A region is given by x + y ≤ 7, x ≥ 1 and y ≥ 2…

Step 1. The corners. The lines $x = 1$ and $y = 2$ meet at $(1, 2)$. The lines $x = 1$ and $x + y = 7$ meet at $(1, 6)$. The lines $y = 2$ and $x + y = 7$ meet at $(5, 2)$. All three obey every rule.

Step 2. The values. At $(1, 2)$, $Z = 3 - 4 = -1$. At $(5, 2)$, $Z = 15 - 4 = 11$. At $(1, 6)$, $Z = 3 - 12 = -9$.

Step 3. Compare. The largest is $11$ and the smallest is $-9$.