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)
A: (1,4.5)
B: (2,4)
C: (6,6)
D: (−2,6)

Answer 1

To create a two-variable inequality based on Kiani's shopping scenario, we can let:

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

The total cost of the hardbacks and paperbacks must not exceed Kiani's budget of $60. Thus, we can write the inequality as:

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

Next, we need to consider the constraints of the problem:

1. **Non-negativity constraint:** Kiani cannot buy a negative number of books. Therefore:
- \( x \geq 0 \)
- \( y \geq 0 \)

Combining these constraints, we have the system:

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

Now let's evaluate the given points to see which represents a viable solution:

A: \( (1, 4.5) \)

\[
12(1) + 5(4.5) = 12 + 22.5 = 34.5 \leq 60 \quad \text{(valid)}
\]

B: \( (2, 4) \)

\[
12(2) + 5(4) = 24 + 20 = 44 \leq 60 \quad \text{(valid)}
\]

C: \( (6, 6) \)

\[
12(6) + 5(6) = 72 + 30 = 102 \leq 60 \quad \text{(invalid)}
\]

D: \( (-2, 6) \)

\[
\text{(not valid since x cannot be negative)}
\]

Based on the evaluations, the viable solutions are:

- A: (1, 4.5) is valid.
- B: (2, 4) is valid.

C and D are not valid solutions.

Thus, both A and B are viable points that satisfy the constraints and the inequality. If we need to choose only one viable solution from the options, we can select either A or B.