Inequalities in Two Variables Quick Check

To model Kiani's shopping situation with a two-variable inequality, we can define the variables as follows:

Let:
- \( x \) = number of hardbacks
- \( y \) = number of paperbacks

Given the costs:
- Hardbacks cost $12 each
- Paperbacks cost $5 each

Kiani's budget constraint can be expressed as:

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

Next, we must determine the constraints of the problem, which include:
1. The non-negativity constraints since Kiani cannot buy a negative number of books:
\[
x \geq 0
\]
\[
y \geq 0
\]

Now we will evaluate the provided points to see which represent viable solutions to the inequality \( 12x + 5y \leq 60 \).

1. **Point \( (-2, 6) \)**:
\[
12(-2) + 5(6) = -24 + 30 = 6 \quad \text{(Valid, but } x \text{ is negative)}
\]
**Not a viable solution.**

2. **Point \( (1, 4.5) \)**:
\[
12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad \text{(Valid)}
\]
**Viable solution.**

3. **Point \( (2, 4) \)**:
\[
12(2) + 5(4) = 24 + 20 = 44 \quad \text{(Valid)}
\]
**Viable solution.**

4. **Point \( (6, 6) \)**:
\[
12(6) + 5(6) = 72 + 30 = 102 \quad \text{(Not valid, exceeds budget)}
\]
**Not a viable solution.**

From the checks above, the viable points are \( (1, 4.5) \) and \( (2, 4) \).

**Conclusion**: The answer to the question is that the points \( (1, 4.5) \) and \( (2, 4) \) are viable solutions, but among the options listed, only \( (1, 4.5) \) explicitly shows as a viable solution adhering to the constraints.