So I am really bad at inequalities. Last month I learned something so I will share it here yeah. I will probably upload all my course notes somewhere after I’m done with this course.

I think in dealing with inequalities, what I think about most is how do I make the form more pleasant. But it is only last month that I started paying attention to what values I am actually removing from consideration. Suppose you want to manipulate the expression $\frac{x^2}{1+x^3}$ for small positive $x$. Let’s see some examples.

Maximally formally beautiful:

\[\frac{x^2}{1+x^3} \leq 1.\]Formally good but not quantitatively good:

\[\frac{x^2}{1+x^3} \leq \frac{1}{x}.\]Formally good and quantitatively good:

\[\frac{x^2}{1+x^3} \leq x^2.\]Formally bad and quantitatively bestest:

\[\frac{x^2}{1+x^3} \leq \frac{x^2}{1+x^3}.\]Yeah thats all I have to say.