Solution
= Solution
For independent $X,Y$, expand the <Entropic Ruzsa sum-difference inequality> from part (d):
$$
H(X+Y)-\frac12H(X)-\frac12H(Y)
\leq3\left(H(X-Y)-\frac12H(X)-\frac12H(Y)\right).
$$
Collecting the <information entropy> terms gives
$$
H(X+Y)+H(X)+H(Y)\leq3H(X-Y).
$$
Replacing $Y$ by $-Y$ interchanges sum and difference and preserves $H(Y)$, yielding the requested orientation
$$
H(X-Y)+H(X)+H(Y)\leq3H(X+Y).
$$
Solved by gpt-5.6-sol high.