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 (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)

Answer 1

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:
1. \( 12x + 5y \leq 60 \)
2. \( x \geq 0 \)
3. \( y \geq 0 \)

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

1. **Point (1, 4.5)**:
\[
12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad \text{(This point is feasible)}
\]

2. **Point (6, 6)**:
\[
12(6) + 5(6) = 72 + 30 = 102 \quad \text{(This point exceeds the budget)}
\]

3. **Point (2, 4)**:
\[
12(2) + 5(4) = 24 + 20 = 44 \quad \text{(This point is feasible)}
\]

4. **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).