Absolute Value Equation Pairs

Rewrite as $|a - 2| = 2|x| - x^2 = 2|x| - |x|^2 = |x|(2 - |x|)$. Since $|a - 2| \ge 0$, we must have $|x|(2 - |x|) \ge 0$. Since $|x| \ge 0$, we need $2 - |x| \ge 0 \implies |x| \le 2$. Since $x$ is an integer, $|x|$ can be $0, 1, 2$.

Case 1: $|x| = 0 \implies x = 0$. $|a - 2| = 0 \implies a = 2$. Pair: $(0, 2)$ — 1 solution.

Case 2: $|x| = 1 \implies x = 1$ or $x = -1$. $|a - 2| = 1(2 - 1) = 1 \implies a - 2 = 1$ or $-1 \implies a = 3$ or $a = 1$. Pairs: $(1, 3), (1, 1), (-1, 3), (-1, 1)$ — 4 solutions.

Case 3: $|x| = 2 \implies x = 2$ or $x = -2$. $|a - 2| = 2(2 - 2) = 0 \implies a = 2$. Pairs: $(2, 2), (-2, 2)$ — 2 solutions.

Total pairs $= 1 + 4 + 2 = 7$.