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?

Because $$(A-B)^{-1}\not=A^{-1}-B^{-1}.$$
Note that we have$$(\color{red}{A-B})^{-1}=\frac{1}{\color{red}{A-B}}.$$
For your case, we have
$$\left(\dfrac{4}{3}-\dfrac{3}{2}\right)^{-1}=\dfrac{1}{\dfrac{4}{3}-\dfrac{3}{2}}$$
