Lil Bit Complex

We note that:

$\begin{cases} |x+y| + |x-y| \le 4 & \Rightarrow -2 \le y \le 2 \\ |x| \le 1 & \Rightarrow -1 \le x \le 1 \\ y \ge \sqrt{x^2 -2x+1} & \Rightarrow \quad y \ge 1-x \end{cases}$

Therefore, the region is a right-angle triangle bounded by $y = 2$ , $x=1$ and $y=1-x$ as shown in the figure below. The area is then $\dfrac{1}{2}\times 2 \times 2 = \boxed{2}$ .