What is the meaning of the symbol $\not\geq$, and why would it be preferred to $<$?
$a\ge b$ means $a>b$ or $a=b$. And $a\not\ge b$ is the negation thereof, i.e., neither $a>b$ nor $a=b$. If we were talking about a total order, this would be equivalent to $a<b$. But maybe we are not.
Mathematical Symbols are there to express mathematical ideas in a short, precise and understandable way. If you want to express that "a is not greater than or equal to b", it could be argued that $a \ngeq b$ is the most direct expression for that.
The other symbol, $a \gneq b$ means "a is greater but NOT(!) equal to b". The crossed out line of the equal sign can be understood as an emphasis on the "not equal" part. From a purely logical standpoint, $a \gneq b$ and $a > b$ mean exactly the same.
Without any context it is difficult to answer.
Anyway generally speaking let's say that an order relation has been defined in a set whereby two elements of which may be in such a relation (and this order relation has been given this symbol $\ge$),
$a\ngeq b$ means either of the two:
- $b \ge a$ and $a\ne b$
- $a$ and $b$ are not comparable at all.
Instead $a\gneq b$ means both of the these two are satisfied:
- $a \ge b$
- $a \neq b$
Usually in a general context (that is, not with reals) when for an order relation a symbol like this $\ge$ is introduced that is like that used for "greater then or equal to" total order relation among reals, all the other symbols that are usually used with reals like these $<$, $>$, $\le$ are avoided, unless a definition for each is provided.