A region is given by x + y ≤ 8, x + 3y ≤ 18, 3x + y ≤ 18, x ≥ 0 and y ≥ 0…

Step 1. The corners on the axes. $(0, 0)$ is feasible. On the $x$ axis the lines give $(8, 0)$, $(18, 0)$ and $(6, 0)$. Only $(6, 0)$ obeys $3x + y \le 18$ and $x + y \le 8$, so it is the corner. On the $y$ axis the lines give $(0, 8)$, $(0, 6)$ and $(0, 18)$. Only $(0, 6)$ obeys every rule.

Step 2. The slanted pairs. $x + y = 8$ and $x + 3y = 18$ meet at $(3, 5)$. Check: $3(3) + 5 = 14 \le 18$, so it is a corner. $x + y = 8$ and $3x + y = 18$ meet at $(5, 3)$. Check: $5 + 3(3) = 14 \le 18$, so it is a corner. The pair $x + 3y = 18$ and $3x + y = 18$ meet at $(4.5, 4.5)$, but then $x + y = 9$ is more than $8$, so that point fails.

Step 3. Count and add. The corners are $(0, 0)$, $(6, 0)$, $(5, 3)$, $(3, 5)$ and $(0, 6)$. That is $5$ corners, and $0 + 6 + 5 + 3 + 0 = 14$.