=x−64x​

yyy varies directly as xxx . If y=172y=\frac{17}{2}y=217​ when x=17x=17x=17 , find yyy when x=16x=16x=16 .

4x+73x\frac{4}{\mathrm{x}}+\frac{7}{3 \mathrm{x}}x4​+3x7​

Simplify the complex fraction. - 1x−21x−6\frac{\frac{1}{x}-2}{\frac{1}{x}-6}x1​−6x1​−2​

Find the LCD of the given rational expressions. - 7x2−2x−24,76x+24\frac{7}{x^{2}-2 x-24}, \frac{7}{6 x+24}x2−2x−247​,6x+247​

18x−2=1+20x+2\frac{18}{x-2}=1+\frac{20}{x+2}x−218​=1+x+220​

Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 245 miles in the same time that Dana travels 215 miles. If Chuck's rate of travel is 6mph6 \mathrm{mph}6mph more than Dana's, and they travel the same length of time, at what speed does Chuck travel?

x2−94x2−25÷x−35−2x\frac{x^{2}-9}{4 x^{2}-25} \div \frac{x-3}{5-2 x}4x2−25x2−9​÷5−2xx−3​

Find the domain of the rational function - r(x)=x2+4x−32x2−9r(x)=\frac{x^{2}+4 x-32}{x^{2}-9}r(x)=x2−9x2+4x−32​

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

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)=x+8,g(x)=x2+5x−244x2+24x−64f(x)=x+8, g(x)=\frac{x^{2}+5 x-24}{4 x^{2}+24 x-64}f(x)=x+8,g(x)=4x2+24x−64x2+5x−24​

A boat can go 180 miles downstream in the same time it can go 120 miles upstream. The speed of the current is 4 miles per hour. Find the speed of the boat in still water.

25x−5+9x−5\frac{25}{x-5}+\frac{9}{x-5}x−525​+x−59​

Determine whether the two binomials are an opposites. - x−5x-5x−5 and 5−x5-x5−x

Solve for the specified variable - S=Hm(t1−t2)S=\frac{H}{m\left(t_{1}-t_{2}\right)}S=m(t1​−t2​)H​ for t2t_{2}t2​

Solve for the specified variable - I=ER+rI=\frac{E}{R+r}I=R+rE​ for rrr

Find the domain of the rational function - r(x)=x2+x−36x2−17x+72r(x)=\frac{x^{2}+x-36}{x^{2}-17 x+72}r(x)=x2−17x+72x2+x−36​

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

Determine whether the two binomials are an opposites. - $3 x-6$ and $3 x+6$

Find all values of the variable for which the rational expression is undefined. - $\frac{7}{x+8}$

Simplify the complex fraction. - $\frac{\frac{x}{6}-\frac{3}{x}}{\frac{3}{x}+\frac{1}{2}}$

For the given rational functions $f(x)$ and $g(x), f(x)$ - $g(x)$ . - $f(x)=\frac{24}{x-6}, g(x)=\frac{4 x}{x-6}$

$y$ varies directly as $x$ . If $y=\frac{17}{2}$ when $x=17$ , find $y$ when $x=16$ .

$\frac{4}{\mathrm{x}}+\frac{7}{3 \mathrm{x}}$

Simplify the complex fraction. - $\frac{\frac{1}{x}-2}{\frac{1}{x}-6}$

Find the LCD of the given rational expressions. - $\frac{7}{x^{2}-2 x-24}, \frac{7}{6 x+24}$

$\frac{18}{x-2}=1+\frac{20}{x+2}$

Chuck and Dana agree to meet in Chicago for the weekend. Chuck travels 245 miles in the same time that Dana travels 215 miles. If Chuck's rate of travel is $6 \mathrm{mph}$ more than Dana's, and they travel the same length of time, at what speed does Chuck travel?

$\frac{x^{2}-9}{4 x^{2}-25} \div \frac{x-3}{5-2 x}$

Find the domain of the rational function - $r(x)=\frac{x^{2}+4 x-32}{x^{2}-9}$

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

For the given functions $f(x)$ and $g(x)$ , find $f(x) \cdot g(x)$ . - $f(x)=x+8, g(x)=\frac{x^{2}+5 x-24}{4 x^{2}+24 x-64}$

$\frac{25}{x-5}+\frac{9}{x-5}$

Determine whether the two binomials are an opposites. - $x-5$ and $5-x$

Solve for the specified variable - $S=\frac{H}{m\left(t_{1}-t_{2}\right)}$ for $t_{2}$

Solve for the specified variable - $I=\frac{E}{R+r}$ for $r$

Find the domain of the rational function - $r(x)=\frac{x^{2}+x-36}{x^{2}-17 x+72}$