Guess and Check

$\begin{aligned} y & = \frac {x+3}{x-1} = \frac {x-1+4}{x-1} = 1 + \color{#3D99F6}{\frac 4{x-1}} \end{aligned}$

For $y$ to be an integer, $\color{#3D99F6}{\dfrac 4{x-1}}$ must be an integer, then $x-1$ must be a factor of 4, which are -4, -2, -1, 1, 2, and 4. Therefore the number of integer $x$ such that $y$ is also an integer is $\boxed{6}$ .