Minimise Z = 2x + 3y subject to x + 2y ≥ 8, 3x + y ≥ 9, x ≥ 0 and y ≥ 0…

Step 1. The values. At $(0, 9)$, $Z = 27$. At $(2, 3)$, $Z = 4 + 9 = 13$. At $(8, 0)$, $Z = 16$. The smallest is $M = 13$.

Step 2. The test. The half-plane to check is $2x + 3y < 13$. The two edges meeting at $(2, 3)$ are $x + 2y = 8$, with slope $-\dfrac{1}{2}$, and $3x + y = 9$, with slope $-3$. The line $2x + 3y = 13$ has slope $-\dfrac{2}{3}$, which lies between $-3$ and $-\dfrac{1}{2}$. So this line touches the region only at the corner $(2, 3)$, and the whole region lies on the side with $Z \ge 13$.

Step 3. Conclude. The half-plane $Z < 13$ has no point in common with the region, so $13$ is the minimum.