graph and find the max and min: 1<y<4 , 4y-6x>-32 , 2y>-x+4 , f(x,y)=-6x+3y

Question

i need to graph and find the max and min of the functions or region.. the part im having the most trouble with is 1<y<4. i know how to graph that but i dont understand it with this problem. i dont know what should be verticals because this gives more verticals than it should.

Answer 1

1<y<4 is the region between two horizontal lines. The region enclosed by the given inequalities is a trapezoid ABCD with A(-4,4), B(8,4), C(6,1), D(2,1).

A is the intersection of y=4 and 2y=-x+4; 8=-x+4, x=-4;

B is the intersection of y=4 and 4y-6x=-32; 16-6x=-32, 48=6x, x=8;

C is the intersection of y=1 and 4y-6x=-32; 4-6x=-32, 36=6x, x=6;

D is the intersection of y=1 and 2y=-x+4; 2=-x+4, x=2.

f(x,y)=-6x+3y, so:

A:f(-4,4)=24+12=36; B:f(8,4)=-48+12=-36; C:f(6,1)=-36+3=-33; D:f(2,1)=-12+3=-9.

B represents the minimum while A represents the maximum.

Trapezoid dimensions: AB=12, CD=4, height is 3.