How to prove an Inequality I'm beginner with proofs and I got the follow exercise:

Prove the inequality $$(a + b)\Bigl(\frac{1}{a} + \frac{4}{b}\Bigr) \ge 9$$ when $a > 0$ and $b > 0$. Determine when the equality occurs.

I'm lost, could you guys give me a tip from where to start, or maybe show a good resource for beginners in proofs ? 
Thanks in advance.

Simple algebra yields
$$
1+\frac{4a}{b}+\frac{b}{a} +4=5+\frac{4a}{b}+\frac{b}{a}
$$
If $a \geq b $ then the inequality is obvious. The second case is a bit trickier. The inequality then reduces to proving that (writing $0<\frac{a}{b}=\epsilon<1$)
$$
4 \epsilon^2-4 \epsilon+1 \geq 0
$$
Which holds for all $\epsilon$ and the equality is at $2a=b$.
