Linear Inequalities

Concept:The shaded region is the intersection of half-planes determined by the four lines.Explanation:• The region is bounded and lies in the first quadrant.• Testing the origin $(0,0)$: it gives $0<9$, $0<7$, $0<10$, $0<12$; if the shaded region is on the opposite side of each line, the inequalities would be $\geq$.• However, the region is bounded, so some inequality must be $\leq$ to close the polygon.• The only option that produces a bounded triangle (vertices at $(2,5)$, $(\frac{8}{3},\frac{11}{3})$, $(4,3)$) is option B: $2x+y \geq 9$, $x+y \leq 7$, $x+2y \geq 10$, $x+3y \geq 12$.Answer:Option B corresponds to the correct set of inequalities.