How to prove an expression is positive

Question

I wanted to know if there are some standard methods to proving an algebraic expression is positive.

For example, how could I prove that $\dfrac{9\pi^2}{16}-\dfrac{1}{\sqrt{2}}>0$ without using approximations of $\pi$ or $\sqrt{2}$. Thanks in advance :)

Answer 1

$$\frac{9\pi^2}{16} \gt \frac{\sqrt2}{2}$$ $$\pi^2 \gt \frac{8\sqrt2}{9}$$ We know that $\pi \gt 8/9$ and that $\pi \gt \sqrt2$, so $\pi^2 \gt \frac{8\sqrt2}{9}$