Kiran is solving \( \frac{2x-3}{x-1} = \frac{2}{x(x-1)}\) for \(x\), and he uses these steps:

\(\begin{align} \frac{2x-3}{x-1} &= \frac{2}{x(x-1)}\\[1.5ex] (x-1)\left(\frac{2x-3}{x-1} \right) &= x(x-1) \left( \frac{2}{x(x-1)} \right)\\[1.5ex] 2x-3 &= 2 \\[1.5ex] 2x &= 5 \\[1.5ex] x &= 2.5 \\ \end{align} \)

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).

Identify all values of \(x\) that make the equation true.

1. \( \displaystyle x=\frac{25}{x}\)
2. \( \displaystyle x+2= \frac{6x-3}{x}\)
3. \( \displaystyle \frac{x}{x^2} = \frac{3}{x}\)
4. \( \displaystyle \frac{6x^2+18x}{2x^3} = \frac{5}{x}\)

Rewrite the rational function \( g(x) = \frac{x-9}{x}\) in the form \( g(x) = c + \frac{r}{x}\), where \(c\) and \(r\) are constants.

Elena has a boat that can 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 \(r\) miles per hour. Write an equation that will help her solve for \(r\).