Constraints:
(1) 4x-5y≤50,
(2) -x+2y≥24
(3) x+y≤80
Vertices:
(2)+(3)=3y=104, y=104/3, x=136/3
(1)+4(2)=3y=146, y=146/3, x=220/3
4(3)-(1)=9y=270, y=30, x=50
Let's assume that neither x nor y can be negative, otherwise there would be no definable minimum for g. Therefore we must assume that the x and y axes form a boundary with the given lines. These boundaries will include the y intercepts (0,12), (0,80) only. The feasibility region is a triangle with vertices:
(0,80), (136/3,104/3), (0,12). Other intercepts including all the x intercepts lie outside the feasibility region.
Taking each vertex we can calculate g:
(0,80): g=640; (136/3,104/3): g=1240/3; (0,12): g=96.
So x=0, y=12 produces the minimum g.
Now we need to check that this satisfies all the constraints:
(1) 4x-5y=-60 which is less than 50; OK
(2) -x+2y=24; OK
(3) x+y=12 which is less than 80; OK
If the axes are not constraints then only constraint (2) would apply and there would be no definable minimum.