Q18: Number of corner points for $x - y \ge 0$⁠, $2y \le x + 2$⁠, $x, y \ge 0$

A corner point is a point where two boundary lines meet and which satisfies every constraint. So we find where each pair of lines meets, then test each point.

$x - y = 0$ is the same as $y = x$⁠, which is why we can put $y = x$ into the other equation.

Then $y = x$ gives $y = 2$⁠.

Where $2y = x + 2$ meets the axes:

At $(0, 1)$⁠: $x - y = -1$⁠, which is negative. At $(-2, 0)$⁠: $x = -2$⁠, which is negative. So neither point is in the feasible region.

$(2, 2)$ and $O(0, 0)$ satisfy all four constraints. For example, at $(2, 2)$⁠: $x - y = 0$ and $2y = 4 = x + 2$⁠.

The region lies above $y = 0$⁠, below $y = x$ and below $y = \frac{x + 2}{2}$⁠. Its edges run from $O$ along $y = x$ to $(2, 2)$⁠, then along $2y = x + 2$ for ever, and from $O$ along the $x$-axis for ever.

Points such as $(10, 0)$ satisfy every constraint, so the region is unbounded. Its only corners are $O(0, 0)$ and $(2, 2)$⁠, so there are 2.