This question comes from "Probability Theory" by Achim Klenke, pg. 198. Any hints are appreciated.
So, you know the value of $X$, say $X=x$ for a given real number $x$ such that $|x|\leqslant1$, and you want to define $Y$ such that $Y=1$ with probability $\frac12(1+x)$ and $Y=-1$ with probability $\frac12(1-x)$.
Assume that you draw an independent random variable $U$ uniform on $(-1,1)$ and that you define $Y$ as $Y=+1$ if $U\in B_x$ and $Y=-1$ if $U\notin B_x$ for a given Borel set $B_x\subseteq(-1,1)$. Then $$ \mathrm E(Y\mid X=x)=(+1)\cdot\tfrac12|B_x|+(-1)\cdot\tfrac12|(-1,1)\setminus B_x|=|B_x|-1, $$ hence this construction yields a solution if $|B_x|=1+x$ for every $x$. Can you finish?