Sum of Big O - which one is it? On the wikipedia page, we have the following property:
If $f_1(x) = O(g_1(x))$ and $f_2(x) = O(g_2(x))$ then $f_1(x) + f_2(x) = O(|g_1(x)| + |g_2(x)|)$. 
But in my textbook, I also see the following sum property:
If $f_1(x) = O(g_1(x))$ and $f_2(x) = O(g_2(x))$ then $f_1(x) + f_2(x) = O(\max(g_1(x), g_2(x)))$. 
Are these two properties equivalent? If not, which sum property do I use in general?

They are equivalent (assuming that $g_1,g_2$ are assumed non-negative in your textbook). Note that
$$
\max(a,b)\leq a+b \leq 2\max(a,b)
$$
for any $a,b\geq 0$, and that the constant $2$ can be "hidden" in the $O(\cdot)$ notation.
