Meaning of tightness of inequality I'd like to better understand the idea of tightness of an inequality.  I found this helpful post but would like to know more.  For example, is tightness only changed by modifying coefficients in a linear equation?  
For continuity, here's the example used in the post mentioned:

$$ (x,y>0) \\ x^2+y^2\geq Kxy $$ Is "tight" for K=2. If K>2 then it
  fails. If K<2 then it can be improved to chose K=2.


An inequality is tight if there is some choice of the variables involved for which equality holds.  Otherwise it is not.  For instance, for positive $x, y$ the inequality
$$x^2 + y^2 \geq 2 x y$$
is true, and moreover if you take $x = y = 1$ then both sides equal 2, so this inequality is tight.
The inequality
$$x^2 + y^2 \geq x y$$
is also true for positive $x, y$, but it is not tight, since we always have (for instance)
$$x^2 + y^2 \geq 2 x y > x y$$
There can be many ways to make an inequality which isn't tight into one which is. For instance the (true) inequality
$$x + 1 \geq x$$
is not tight.  We could "modify" it to the tight inequality
$$x \geq x.$$
(There's no formal notion of modifying an inequality to another one; it's just meant in the loose, intuitive sense.)
