Question

1. 4.4.3 test (cst): systems of equations and inequalities zane is making fruit punch in a 60 - cup punch bowl for a party. he plans to add at least 16 ounces of fruit juice to the bowl before adding ginger ale. identify the graph that represents the amounts of fruit juice and ginger ale he can use, and then identify a valid solution. (the graph shows the overlapping solution regions only, not the shading for each inequality.)

a. valid solution: (12, 16)
graph for option a

b. valid solution: (20, 12)
graph for option b

c. valid solution: (12, 16)
graph for option c

d. valid solution: (16, 12)
graph for option d

Explanation:

To solve this, we analyze the constraints: total punch is 80 cups (so fruit juice \( x \) + ginger ale \( y \leq 80 \)) and at least 16 cups of fruit juice (\( x \geq 16 \)). A valid solution must satisfy both.

Analyze Option A: (12, 18)

• Check \( x \geq 16 \): \( 12 \geq 16 \) is false. So invalid.

Analyze Option B: (20, 12)

• Check \( x \geq 16 \): \( 20 \geq 16 \) (true).
• Check \( x + y \leq 80 \): \( 20 + 12 = 32 \leq 80 \) (true).
• The graph for B has the blue region (solution region) where \( x \geq 16 \) and \( x + y \leq 80 \), so (20,12) lies in it.

Analyze Option C: (12, 16)

• Check \( x \geq 16 \): \( 12 \geq 16 \) (false). Invalid.

Analyze Option D: (18, 7)

• The graph for D does not show the \( x \geq 16 \) constraint (no vertical line at \( x = 16 \)), so it’s inconsistent with the “at least 16 cups of fruit juice” rule. Invalid.

Answer:

B. Valid solution: (20, 12)