Why is $\frac{1}{4/3} - \frac{1}{3/2}$ not the same as $\bigl(\frac{4}{3} - \frac{3}{2}\bigr)^{-1}$ If you have the problem:$$\frac{1}{\frac{4}{3}} - \frac{1}{\frac{3}{2}} =?$$
Why can't you change the problem to $(\frac{4}{3} - \frac{3}{2})^{-1}$   and get the same answer?
In the first scenario, the answer is $1/12$
In the second scenario, the answer is $-6$
Why doesn't this work?

The original problem is $$\frac{1}{x}-\frac{1}{y}=\frac{y-x}{xy}$$
for specific values of $x,y$.  The modified problem is $$\frac{1}{x-y}$$
In order for them to be equal, $$\frac{y-x}{xy}=\frac{1}{x-y}$$
cross-multiplying, we get $$-(x-y)^2=xy$$
Expanding, we get $$-x^2+2xy-y^2=xy$$
or $$x^2-xy+y^2=0$$
The only real solutions this has are $x=0, y=0$, which are excluded since we are dividing by $x,y$.  Hence this proposed simplification is never correct, even accidentally.
