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)

Part A) See Below.

Part B) No.

Part C) E.G. (1, 4).

Sarah bought 1 cupcake and 4 fudges.

This is within her budget and it allows her to feed at least four siblings.

Step-by-step explanation:

We have the system of inequalities:

Where x represents the numbe of cupcakes and y represent the number of fudges that is purchased by Sarah.

Part A)

Please refer to the first graph.

Since we have a pair of linear equations, our system of inequalities is a linear pair of inequalities.

The first equation (in red) can be rewritten as:

Hence, the shaded portion is below the line.

The second equation (in blue) can be rewritten as:

Hence, the shaded portion is above the line.

And since both inequalities are “or equal to,” the lines are not dotted but are rather shaded in.

The solution set is where the two shaded regions intersect; it is the darkest region in the graphs shown.

Practically and contextually speaking, neither x and y can be negative. This is because we can’t purchase negative cupcakes/fudges. So, a better representation of this will be the second graph.

Part B)

Looking at the graph, we can determine that (8, 10) is not in the solution area for the system. They are not within the area where the two shaded portions intersect.

Mathematically, the point do not satisfy both inequalities. We have:

If we substitute 8 for x and 10 for y, we acquire:

So:

This is not a true statement, so it does not satisfy the first inequality.

(Contextually, this means that Sarah does not have enough money.)

For the second inequality:

This is a true statement, but since it does not satisfy the first inequality, (8, 10) is not a feasible point.

Part C)

We can choose any point in the solution area. Let’s pick (1, 4).

So, x=1 and y=4.

This means that Sarah purchased 1 cupcake and 4 fudges.

This will cost Sarah $2(1)+$1(4)=$6. So, Sarah can indeed purchase this.

And the total number of items bought is 1+4=5, enough to feed at least 4 siblings.

Another thing to note is that both x and y must be integers, since we cannot buy 1.5 cupcakes. So, we have to further limit our solution set to only integer data points.

Also keep in mind that points on the line within the shaded area will also work. For instance, (0, 4) (0 cupcakes and 4 fudges) can also satisfy our inequalities and Sarah’s needs.

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