Question

Question 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 (2,4) left parenthesis 2 comma 4 right parenthesis (1,4.5) left parenthesis 1 comma 4.5 right parenthesis (6,6) left parenthesis 6 comma 6 right parenthesis (−2,6)

Answer 1

To model Kiani's shopping situation, we can denote the number of hardbacks bought as \( x \) and the number of paperbacks bought as \( y \). Given the prices of the books and Kiani's budget, we can establish the following inequality:

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

Constraints of the Problem:

1. Non-negativity: Kiani cannot buy a negative number of books, so: \[ x \geq 0 \quad \text{and} \quad y \geq 0 \]

Now we will check which of the provided points is a viable solution:

1. Point (2, 4): \[ 12(2) + 5(4) = 24 + 20 = 44 \leq 60 \quad \text{(Valid)} \]

2. Point (1, 4.5): \[ 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \leq 60 \quad \text{(Valid)} \]

3. Point (6, 6): \[ 12(6) + 5(6) = 72 + 30 = 102 \not\leq 60 \quad \text{(Invalid)} \]

4. Point (-2, 6): \[ \text{(Invalid because } x \text{ must be non-negative)} \]

Conclusion:

The viable solutions from the given options are (2, 4) and (1, 4.5).

Hence, the points that represent viable solutions are (2, 4) and (1, 4.5).