*NOT* [quantifier]. Why is *NOT* [for every] equivalent to [there is]? Given a proposition such as, "For every real number $x \ge 2$, $x^2 + x - 6 \ge 0$", I am told that the negation, "NOT [For every real number $x \ge 2$, $x^2 + x - 6 \ge 0$]", would be "There is a real number $x \ge 2$ such that $x^2 + x - 6 < 0$".
I am specifically confused with regards to why NOT [for every] is equivalent to [there is] rather than [for none]? It seems logical to me that the negation of everything (for all) should actually be equivalent to nothing?
I would greatly appreciate it if someone could please take the time to clarify this concept.

You do not have equivalence between 'not for every' and 'there is'. You also need to negate the open sentence which is in the scope of the quantifiers. And you in fact did that: you turned the inequality around.
The general rule is that $\neg\forall xP(x)$ is equivalent to $\exists x\neg P(x)$. And that is quite intuitive: that not everything is a $P$-thing is the same as saying that there is a non-$P$-thing.
A little more detail: Your statement has more logical structure then that, as it contains a conditional. The statement has the form $\neg\forall x(Q(x)\to R(x))$ with $Q(x)$ being $x\ge2$ and $R(x)$ being $x^2 + x - 6 \ge 0$. This is equivalent to $\exists x\neg(Q(x)\to R(x))$ which is again equivalent to $\exists x(Q(x)\wedge\neg R(x))$. And $\neg R(x)$ is equivalent to $x^2 + x - 6 < 0$.
