Is the number $\left|2^{3000}-3^{2006}\right|$ bigger or smaller than $\frac{1}{2}$ ?
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 $\left|2^{3000}-3^{2006}\right|>\frac{1}{2}$.