TITLE: Negating A Mathematical Statement
QUESTION [18 upvotes]: Regard this statement $ x \ge 0$. According to my teacher, by negating this statement, it will become $ x < 0$. Why is this so; why does the $\ge$ morph into $<$, and not into $\le$?

REPLY [9 votes]: logically, there is no "morphing" going on here; your teacher is simply restating the same thing meant by $\,(x \ngeq 0)\,$, but just put differently $\,(x \lt 0)\,$: 
NOT $\geq$ means: 
"is NOT (greater than OR equal to)", $\iff$ ("is NOT greater than" AND "is NOT equal to")*... 
...which leaves us with ("is less than") 
that is "NOT $\geq$" must mean "is less than"
Formally, this is an application of (*) DeMorgan's Law (recall: $\lnot(p \lor q) \equiv \lnot p \land \lnot q$), and of the Trichotomy Law: between any two real numbers $a, b$, one and only one of the following relations holds:

$a \lt b $
$a = b$
$a \gt b$

The negation of $\,x\geq0\,$ ($x\ngeq0$) rules out $x = 0$ and rules out $x \gt o$ leaving us with just $x < 0$