Linear Inequalities

Concept:The shaded region is the intersection of half-planes determined by the given lines and axes.Explanation:Check the origin $(0,0)$, which is often inside the shaded region if it satisfies all inequalities.For $L_1: 15x+6y=300$, at origin $0 \le 300$, so the inequality is $15x+6y \le 300$.For $L_2: 5x+4y=120$, at origin $0 \le 120$, so the inequality is $5x+4y \le 120$.For $L_3: x+2y=50$, at origin $0 \le 50$, so the inequality is $x+2y \le 50$.Also, $x \ge 0$ and $y \ge 0$ because the region is in the first quadrant.These match option A.Answer:Option A: $15x+6y \le 300$, $5x+4y \le 120$, $x+2y \le 50$, $x \ge 0, y \ge 0$.