Kiani is shopping at a bookstore with a budget of $60 . Hardbacks cost $12 and paperbacks cost $5 . Write a twovariable inequality and determine the constraints of the problem. Which of the following points represents a viable solution? A:(6,6) B:(-2,6) C: (-1,4.5)
D:(2,4)
Let x represent the number of hardbacks and y represent the number of paperbacks.
The cost of hardbacks is $12 times the number of hardbacks: 12x.
The cost of paperbacks is $5 times the number of paperbacks: 5y.
The total cost of the books must be less than or equal to the budget of $60: 12x + 5y ≤ 60.
The constraints of the problem are:
- Number of hardbacks (x) and paperbacks (y) must be non-negative since negative books don't exist: x ≥ 0, y ≥ 0.
- The total cost of the books must be less than or equal to the budget: 12x + 5y ≤ 60.
Now let's check which of the given points are a viable solution:
A: (6,6)
Plugging in the values:
12(6) + 5(6) = 72 + 30 = 102 (which is greater than 60)
This point does not represent a viable solution.
B: (-2,6)
Plugging in the values:
12(-2) + 5(6) = -24 + 30 = 6 (which is less than 60)
However, since the number of hardbacks (x) must be non-negative (x ≥ 0), this is not a viable solution.
C: (-1,4.5)
Plugging in the values:
12(-1) + 5(4.5) = -12 + 22.5 = 10.5 (which is less than 60)
However, since the number of hardbacks (x) must be non-negative (x ≥ 0) and the number of paperbacks (y) must be an integer, this is not a viable solution.
D: (2,4)
Plugging in the values:
12(2) + 5(4) = 24 + 20 = 44 (which is less than 60)
This point represents a viable solution.
Therefore, the viable solution is D: (2,4).