What is the algebraic way to solve inequalities like this: $|x+4|<2x$?

Can I rewrite this as $-2x<x+4<2x$ or [$x+4<2x$ and $-(x+4)<2x$]?

I found out that the answer was $x>4$, but how can I do it algebraically instead of by inspection? I also did some algebra that lead me to $x>{-4\over 3}$ when I considered $-(x+4)<2x$, this can't be right.