Given a Utility function, U(x,y), why is multiplying U(x,y) * x not a monotonic transformation? If $x$, is a commodity and it is s.t. $x \geq 0$ (it is always nonnegative) how will multiplying it times a utility function $U(x,y)$ NOT yield a simple monotonic transformation?
Because $x$ is not a constant, and therefore multiplying $U(x,y)$ by $x$ does not necessarily preserve the ordering between bundles.

For instance, consider the function $U(x,y)=x+y$, and the bundles $(x_0,y_0)=(2,0)$ and $(x_1,y_1)=(1,2)$. We have
\begin{equation*}
U(x_0,y_0)=2 &lt; U(x_1,y_1)=3
\end{equation*}
but
\begin{equation*}
x_0 U(x_0,y_0) = 4 &gt; x_1 U(x_1,y_1) = 3
\end{equation*}
Hence multiplying $U$ by $x$ changes the preferences, therefore is not a monotonic transformation.

As a side note, multiplying the utility function by a constant provides a monotonic transformation if the constant is positive, and not only nonnegative as you assumed (if the constant is zero the orderings are obviously not preserved).