Q17: Corner points for $x \ge 0$⁠, $y \ge 0$⁠, $x + y \ge 4$

Put $y = 0$ in $x + y = 4$ to get $x = 4$⁠, the point $(4, 0)$⁠. Put $x = 0$ to get $y = 4$⁠, the point $(0, 4)$⁠.

To shade an inequality, test one point not on the line. The origin is the easiest. If it fails, the region is the other side of the line.

$x \ge 0$ and $y \ge 0$ keep us in the first quadrant.

A corner point is a vertex of the feasible region: a point where two boundary lines meet and which satisfies every constraint.

The boundary lines $x = 0$⁠, $y = 0$ and $x + y = 4$ meet in pairs at $(0, 4)$⁠, $(4, 0)$ and $(0, 0)$⁠. The point $(0, 0)$ is not in the region, as the test showed, so it is not a corner point.

Watch out: an unbounded region can still have corner points. Here there are two.