Linear Inequalities

Concept:The shaded region is the intersection of half-planes defined by the given lines, with the inequalities determined by testing a point inside the region.Explanation:• The lines are $L_1: 5x+3y=30$, $L_2: x+y=9$, $L_3: y=\frac{x}{3}$, $L_4: y=\frac{x}{2}$.• The shaded region lies above $L_1$ (since the origin $(0,0)$ gives $0<30$, but the region is away from the origin, so $5x+3y \ge 30$).• It lies below $L_2$ (since origin gives $0<9$, and region includes points near the origin? Actually, typical feasible region is bounded, so $x+y \le 9$).• It lies above $L_3$ (since $y=\frac{x}{3}$; test point common to region, e.g., $(6,3)$ gives $3 \ge 2$, so $y \ge \frac{x}{3}$).• It lies below $L_4$ (since $y=\frac{x}{2}$; test $(6,3)$ gives $3 \le 3$, so $y \le \frac{x}{2}$).• Also $x \ge 0, y \ge 0$.• Option B matches these inequalities exactly.Answer:Option B: $5x+3y \ge 30$, $x+y \le 9$, $y \ge \frac{x}{3}$, $y \le \frac{x}{2}$, $x \ge 0$, $y \ge 0$.