Question 1187163
x is the number of cupcakes.
y is the number of fudge pieces.


2x + y <= 8 is the inequality for the cost.


x + y >= 4 is the inequality for the number of cupcakes or pieces of fudge that she wants to buy.


using the desmos.com calculator, you would graph the OPPOSITE of the inequalities.
the area on the graph that is NOT shaded is the feasible region.


the graph looks like this:


<img src = "http://theo.x10hosting.com/2022/010901.jpg" >



the solution set is all integral coordinate points that are either on the lines x + y = 4 or 2x + y = 8 or x = 0 or anywhere in the unshaded area of the graph.


specifically, the coordinate points that are acceptable are:
(0,8)
(0,7)
(0,6)
(0,5)
(0,4)
(1,6)
(1,5)
(1,4)
(1,3)
(2,4)
(2,3)
(2,2)
(3,2)
(3,1)
(4,0)


the point (8,10) is not included in the solution set because it is not in the unshaded area on the graph.


it does not meet the constraint requirements.
(8,10) means x = 8 and y = 10
x + y = 18 is satisfied because the constraint is x + y >= 4.
the constraint 2x + y <= 8 is not satisfied because 2x + y = 26 which is NOT smaller than or equal to 8.
ALL the constraints have to be satisfied at each coordinate point.
otherwise that coordinate point is not acceptable.


take any of the points that are feasible and test them out.
you will find that each of them meets ALl of the constraints.
for example:
(4,0) meets the constraints because x + y >= 4 and 2x + y <= 8
this cross point would buy 4 cupcakes and no pieces of fudge.


at the other extreme, (0,8) is feasible because x + y >= 4 and 2x + y <= 8.
this cross point would buy 8 pieces of fudge and no cupcakes.


the real world context has been explained for both of these points.


the real world context for (8,10) would be explained as:
x + y = 8 cupcakes and 10 pieces of fudge, the combination of which are greater than or equal to 4.
2x + y =  16 + 10 = 26 dollars which is NOT less than or equal to 8.