6. To find out which combinations of sketchbooks and paint sets Alex can purchase with her $96.00 gift card, we can set up an equation based on the cost of each item:
Let:
- \( x \) = number of sketchbooks
- \( y \) = number of paint sets
The cost equation for the sketchbooks and paint sets is: \[ 12x + 15y = 96 \]
Now, we can evaluate each option:
A) 2 sketchbooks and 5 paint sets \[ 12(2) + 15(5) = 24 + 75 = 99 \] (exceeds $96)
B) 1 sketchbook and 4 paint sets \[ 12(1) + 15(4) = 12 + 60 = 72 \] (under $96)
C) 4 sketchbooks and 3 paint sets \[ 12(4) + 15(3) = 48 + 45 = 93 \] (under $96)
D) 3 sketchbooks and 4 paint sets \[ 12(3) + 15(4) = 36 + 60 = 96 \] (exactly $96)
From the calculations, the only combination that allows Alex to exactly spend $96.00 is: D) 3 sketchbooks and 4 paint sets.
7. To determine if the point (0, -3) is a solution to the inequality \( y > \frac{4}{5}x - 3 \), we first plug in the x-coordinate of the point (which is 0) into the inequality:
-
Substitute \( x = 0 \): \[ y > \frac{4}{5}(0) - 3 \] \[ y > -3 \]
-
Now we evaluate the y-coordinate of the point (0, -3): \[ -3 > -3 \] (this is false)
Because \( -3 \) does not satisfy the inequality \( y > -3 \), it means:
The point (0, -3):
- Does not satisfy the inequality \( y > \frac{4}{5}x - 3 \).
Thus, the answer is: A) No, because (0,−3) falls on the dashed boundary line.