Question

Kiani shopping at bookstore with a budget of $60. Hardbacks costs $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?

Answer 1

To represent Kiani's shopping scenario, we can define the following variables:

- Let \( x \) be the number of hardbacks purchased.
- Let \( y \) be the number of paperbacks purchased.

The cost of hardbacks is $12 each, and the cost of paperbacks is $5 each. Given Kiani's budget constraint of $60, we can write the following inequality to represent the total cost:

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

### Constraints of the problem:

1. Non-negativity constraints: Since Kiani cannot purchase a negative number of books:
- \( x \geq 0 \)
- \( y \geq 0 \)

2. The budget constraint we already established:
- \( 12x + 5y \leq 60 \)

### Summary of Constraints:
- \( x \geq 0 \)
- \( y \geq 0 \)
- \( 12x + 5y \leq 60 \)

### Viable Solution Points:
To determine which points are viable solutions, you would need to test specific points \( (x, y) \) to see if they satisfy all constraints, including the budget constraint.

For example:
- Point \( (0, 12) \):
- \( 12(0) + 5(12) = 60 \) (valid, exactly the budget)
- Point \( (5, 6) \):
- \( 12(5) + 5(6) = 60 + 30 = 90 \) (invalid, exceeds the budget)
- Point \( (3, 0) \):
- \( 12(3) + 5(0) = 36 \) (valid, within the budget)
- Point \( (2, 6) \):
- \( 12(2) + 5(6) = 24 + 30 = 54 \) (valid, within the budget)

Please provide specific points if you want to check whether they're viable solutions.