If you take the reciprocal in an inequality, would it change the $>/< $ signs? Example:$$-16<\frac{1}{x}-\frac{1}{4}<16$$ 
In the example above, if you take the reciprocal of $$\frac{1}{x}-\frac{1}{4} = \frac{x}{1}-\frac{4}{1}$$
would that flip the $<$  to $>$ or not?
In another words, if you take the reciprocal of $$-16<\frac{1}{x}-\frac{1}{4}<16$$   would it be like this: $$\frac{1}{-16}>\frac{x}{1}-\frac{4}{1}>\frac{1}{16}$$

It depends if $x$ and $y$ are the same sign.
Case 1: $0 < x < y$ then $0 < x(1/y) < y(1/y)$ and $0 < x/y < 1$ and $0 < x/y(1/x) < 1 (1/x)$ so $0 < 1/y < 1/x$.
If both positive, flip.
Case 2: $x < 0 < y$ then $x/y < 0 < 1$.  Then as $x < 0$ we flip when we do $x/y*(1/x) > 0 > 1*(1/x)$ so $ 1/y > 0 > 1/x$ so $1/x < 0 < 1/y$.  Don't flip.
Case 3:  $x < y < 0$ then $x/y > 1 > 0$ and $1/y < 1/x < 0$.  Flip if they are the same sign.
But FOR THE LOVE OF GOD!!!!!!! the reciprical of $1/x - 1/4$ is !!!!!!!NOT!!!!!! $x/1 - 4/1$!!!!!!!!
It is $\frac{1}{1/x - 1/4} = \frac{4x}{4 - x}$. 
