Q.$2x+3y\le6,\ x+4y\le4,\ x\ge0,\ y\ge0$.

Step 1. Test the origin in both slanted inequalities: $0\le6$ and $0\le4$, both true, so the origin is included -- vertex $(0,0)$.

Step 2. Along $y=0$: $2x+3y\le6\Rightarrow x\le3$; $x+4y\le4\Rightarrow x\le4$. The binding one is $x\le3$, giving vertex $(3,0)$ (on the line $2x+3y=6$; check $x+4y\le4$ there: $3\le4$ ✓).

Step 3. Along $x=0$: $3y\le6\Rightarrow y\le2$; $4y\le4\Rightarrow y\le1$. The binding one is $y\le1$, giving vertex $(0,1)$ (on the line $x+4y=4$; check $2x+3y\le6$ there: $3\le6$ ✓). …

Find the axis intercepts of each binding constraint and the intersection …

• Using the non-binding intercept on an axis (e.g. $x=4$ instead of $x=3$) instead of checking which constraint is tighter. …