A small factory makes bicycles and scooters…

Step 1. The model. Maximise $Z = 30x + 20y$ dollars subject to $2x + y \le 100$, $x + 2y \le 80$, $x + y \le 55$, $x \ge 10$ and $y \ge 0$.

Step 2. The corners. On $y = 0$, the rule $2x + y \le 100$ stops $x$ at $50$, so $(10, 0)$ and $(50, 0)$ are corners. The line $2x + y = 100$ meets $x + y = 55$ at $x = 45$, $y = 10$. The line $x + y = 55$ meets $x + 2y = 80$ at $y = 25$, $x = 30$. The line $x + 2y = 80$ meets $x = 10$ at $y = 35$. All five points obey every rule.

Step 3. The profit. At $(10, 0)$, $Z = 300$. At $(50, 0)$, $Z = 1500$. At $(45, 10)$, $Z = 1350 + 200 = 1550$. At $(30, 25)$, $Z = 900 + 500 = 1400$. At $(10, 35)$, $Z = 300 + 700 = 1000$.

Step 4. Say it. Make $45$ bicycles and $10$ scooters. The profit is $1550$ dollars.