Negating A Mathematical Statement 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 $\le$?

There is no "morphing", and this is not just a game played arbitrarily with squiggles on the paper. The symbols _mean_ things, and you can reason out their behaviors if you understand the meanings. $x\ge 0$ means that $x$ is equal to or greater than zero. Negating the statement means constructing a statement whose _meaning_ is "$x$ is **not** equal to or greater than zero".

Which of $x<0$ and $x\le 0$ _means_ "$x$ is not equal to or greater than zero"? It can't be $x\le 0$, because that means that $x$ is less than or equal to zero, and we are trying to say that it is _not_ equal to zero.

$x<0$ is correct, because if $x$ is not greater than or equal to zero, then it must be less than zero, and that is exactly what $x<0$ means.