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)

Question

Responses

(1,4.5)
left parenthesis 1 comma 4.5 right parenthesis

(6,6)
left parenthesis 6 comma 6 right parenthesis

(−2,6)
left parenthesis negative 2 comma 6 right parenthesis

(2,4)

Answer 1

To form a two-variable inequality for Kiani's spending at the bookstore, we can let \( x \) represent the number of hardbacks and \( y \) represent the number of paperbacks she buys.

Given:

• Hardbacks cost $12 each.
• Paperbacks cost $5 each.
• Kiani has a budget of $60.

The inequality representing her budget constraint is:

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

Constraints of the problem:

1. Both \( x \) and \( y \) must be non-negative since Kiani cannot buy a negative quantity of books. Thus:
   • \( x \geq 0 \)
   • \( y \geq 0 \)

This results in the following constraints:

• \( 12x + 5y \leq 60 \)
• \( x \geq 0 \)
• \( y \geq 0 \)

Now, we'll evaluate the given points to determine which one represents a viable solution that satisfies all the constraints:

1. (1, 4.5):

   • \( 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \) (This is within the budget.)
   • \( x \geq 0 \) and \( y \geq 0 \) are satisfied.
   • Viable solution.

2. (6, 6):

   • \( 12(6) + 5(6) = 72 + 30 = 102 \) (This exceeds the budget.)
   • Not a viable solution.

3. (−2, 6):

   • Negative quantity of hardbacks (not possible).
   • Not a viable solution.

4. (2, 4):

   • \( 12(2) + 5(4) = 24 + 20 = 44 \) (This is within the budget.)
   • \( x \geq 0 \) and \( y \geq 0 \) are satisfied.
   • Viable solution.

Conclusion: The points that are viable solutions based on the given conditions are (1, 4.5) and (2, 4).

Therefore, the answer is:

• (1, 4.5) or (2, 4) represents a viable solution.

Answer 2

Pick one