Let $\lambda \in \mathbb{R}$ and let the equation $E$ be $|x|^2 - 2|x| + |\lambda - 3| = 0$. Then the largest element in the set $S = \{x + \lambda : x \text{ is an integer solution of } E\}$ is ______.
Step-by-Step Solution
Key Concept: Rewrite the equation as $(|x|-1)^2 = 1 - |\lambda - 3|$. For real solutions the RHS must be $\geq 0$, so $|\lambda - 3| \leq 1$. Find integer values of $x$ for each valid $\lambda$ and maximize $x + \lambda$.
$(|x|-1)^2 + |\lambda-3| = 1$. At $\lambda = 3$: $x = 0$ or $x = 2$ (and $-2, 0$), giving $x + \lambda \leq 5$. At $\lambda = 4$ or $\lambda = 2$: $|x|=1$, so $x = \pm 1$, giving $x + \lambda = 5$ or $3$. Maximum value is $5$.
Correct Answer: 5