Symbol/notation/strategy for figuring out an unknown inequality? Let's say there's some relationship I'm trying to figure out between two values (e.g. is A > B or is B > A), and I want to use math to prove the relationship. Is there a common convention for how to do something like this? Which inequality symbol would I use between the two values if I don't know which value is bigger than which?
Here's a very simplistic example:
Say I'm not sure which of these is greater:
$\text{A}=\frac{3}{x^2+1}$
$\text{B}=\frac{3}{x^2}$
And I want to start using math to figure this out. Initially I set up an equation such as:
$\frac{3}{x^2+1} ? \frac{3}{x^2}$
And then I solve it:
$3x^2 ? 3\left(x^2+1\right)$
$3x^2?3x^2+3$
$0?3$
At this point, it's obvious that the $?$ symbol should be $<$. So I can now go backwards and replace each $?$ with a $<$, ultimately getting the following:
$\frac{3}{x^2+1} < \frac{3}{x^2}$
I just arbitrarily chose the ? symbol for this example. Is there a standard symbol for doing something like this?
Additionally, if I were to multiply either side by a negative value, I would need to remember to flip the symbols at that step. Is there any easy way to "mark" such a step so that I don't forget to flip the sign?
This is especially problematic if you are multiplying by a term which may be negative. In this specific example, $x^2$ and $x^2+1$ are always positive, so it's not a problem. However, if one of these terms was $x^2-1$ it would be a lot trickier... You would need need some kind of note such as: If$x^2-1<0$flip the sign here.
 A: The general way to do such a thing will be to start with $A-B$ and proceed to show it is positive or negative.
For instance, in the case you have given us, 
$$\begin{align}A-B&= \dfrac{c}{x^2-1}-\dfrac{c}{x^2}\\&=c\cdot \left(\dfrac{x^2-x^2+1}{x^2(x^2-1)}\right)\\&=\dfrac{c}{x^2}\cdot \dfrac{1}{x^2-1}\end{align}$$
Now the resulting expression $A-B$ is positive when $c(x^2-1)\ge 0$ and negative otherwise.
(Note that $A$ in not defined when $x= \pm 1$, and this is exempted from the domain of the problem.)

During the second reading, I find that you have made a mistake in what you have shown us. It happens in the step where you cross multiply $x^2$ to one side and $x^2-1$ to the other. 

While it is true that $x^2 \ge 0$, it is not always true that $x^2-1 \ge 0$. This in fact fails, when $|x| \le 1$. 

Hope this is convincing that this method is more comfortable than inventing Exotic Symbols.
A: This is entirely concerned with your  specific example, but is a good thing to remember when working with inequalities:
If $A>B>0$, then dividing a positive quantity $C$ by $A$ will produce something smaller than dividing $C$ by $B$.
So ${3\over x^2+1}<{3\over x^2}$.
Informally "dividing by something smaller gets you something bigger".
