### Part A: Describe the Graph of the System of Inequalities
Let's analyze and describe the graph for each inequality separately.
#### 1. Inequality [tex]\( x + 3y \leq 8 \)[/tex]:
- This inequality can be rewritten as [tex]\( y \leq \frac{8 - x}{3} \)[/tex].
- The line [tex]\( x + 3y = 8 \)[/tex] has a negative slope.
- To graph the line, you can find the intercepts. When [tex]\( x = 0 \)[/tex], [tex]\( y = \frac{8}{3} \approx 2.67 \)[/tex]. When [tex]\( y = 0 \)[/tex], [tex]\( x = 8 \)[/tex].
- The line will pass through (0, 2.67) and (8, 0).
- The region below this line is shaded, as [tex]\( y \)[/tex] must be less than or equal to [tex]\( \frac{8 - x}{3} \)[/tex].
#### 2. Inequality [tex]\( x + y \geq 2 \)[/tex]:
- This inequality can be rewritten as [tex]\( y \geq 2 - x \)[/tex].
- The line [tex]\( x + y = 2 \)[/tex] also has a negative slope.
- To graph the line, you find the intercepts. When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. When [tex]\( y = 0 \)[/tex], [tex]\( x = 2 \)[/tex].
- The line will pass through (0, 2) and (2, 0).
- The region above this line is shaded, as [tex]\( y \)[/tex] must be greater than or equal to [tex]\( 2 - x \)[/tex].
#### Solution Set:
- The solution set for this system of inequalities is the overlapping shaded region of both inequalities.
- The combination of shaded areas represents all the possible combinations of servings of dry food (x) and wet food (y) that Michelle can buy such that the total cost is within her budget (\[tex]$8) and she can feed at least two dogs.
### Part B: Is the Point (8, 2) Included in the Solution Area?
We need to check if the point \((8, 2)\) satisfies both inequalities.
#### For \( x + 3y \leq 8 \):
\[ 8 + 3 \cdot 2 = 8 + 6 = 14 \leq 8 \]
This simplifies to \( 14 \leq 8 \), which is false.
Since the first inequality is not satisfied, the point \((8, 2)\) does not lie in the solution area. Therefore, \((8, 2)\) is not included in the solution area.
### Part C: Choose a Point in the Solution Set and Interpret What It Means in Terms of the Real-World Context
Let's choose the point \((2, 2)\).
#### Check the inequalities:
1. For \( x + 3y \leq 8 \):
\[ 2 + 3 \cdot 2 = 2 + 6 = 8 \leq 8 \]
The first inequality is satisfied.
2. For \( x + y \geq 2 \):
\[ 2 + 2 = 4 \geq 2 \]
The second inequality is also satisfied.
#### Interpretation:
The point \((2, 2)\) means that Michelle can buy 2 servings of dry food and 2 servings of wet food. This combination will cost her:
\[ 2 \cdot 1 + 2 \cdot 3 = 2 + 6 = \$[/tex]8 \]
With this combination, Michelle utilizes all her \$8 budget and is able to feed at least two dogs at the animal shelter.
