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

1. \( \displaystyle \frac{2x+1}{x}=\frac{1}{x-2}\)

2. \( \displaystyle \frac{1}{x+2}=\frac{2}{x-1}\)

3. \( \displaystyle \frac{x+3}{1-x} = \frac{x+1}{x+2}\)

4. \( \displaystyle \frac{x+2}{x+8}= \frac{1}{x+2}\)

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).