Distinct integer solutions of modulus equation

We are given the equation $|x + y| + |x - y| = 2$.

Using the identity $|a+b| + |a-b| = 2 \max(|a|, |b|)$, we can rewrite the equation as: $2 \max(|x|, |y|) = 2 \implies \max(|x|, |y|) = 1$

Since $x$ and $y$ must be integers, $x, y \in \{-1, 0, 1\}$ such that at least one of $|x|$ or $|y|$ equals $1$.

The total number of pairs $(x, y)$ formed by elements from $\{-1, 0, 1\}$ is $3 \times 3 = 9$.

Only the pair $(0, 0)$ gives $\max(|0|, |0|) = 0 \neq 1$.

Therefore, there are $9 - 1 = 8$ integer solutions.