ASAPPP PLSSS i need someone so answer this 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)

Part A

The graph of x + 3y ≤ 8 will involve the boundary line x+3y = 8. This boundary line is solid because of the "or equal to" portion of the inequality sign.

Two points on the line x+3y = 8 are (2,2) and (5,1)

We shade below this boundary line because (0,0) makes x + 3y ≤ 8  to be true. Note how (0,0) is below the boundary.

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As for x + y ≥ 2, this involves the boundary x+y = 2

The boundary is solid for similar reasoning as before.

The boundary line x+y = 2 goes through the points (0,2) and (2,0)

We shade above the boundary line.

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Refer to the graph below for a visual summary of part A.

In that diagram, regions B and C are the full shaded region of x + 3y ≤ 8

While regions A and B are the full shaded region of x + y ≥ 2

When overlapping those two full regions, we have region B in common for both.

In short, region B is the solution set of the entire system of inequalities.

If a point is in region B, then it's considered a solution point.

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Part B

If we plot (8,2) in the graph we made for part A, then we can see that it is not in the solution set. It's outside of region B.

We can use algebra to verify this.

Plug x = 8 and y = 2 into the first inequality

x + 3y ≤ 8

8 + 3*2 ≤ 8

8 + 6 ≤ 8

14 ≤ 8

Which is false. The point (8,2) makes the first inequality false, so that's enough to prove that it's not a solution to the system.

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Part C

Refer to the diagram below. Pick any point in region B.

Let's say we pick (6,-2)

While this is a solution point (I'll let you confirm this with both inequalities), it's not very realistic. Why not? Because y = -2 isn't possible in reality. We cannot buy negative 2 servings of wet food.

So we must insist that x and y are not negative.

That restricts region B further such that we're in the upper right quadrant (when x ≥ 0 and y ≥ 0)

With that further restriction in mind, let's pick the point (3,1)

This point says Michelle can buy 3 servings of dry food, and 1 serving of wet food.

The total number of servings is 3+1 = 4 which means she can feed four dogs. So that takes care of the second inequality. Also, note how x+3y = 3+3(1) = 6 which is less than $8, so she's under budget and can afford to get this combo. This confirms the first inequality.

for the points

step-by-step explanation:

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