We’ve had at least one thread on people’s favorite equations, so it seems fair to do the same for inequalities.
Mine is the very simple 1 - x < e[sup]-x[/sup], which is used all over the place in probability theory. It’s very simple to prove, as well: 1 - x is the tangent line to e[sup]-x[/sup] at x = 0. Since e[sup]-x[/sup] is a convex function, it lies above its tangent line at any point. The result follows immediately.
Triangle inequality.
Racial! Oh, wait…
Chebyshev’s Inequality.
But that’s only because I like saying “Chebyshev” (I have a thing with fun words/names to say!)
Otherwise I don’t enjoy math enough to have favourite anythings!
Given x = y
Assume x = 1, y=2
Ergo 1 = 2
You gotta love it! 
^ Oh dear. 
As for me, I’ll be the the boring one who says Cauchy–Schwarz.
Oh, that’s not boring! I actually came in to post about the Cauchy-Riemann equations, confusing them with Cauchy-Schwartz. Whatevs.
In any case, I recall Jensen’s inequality being pretty neat.
ETA: Us set theorists don’t really have many good ones, other than 2^k>k for any cardinal k (Cantor).
Women’s Suffrage.
The imbalance of charge that causes electrons to flow, making this post, and the nerve impulses that created it, possible.
Fermat’s Last Theorem.
We need to end it. Women have suffered enough.
The classic, the Haves versus the Have Nots.
The value of an annuity certain for a term equal to a person’s life expectancy at any given age always exceeds the value of a life annuity for a person of that age.
You forgot to divide by 0!
That wouldn’t change anything. 0!=1. 
Being an algebraist, mine is that when E if a finite field extension of F, then the dimension of E over F is greater than or equal to the number of automorphims of E that fix the elements of F.
HEY! Who let the actuaries in here?
The vagina, no question.
iam > u
Do you mean 0 != 1, or 0! = 1? (It works either way.)