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?

It looks like based on your post that you want to know why $\frac{1}{\frac{4}{3}} - \frac{1}{\frac{3}{2}}$ is not equal to $\frac{1}{\frac{4}{3} - \frac{3}{2}}$.
In general, you can only say a property holds like, in your case, $(a + b)^{-1} = a^{-1} + b^{-1}$ if you can show it holds for every $a$ and $b$.  But as you pointed out in your question, if $a = \frac{4}{3}$ and $b = \frac{3}{2}$, this property doesn't hold.  Therefore, it's not true that $(a + b)^{-1} = a^{-1} + b^{-1}$ holds.  The property is false, as your example perfectly showed.
