Kiani is shopping at a bookstore with a budget of $60. Hardbacks cost $12 and paperbacks cost $5. Write a two-variable inequality and determine the constraints of the problem. Which of the following points represents a viable solution?(1 point) Responses (1,4.5) left parenthesis 1 comma 4.5 right parenthesis (6,6) left parenthesis 6 comma 6 right parenthesis (2,4) left parenthesis 2 comma 4 right parenthesis (−2,6)

Question

GPT-4o mini · Answer

To model the situation, we can define two variables:

Let $ x $ be the number of hardbacks Kiani buys.
Let $ y $ be the number of paperbacks Kiani buys.

The cost of each hardback is $12, and the cost of each paperback is $5. Kiani's total cost must not exceed $60, leading to the inequality:

\[ 12x + 5y \leq 60 \]

The constraints for $ x $ and $ y $ must also include non-negativity since Kiani cannot buy a negative number of books:

\[ x \geq 0 \quad \text{and} \quad y \geq 0 \]

Summary of the Constraints:

$ 12x + 5y \leq 60 $
$ x \geq 0 $
$ y \geq 0 $

Next, we will check the provided points to see which one is a viable solution.

Point (1, 4.5): \[ 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad \text{(This point is feasible)} \]
Point (6, 6): \[ 12(6) + 5(6) = 72 + 30 = 102 \quad \text{(This point exceeds the budget)} \]
Point (2, 4): \[ 12(2) + 5(4) = 24 + 20 = 44 \quad \text{(This point is feasible)} \]
Point (−2, 6): \[ \text{Not feasible since } x \text{ cannot be negative.} \]

Conclusion:

The viable solutions among the given points are (1, 4.5) and (2, 4). The only point that fits all constraints while also satisfying the inequality is:

Point (1, 4.5)
Point (2, 4)

Thus, (1, 4.5) and (2, 4) both represent viable solutions under the given constraints. However, if you need to choose just one, you can mention either (1, 4.5) or (2, 4).