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) The graph in the attached figure (see the explanation)

Part B) The ordered pair is not included in the solution area for the system

Part C) Point (2,3). Sarah buys five siblings. Two cupcakes and three fudge

Step-by-step explanation:

Part A)

Let

x ----> is the number of cupcakes

y ---> is the number of fudge

we have

----> inequality A

The solution of the inequality A is the shaded area below the solid line

The slope of the solid line is negative

The x-intercept of the solid line is the point (4,0)

The y-intercept of the solid line is the point (0,8)

----> inequality B

The solution of the inequality B is the shaded area above the solid line

The slope of the solid line is negative

The x-intercept of the solid line is the point (4,0)

The y-intercept of the solid line is the point (0,4)

using a graphing tool

The solution of the system of inequalities is the shaded area between the positive integers (whole numbers) of the two solid lines

see the attached figure

Part B) Is the point (8, 10) included in the solution area for the system? Justify your answer mathematically

we know that

If a ordered pair is a solution of the system of inequalities, then the ordered pair must satisfy both inequalities of the system

For x=8, y=10

Substitute the value of x and the value of y of the ordered pair in each inequality

Verify inequality A

 ----> is not true

so

The ordered pair not satisfy the inequality A

therefore

The ordered pair is not included in the solution area for the system

Part C) Choose a point in the solution set and interpret what it means in terms of the real-world context

I choose the point (2,3)

The point (2,3) is a solution because is included in the solution area for the system

That means ----> Sarah buys five siblings. Two cupcakes and three fudge

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

the correct answer would be:

220 centimeters

to get this you can use the pythagorean theorem

hope this

step-by-step explanation: