$\text{Find } f(x+1) \text{ by substituting } (x+1) \text{ in for every } x \text{ in the } f(x)\text{ function. Set that equal to } \frac{1}{2}.$

$f(x + 1) = \frac{x+1+1}{x+1-2} = \frac{1}{2}$

$\ \ \ \ \ \ \ \ \ \ \ \ \Rightarrow \frac{\textit{x} + 2}{\textit{x} - 1} = \frac{1}{2}$

Now solve for $x$.