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.

We can play with the inequality: let's suppose it was true for now, and perform a series of reversible steps and see what the inequality would imply. So we begin by multiplying everything out, which gives:
$1 + 4 + \dfrac{b}{a} + \dfrac{4a}{b} \geq 9 \Leftrightarrow \dfrac{b}{a} + \dfrac{4a}{b} \geq 4$. (1)
Now, when working with inequalities there are some common results that you use; one of these is the AM-GM inequality which states:
$x + y \geq 2\sqrt{xy}$,
where $x$ and $y$ are nonnegative. (Square both sides and collect the terms to one side to see why this is true).
Letting $x = b/a$ and $y = 4a/b$ yields that
$\dfrac{b}{a} + \dfrac{4a}{b} \geq 2\sqrt{\dfrac{b}{a}\cdot \dfrac{4a}{b}} = 2\sqrt{4} = 4$,
which is what we wanted to show in line (1). 
Since we can reverse all of our steps (we expanded and subtracted 5; to reverse this add 5 and then factor), we have proven the original inequality. (Note: a formal solution would start with this last line and work "upwards.")
