how do i get the answer to this question?---**Question Stem:**
What is the area of the region in the coordinate plane defined by
$||x| - 1| + ||y| - 1| \leq 1?$
**Options:**
(A) 2
(B) 8
(C) 4
(D) 15
(E) 12
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Let's analyze the inequality ||x| - 1| + ||y| - 1| ≤ 1. This expression contains nested absolute values, which means we need to consider different cases based on the signs of the expressions inside. The key insight is to understand how |x| - 1 and |y| - 1 behave in different regions of the coordinate plane.
To solve this problem, we need to analyze when |x| - 1 and |y| - 1 are positive or negative. For |x| - 1, it's positive when x is greater than or equal to 1, or when x is less than or equal to negative 1. It's negative when x is between negative 1 and 1. The same pattern applies to |y| - 1. These critical lines at x equals plus or minus 1, and y equals plus or minus 1, divide the plane into nine regions.
Now let's analyze the inequality in each region. In region 1, where both x and y are greater than or equal to 1, both |x| - 1 and |y| - 1 are non-negative, so the outer absolute values don't change the signs. This gives us x - 1 plus y - 1 less than or equal to 1, which simplifies to x plus y less than or equal to 3. In region 2, where x is between -1 and 1 but y is greater than or equal to 1, we get 1 minus |x| plus y minus 1 less than or equal to 1, which simplifies to |x| plus y greater than or equal to 1.
By analyzing all nine regions and using symmetry, we find that the solution region forms a diamond shape centered at the origin. The vertices of this diamond are at (0, 2), (2, 0), (0, -2), and (-2, 0). This diamond is actually a square rotated 45 degrees. To find the area, we can use the formula for the area of a rhombus: one half times the product of the diagonals. Both diagonals have length 4, so the area is one half times 4 times 4, which equals 8.
Let's summarize our solution. We analyzed the inequality ||x| - 1| + ||y| - 1| ≤ 1 by breaking it down into cases based on the signs of |x| - 1 and |y| - 1. This created nine regions in the coordinate plane. By solving the inequality in each region and using symmetry, we found that the solution forms a diamond shape with vertices at (2, 0), (0, 2), (-2, 0), and (0, -2). Using the formula for the area of a rhombus, which is one half times the product of the diagonals, we calculated the area as 8. Therefore, the answer is B.