Let λ ∈ ℝ and let the equation E be | x | ^ 2 - 2 | x | + | λ - 3 |...

Given:

$| x{|}^{2} - 2 | x | + | \lambda - 3 | = 0$

$⇒ | \lambda - 3 | = \left|x\right|\left(2 - \left|x\right|\right)$

At $\lambda = 3$, $x = 0$ and $2$,

At $\lambda = 4$ or $2$, then,

$x = 1$ or $- 1$

So, maximum value of $x + \lambda = 5$ when $x = 1$ and $\lambda = 4$.