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?
(2,4)
(-2,6)
(1,4.5)
(6,6)

Answer 1

To write a two-variable inequality for Kiani's shopping situation, we need to define the variables:

Let \( x \) represent the number of hardbacks bought and \( y \) represent the number of paperbacks bought.

The costs in dollars can be expressed as:
- The cost for hardbacks is \( 12x \).
- The cost for paperbacks is \( 5y \).

Given Kiani's budget constraint of $60, we can create the following inequality:

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

### Constraints of the Problem:
1. Since Kiani cannot buy a negative number of books, we also have:
\[ x \geq 0 \]
\[ y \geq 0 \]

These constraints indicate that the number of hardbacks and the number of paperbacks must be non-negative.

### Checking the Viable Solutions:
Now, we need to check which of the provided points (2,4), (-2,6), (1,4.5), and (6,6) satisfies the inequality \( 12x + 5y \leq 60 \) and the non-negativity constraints.

1. **Point (2, 4)**:
\[
12(2) + 5(4) = 24 + 20 = 44 \quad \Rightarrow \quad 44 \leq 60 \quad (\text{Valid})
\]

2. **Point (-2, 6)**:
\[
\text{Not valid since } x = -2 \text{ is negative.}
\]

3. **Point (1, 4.5)**:
\[
12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad \Rightarrow \quad 34.5 \leq 60 \quad (\text{Valid})
\]

4. **Point (6, 6)**:
\[
12(6) + 5(6) = 72 + 30 = 102 \quad \Rightarrow \quad 102 \leq 60 \quad (\text{Not valid})
\]

### Viable Solutions:
The points that satisfy all constraints are:
- **(2, 4)**
- **(1, 4.5)**

Thus, the viable solutions from the provided options are **(2, 4)** and **(1, 4.5)**.