Area of Region Defined by Inequalities

$|x| + |y| \le 2$ forms a square with vertices at $(2,0), (0,2), (-2,0), (0,-2)$, total area $= 8$. $|x| \ge 1$ restricts $x \le -1$ or $x \ge 1$. The region is composed of two symmetrical triangles on the left and right. In the first quadrant ($x \ge 1$): $x + y \le 2 \implies y \le 2 - x$. For $x \in [1, 2]$, $y$ goes from $0$ to $2-x$. This is a right triangle with base $1$ and height $1$, area $= 0.5$. By 4-fold symmetry across quadrants, total area $= 4 \times 0.5 = 2$.