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
.