Question 1187163: Sarah has $8 and wants to buy a combination of cupcakes and fudge to feed at least four siblings. Each cupcake costs $2, and each piece of fudge costs $1.
This system of inequalities models the scenario:
2x + y ≤ 8
x + y ≥ 4
Part A: Describe the graph of the system of inequalities, including shading and the types of lines graphed. Provide a description of the solution set. (4 points)
Part B: Is the point (8, 10) included in the solution area for the system? Justify your answer mathematically. (3 points)
Part C: Choose a point in the solution set and interpret what it means in terms of the real-world context. (3 points)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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:
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.
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