Rational Expressions and Equations

\(\text{Which expression is equivalent to }\frac{8x(x-7)-3(x-7)}{2x-14},\text{ where }x > 7?\)

\(\text{Which expression is equivalent to }\frac{7x(x-9)-2(x-9)}{3x-27},\text{ where }x > 9?\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~f(x)=x^{3}-9x\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~g(x)=x^{2}-2x-3\\
&\text{Which of the following expressions is equivalent to }\frac{f(x)}{g(x)},\text{ for } x > 3 ?\end{aligned}\)

\(\text{Which expression is equivalent to }\frac{42a}{k}+42ak,\text{ where } k > 0?\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{1}{2x+1}+5\\&\text{Which of the following is equivalent to the expression above for }x > 0?\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{2x+6}{(x+2)^{2}}-\frac{2}{x+2}\\\\
&\text{The expression above is equivalent to }\frac{a}{(x+2)^{2}},\text{ where }a\text{ is a positive constant and }x\neq-2.\\
&\text{What is the value of }a?\end{aligned}\)

\(\text{Which expression is equivalent to }\frac{20\pi^2}{\pi^2-x^2}-\frac{20\pi}{x+\pi}?\)

\(\text{Which expression is equivalent to }\frac{y+12}{x-8}+\frac{y(x-8)}{x^{2}y-8xy}?\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{x^2-1}{x-1}=-2\\&\text{What are all values of }x\text{ that satisfy the equation above?}
&\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{x}{x-3}=\frac{2x}{2}\\
&\text{Which of the following represents all the possible values of }x\text{ that satisfy }\\&\text{the equation above?}\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~\frac{1}{x}+\frac{1}{x-3}=\frac{6}{x^{2}-3x}\\&\text{What are all the solutions to the given equation?}
\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{x}{(x-1)(x+1)}=\frac{81}{56x}\\&\text{Which of the following is a solution to the given equation?}
\end{aligned}\)

\(\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{(x-7)(x-9)}{x-4}=0\\&\text{What is the sum of the solutions to the given equation?}\end{aligned}\)

\(\begin{aligned}
&\text{The function }w\text{ is defined by }w(r)=\frac{1}{r-8}-\frac{r-5}{-r+4.25}.\text{ What is the greatest}\\
&\text{possible value of }r\text{ such that }w(r)=0?\end{aligned}\)

\(\begin{aligned}
&\text{The function }f\text{ is defined by }f(x)=\frac{x^2+ax+b}{2x+c}\text{, where }a,~b,\text{ and }c\text{ are}\\
&\text{constants.The graph of the function }f\text{ in the }xy\text{-plane, where }y=f(x),\text{ does}\\
&\text{not intersect the line }x=4.\text{ If }f(5)=f(6)=0,\text{ what is the value of }a+b+c?\end{aligned}\)

\(\begin{aligned}&\text{In the }xy\text{-plane, the graph of function }f,\text{ where }y=f(x),\text{ has exactly }8~x\text{-intercepts. One of}\\
&\text{these }x\text{-intercepts is }(17,0).\text{ The rational function }g\text{ is defined by }g(x)=\frac{f(x)}{x-17}.\text{ In the }xy\text{-}\\
&\text{plane, how many }x\text{-intercepts does the graph of }y=g(x)\text{ have?}\end{aligned}\)

\(\begin{aligned}
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac{x^{2}}{\sqrt{x^{2}-c^{2}}}=\frac{c^{2}}{\sqrt{x^{2}-c^{2}}}+39\\
&\text{In the given equation, }c\text{ is a positive constant. Which of the following is one of the solutions }\\&\text{to the given equation?}\end{aligned}\)