$\exists c>0$, $\forall A\subseteq \Bbb R_{\ne 0}$ s.t. $|A|=n$, $\exists B\subseteq A$ s.t. $|B|>cn$ & $b_1+2b_2=2b_3+2b_3$ has no solutions in $B$.

Question

EXERCISE 2.7.2 fron Alon and Spencer's The Probabilistic Method.
Prove that there is a positive constant $c$ so that every set $A$ of $n$ nonzero reals contains a subset $B\subseteq A$ of size $|B| > cn$ so that there are no $b_{1},b_{2},b_{3},b_{4}\in B$ satisfying $$b_{1}+2b_{2}=2b_{3}+2b_{4}\,.$$

My idea is to regard the elements of $A$ in modulo $m$ for some $m>0$, then I must find numbers which they are not true in above relation in modulo $m$. This is just the Idea but my problem is to make rigid and exact.