View source for 2023 AMC 10B Problems/Problem 13

==Problem 13==

What is the area of the region in the coordinate plane defined by

<math>| | x | - 1 | + | | y | - 1 | \le 1</math>?

<math>\text{(A)}\ 2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 12</math>

== Solution ==
First consider, <math>|x-1|+|y-1| \le 1.</math>
We can see that it's a square with radius 1 (diagonal 2). The area of the square is <math>\sqrt{2}^2 = 2.</math>

Next, we add one more absolute value and get <math>|x-1|+||y|-1| \le 1.</math> This will double the square reflecting over x-axis.

So now we got 2 squares.

Finally, we add one more absolute value and get <math>||x|-1|+||y|-1| \le 1.</math> This will double the squares reflecting over y-axis.

In the end, we got 4 squares.  The total area is <math>4\cdot2 = </math> <math>\boxed{\text{(B)} 8}</math>

~Minor formatting change: e_is_2.71828