Which of the following is not having feasible region given set of con…

$\text{Check each set of constraints for feasibility.}$

$(A)\;x+2y\le50,\;3x+y\le80,\;x\ge0,\;y\ge0.$

$\text{Both inequalities allow a common region in first quadrant}.$

$(B)\;x+2y\le50,\;3x+y\ge80,\;x\ge0,\;y\ge0.$

$\text{Lines intersect in first quadrant, feasible region exists}.$

$(C)\;x+y\le10,\;2x+y\ge40,\;x\ge0,\;y\ge0.$

$\text{From }x+y\le10 \Rightarrow 2x+y\le20,$

$\text{but }2x+y\ge40\text{ cannot be satisfied simultaneously}.$

$\text{No feasible region}.$

$(D)\;2x+y\le30,\;5x+2y\le100,\;x\ge0,\;y\ge0.$

$\text{Feasible region exists}.$

$\text{Correct option: }(C).$