Q.$2x+y\ge8,\ x+2y\ge8,\ x+y\le6$.

Step 1. Lines: $2x+y=8$, $x+2y=8$, $x+y=6$.

Step 2. $2x+y=8$ meets $x+2y=8$: subtract to get $x-y=0\Rightarrow x=y$; substitute into $2x+x=8\Rightarrow x=\dfrac83$. Vertex $\left(\dfrac83,\dfrac83\right)$; check $x+y=\dfrac{16}3\approx5.33\le6$ ✓.

Step 3. $2x+y=8$ meets $x+y=6$: subtract to get $x=2,\ y=4$. Vertex $(2,4)$; check $x+2y=10\ge8$ ✓.

Step 4. $x+2y=8$ meets $x+y=6$: subtract to get $y=2,\ x=4$. Vertex $(4,2)$; check $2x+y=10\ge8$ ✓. …

Pairwise-intersect the three boundary lines, keeping only intersection points tha …

• Forgetting to check each candidate vertex against the THIRD inequality (not just the two lines used to find it). …