The usual idea is to isolate a term on one side of the equation and find conditions under which one side dominates the other.  For example,

$$ a x^q = b x^p - c$$
If $x$ is large, the left side is going to be larger than the right.  Thus for $x > 0$, 
$b x^p  - c < b x^p \le a x^q$ if $x \ge (b/a)^{1/(q-p)}$
 so an upper bound on positive roots is $(b/a)^{1/(q-p)}$.  

Similarly, a lower bound on positive roots is $(c/b)^{1/p}$.

For particular cases, you may get tighter bounds using Sturm's theorem.