Maximum Value of Modulus Expression

Analyze critical points $x = -3, 2, 4$.

1. $x \ge 4$: $A = (x+3) + (x-2) - (2x-8) = 9$.
2. $2 \le x < 4$: $A = (x+3) + (x-2) + (2x-8) = 4x - 7 \in [1, 9)$.
3. $-3 \le x < 2$: $A = (x+3) - (x-2) + (2x-8) = 2x - 3 \in [-9, 1)$.
4. $x < -3$: $A = -(x+3) - (x-2) + (2x-8) = -9$. In all cases, $|A| \le 9$. Maximum value = 9.