If $xf \in L^2(\Bbb R)$, is $f \in L^1(\Bbb R)$?

Question

I tried using Holder's inequality with $|f| = |f| \frac{(1+|x|)^{\frac{1}{2}}}{(1+|x|)^{\frac{1}{2}}}$, or variant but I can't seem to make it work.

Any help is appreciated.