Let $x$ and $y$ be real numbers such that $3x + 2y \le 7$ and $2x + 4y \le 8.$  Find the largest possible value of $x + y.$
Dividing the second inequality by 2, we get $x + 2y \le 4.$  Adding the first inequality $3x + 2y \le 7,$ we get
\[4x + 4y \le 11,\]so $x + y \le \frac{11}{4}.$

Equality occurs when $x = \frac{3}{2}$ and $y = \frac{5}{4},$ so the largest possible value of $x + y$ is $\boxed{\frac{11}{4}}.$