Michelle has $8 and wants to buy a combination of dog food to feed at least two dogs at the animal shelter. A serving of dry food costs $1, and a serving of wet food costs $3. This system of inequalities models the scenario:

x + 3y ≤ 8
x + y ≥ 2

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, 2) 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)

We have the system of inequalities:

x + 3y ≤ 8

x + y ≥ 2

One easier way to read this, is to write both of them like linear relationships in the slope-intercept way, we will get:

y ≤ -(1/3)*x + 8/3

y  ≥ 2 - x

a) The graph of y ≤ -(1/3)*x + 8/3 is a solid line:

y = -(1/3)*x + 8/3

And we need to shade all the region below this line, because "y is smaller than or equal to the line"

And for the inequality y  ≥ 2 - x, we will have a solid line y = 2 - x, and we need to shade all the region above the line.

An example of the graph can be seen below.

b) We want to see if (8, 2) is a solution.

We just need to replace x by 8 and y by 2 in both inequalities, then:

first inequality:

8 + 3*2 ≤ 8

8 + 6  ≤ 8

14  ≤ 8

This is false, 14 is not smaller than or equal to 8, then we already can see that (8, 2) is not a solution.

C) A point in the solution set is a point that belongs to the double-shaded region of the graph (Where we only see at the first quadrant, because we need to have x≥0 and y ≥0)

We can see that one possible solution is (2, 1)

Let's check this:

first inequality:

2  + 3*1 ≤ 8

5 ≤ 8

True.

Second inequality:

1 + 2 ≥ 2

3 ≥ 2

True.

Then (2, 1) is a solution, this means that:

x = 2

y = 1

And x is the number of dry food servings that Michelle buys, and y is the number of wet food servings that Michelle buys.

Then the solution (2, 1) means that:

She can buy 2 dry food servings and 1 wet food serving.