How many ordered triples of integers $(x, y, z)$ satisfy the following system of inequalities?
\begin{align*}
-x-y-z\le -2\\
-x+y+z\le 2\\
x-y+z\le 2\\
x+y-z\le 2
\end{align*}
$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 11 \qquad
\textbf{(D)}\ 15 \qquad \textbf{(E)}\ 17 \qquad $
Difficulty: 61/100 - A tricky and nontraditional system of inequalities.
Core Concepts: inequalities, casework
Challenges: It is important to be careful when adding/subtracting inequalities of different directions.
Techniques: Being able to rearrange and add/subtract inequalities to receive bounds, and being able to translate into cases. Exploiting the symmetry in the problem.
Error-prone Steps: Wrong sign when combining inequalities, illegally subtracting inequalities without accounting for direction.
Ideal Time:
Experienced: ≤ 5 min
Intermediate: ≤ 10 min
Beginner: ≤ 15-20 min