2x + y ≤ 8
x + y ≥ 4
From the description, x represents the number of cupcakes bought and y represents the number of fudges bought.
A. To graph the inequalities, first, we need to graph the lines
2x + y = 8
x + y = 4
Substituting x = 0 into the first equation,
2(0) + y = 8
y = 8
Substituting y = 0 into the first equation,
2x + 0 = 8
2x = 8
x = 8/2
x = 4
Then, the line passes through the points (0, 8) and (4, 0)
Substituting x = 0 into the second equation,
0 + y = 4
y = 4
Substituting y = 0 into the second equation,
x + 0 = 4
x = 4
Then, the line passes through the points (0, 4) and (4, 0)
Both lines are solid because of the '≤' and '≥' signs.
Given the sign '≤', we have to shade the area below the line 2x + y = 8
Given the sign '≥', we have to shade the area above the line x + y = 4
The graphical solution is:
B. The point (8, 10) is not included in the solution because it is outside the shaded area
Replacing this point into the inequalities:
2(8) + 10 ≤ 8
16 + 10 ≤ 8
26 ≤ 8
8 + 10 ≥ 4
18 ≥ 4
None of them are true, then the point (8, 10) is not included in the solution
C. From the picture, the point (2, 3) is in the solution. This means that Sarah can buy 2 cupcakes and 3 fudges