Question

6. Alex won a gift card worth $96.00 for winning a design competition. She decided to spend her winnings on sketchbooks and paint sets. Each sketchbook (x) costs $12.00, while each paint set (y) costs $15.00.

Which of the following combinations of sketchbooks and paint sets can Alex purchase, assuming she spends her entire gift card?
A) 2 sketchbooks and 5 paint sets
B) 1 sketchbook and 4 paint sets
​​​​C) 4 sketchbooks and 3 paint sets
D) 3 sketchbooks and 4 paint sets

7. Is the point (0, −3) a solution to the inequality y > 4/5x − 3
A) No, because (0, −3) falls on the dashed boundary line.
B) Yes, because (0, −3) falls on the boundary line.
C) No, because (0, −3) falls within the shaded region.
D) Yes, because (0, −3) falls within the shaded region.

Answer 1

**6. To determine which combinations of sketchbooks and paint sets Alex can purchase using her $96.00 gift card, we need to formulate the equation based on the costs of each item.

Let \( x \) be the number of sketchbooks and \( y \) be the number of paint sets. The total cost can be expressed as:
\[ 12x + 15y = 96 \]

Now we can evaluate each of the options:

A) 2 sketchbooks and 5 paint sets:
\[ 12(2) + 15(5) = 24 + 75 = 99 \] (not valid)

B) 1 sketchbook and 4 paint sets:
\[ 12(1) + 15(4) = 12 + 60 = 72 \] (valid but under budget)

C) 4 sketchbooks and 3 paint sets:
\[ 12(4) + 15(3) = 48 + 45 = 93 \] (valid but under budget)

D) 3 sketchbooks and 4 paint sets:
\[ 12(3) + 15(4) = 36 + 60 = 96 \] (valid and uses entire budget)

Only **D) 3 sketchbooks and 4 paint sets** allows Alex to spend her entire gift card of $96.00.

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**7. To determine whether the point (0, −3) is a solution to the inequality \( y > \frac{4}{5}x - 3 \), we can substitute \( x = 0 \) and \( y = -3 \) into the inequality:**

Substituting into the inequality gives:
\[ -3 > \frac{4}{5}(0) - 3 \]
\[ -3 > -3 \]

This statement is false because -3 is not greater than -3.

Thus, the correct option is:
A) No, because (0, −3) falls on the dashed boundary line (the inequality does not include equality because of the 'greater than' sign).