Solution:

Notice that $\left|2^{3000}-3^{2006}\right|$ is a non-negative integer, so it is either $0$ or bigger than $1/2$. However, this number is not $0$ since $2^{3000} \neq 3^{2006}$, hence $\left|2^{3000}-3^{2006}\right| > \frac{1}{2}$.