[size=150]Identify all values of [math]x[/math] that make the equation true.[/size][br][br][math]\frac{2x+1}{x}=\frac{1}{x-2}[/math]
[math]\frac{1}{x+2}=\frac{2}{x-1}[/math]
[math]\frac{x+3}{1-x}=\frac{x+1}{x+2}[/math]
[math]\frac{x+2}{x+8}=\frac{1}{x+2}[/math]
[size=150]Kiran is solving [math]\frac{2x-3}{x-1}=\frac{2}{x(x-1)}[/math] for [math]x[/math], and he uses these steps:[/size][br][math]\begin{align} \frac{2x-3}{x-1} &= \frac{2}{x(x-1)}\\ (x-1)\left(\frac{2x-3}{x-1} \right) &= x(x-1) \left( \frac{2}{x(x-1)} \right)\\ 2x-3 &= 2\\ 2x &= 5 \\ x &= 2.5 \\ \end{align}[/math][br][br]He checks his answer and finds that it isn't a solution to the original equation, so he writes “no solutions.” Unfortunately, Kiran made a mistake while solving. Find his error and calculate the actual solution(s).
[size=150]Identify all values of [math]x[/math] that make the equation true.[/size][br][br][math]x=\frac{25}{x}[/math]
[math]x+2=\frac{6x-3}{x}[/math]
[math]\frac{x}{x^2}=\frac{3}{x}[/math]
[math]\frac{6x^2+18x}{2x^3}=\frac{5}{x}[/math]
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[/img][br]Is this the graph of [math]g(x)=\text{-}x^4(x+3)[/math] or [math]h(x)=x^4(x+3)[/math]? Explain how you know.
[size=150]Rewrite the rational function [math]g(x)=\frac{x-9}{x}[/math] in the form [math]g(x)=c+\frac{r}{x}[/math], where [math]c[/math] and [math]r[/math] are constants.[/size]
Elena has a boat that would go 9 miles per hour in still water. She travels downstream for a certain distance and then back upstream to where she started. Elena notices that it takes her 4 hours to travel upstream and 2 hours to travel downstream. The river’s speed is [math]r[/math] miles per hour. Write an expression that will help her solve for [math]r[/math].