Inequalities Concerning Primes

Question

I recently read a very good inequality concerning the no of primes $\pi(x)$:

$$\pi(n)>\frac{1}{6}\frac{n}{\log n}\mathrm{\ for\ }n\ge 2$$

Are any other such elementary inequalities concerning the primes?

Answer 1

What is your question? This is well-known stuff. It appears for instance in section 4.5 of Apostol's Introduction to Analytic Number Theory, which also contains an upper bound. You could at least say where you got yours from. (This should have been a comment, not an answer, but I don't have enough reputation for adding comments.)