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 wer food costs $3. This system of inequalities models the scenario: X + 3y = 8 x + y 22 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)

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  A) x +3y ≤ 8; shading below the solid line with x-intercept 8 and y-intercept 8/3. x +y ≥ 2; shading above the solid line with intercepts 2. The solution set is the quadrilateral between the two lines in the first quadrant.

  B) no 8+3(2) > 8 so does not satisfy the first inequality

  C) (5, 1) means Michelle can buy 5 servings of dry food and 1 serving of wet food

Step-by-step explanation:

A) The two inequalities of interest are ...

We are only interested in non-negative x- and y-values, though these two inequalities do not reflect that. A complete system would add x ≥ 0, y ≥ 0.

The boundary line of the first inequality is ...

  x +3y = 8

This has an x-intercept of 8, and a y-intercept of 8/3. These can be found by solving when y=0 and x=0, respectively. The boundary line can be drawn through these intercept points, or advantage can be taken of the knowledge that the slope of the line is -1/3. That is, points (8, 0), (5, 1), (2, 2), and (-1, 3) will be on the line.

Since the inequalities include the "or equal to ..." case, both boundary lines will be solid lines. The line itself is part of the solution set.

__

The boundary line of the second inequality is ...

  x + y = 2

This is a line with a slope of -1 through the intercept points of (2, 0) and (0, 2).

__

The direction of shading can be determined by considering the relation of either variable to the inequality symbol. Here, in the first inequality, the variables both have positive coefficients and are on the left side of the ≤ symbol. This means that variable values less than those on the line are in the solution set. Shading will be below and to the left of the line.

In the second inequality, both variables have positive coefficients and are on the left side of the ≥ symbol. That means solutions to that inequality are greater than those on the line. Shading will be above and to the right of the line. Shaded areas will overlap in the space between the lines to the lower right of where the lines cross at (-1, 3). The overlapping area in the first quadrant is the solution set.

The graph is shown below.

_____

B) The point (8, 2) is not in the solution set. It is above the line x+3y ≤ 8, so not in the shaded area. When those x, y values are put in the inequality, we get ...

  8 +3(2) = 14 is not at most 8 . . . . the inequality is not satisfied

_____

C) The point (5, 1) is on the upper boundary line of the solution set, so is a possible solution. It means Michelle can buy 5 servings of dry food and 1 serving of wet food with her $8.

c

step-by-step explanation: