Question

Kiani is shopping at a bookstore with a budget of $80. Hardbacks cost $12 and paperbacks cost SS. White a two-variable inequality and determine the constraints of the problem. Which of the following points represents a viable solution? (1 point)

Ο (-2,6)

Ο (2.4)

Ο (1.4.5)

0 (66)

Answer 1

To formulate the problem, let \( x \) represent the number of hardbacks Kiani buys, and let \( y \) represent the number of paperbacks. The following conditions must be met based on Kiani's budget:

1. The cost of the hardbacks is \( 12x \) (since each hardback costs $12).
2. The cost of the paperbacks is \( 5y \) (assuming "SS" means $5 for each paperback).

Kiani's total cost must not exceed $80, which gives us the following inequality:

\[
12x + 5y \leq 80
\]

Additionally, since Kiani cannot buy a negative number of books, we have the following constraints:

\[
x \geq 0 \\
y \geq 0
\]

Now we have a system of inequalities that describe the constraints:

1. \( 12x + 5y \leq 80 \)
2. \( x \geq 0 \)
3. \( y \geq 0 \)

Next, we must evaluate the proposed points to determine which one represents a viable solution (i.e., a point that satisfies all inequalities):

1. \( (-2, 6) \): This point is invalid because \( x \) (the number of hardbacks) cannot be negative.
2. \( (2, 4) \): Check:
- \( 12(2) + 5(4) = 24 + 20 = 44 \) (which is \( \leq 80 \), valid)
- \( 2 \geq 0 \) (valid)
- \( 4 \geq 0 \) (valid)

This point is a viable solution.

3. \( (1, 4.5) \): Check:
- \( 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \) (which is \( \leq 80 \), valid)
- \( 1 \geq 0 \) (valid)
- \( 4.5 \geq 0 \) (valid)

This point is also a viable solution.

4. \( (6, 6) \): Check:
- \( 12(6) + 5(6) = 72 + 30 = 102 \) (which is \( > 80 \), invalid)
- \( 6 \geq 0 \) (valid)
- \( 6 \geq 0 \) (valid)

This point is not a viable solution.

Thus, the viable solutions are \( (2, 4) \) and \( (1, 4.5) \).

In the choices given, the point that represents a viable solution is:

- **(2, 4)**