An upper bound on the LS-category in presence of the fundamental group

We prove that $$\cat X\le \frac{\cd(\pi_1(X))+\dim X}{2}$$ for every CW complex $X$ where $\cd(\pi_1(X))$ denotes the cohomological dimension of the fundamental group of $X$. We obtain this as a corollary of the inequality $$\cat X\le\frac{\cat (u_X)+\dim X}{2}$$ where $u_X:X\to B\pi_1(X)$ is a classifying map for the universal covering of $X$.