Answer:
8 square units
Step-by-step explanation:
The graph of |x| + |y| = 1 is a special case of the superellipse with parameter r = 1. It forms a diamond shape centered at the origin (0, 0) with vertices at (1, 0), (0, 1), (-1, 0), and (0, -1).
The inequality |x| + |y| ≤ 1 includes all points within and on the boundary of this diamond shape. Since this region forms a square rotated 45 degrees, with sides lengths of √2 units, its area is 2 square units.
The inequality |x - 1| + |y - 1| ≤ 1 represents |x| + |y| ≤ 1 translated 1 unit to the right and 1 unit up. Therefore, its center is at (1, 1) and its vertices are at (2, 1), (1, 2), (0, 1), and (1, 0). The area of the region defined by this inequality is also 2 square units, since it is a translation and does not change the size of the region.
The variables |x| and |y| in the inequality ||x| - 1| + ||y| - 1| ≤ 1 imply that the same diamond shape will be reflected into all four quadrants of the xy-plane, creating symmetrical regions centered at (1, 1), (-1, 1), (-1, -1) and (1, -1) in quadrants I, II, III and IV, respectively.
The area of the region defined by |x - 1| + |y - 1| ≤ 1 is 2 square units. Therefore, since the region described by ||x| - 1| + ||y| - 1| ≤ 1 is mirrored in all four quadrants, the total area is four times the area of one diamond, which is 8 square units.
