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) Responses (6,6) left parenthesis 6 comma 6 right parenthesis (−2,6) left parenthesis negative 2 comma 6 right parenthesis (2,4) left parenthesis 2 comma 4 right parenthesis (1,4.5) left parenthesis 1 comma 4.5 right parenthesis

Answer 1

Let's assign variables to represent the books Kiani is buying:

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

The cost of hardbacks is $12 each and the cost of paperbacks is $5 each. Kiani has a budget of $60. Therefore, we can write the inequality representing Kiani's budget:

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

Now, we also need to consider the constraints of the problem. Since Kiani can't buy a negative number of books, we need to impose the following conditions:

1. \( x \geq 0 \) (Kiani cannot buy a negative number of hardbacks)
2. \( y \geq 0 \) (Kiani cannot buy a negative number of paperbacks)

So the system of inequalities that represents the constraints of the problem is:

\[
\begin{align*}
12x + 5y & \leq 60 \\
x & \geq 0 \\
y & \geq 0
\end{align*}
\]

Next, let’s determine which of the provided points represent a viable solution by substituting each point into the inequality \( 12x + 5y \leq 60 \):

1. **Point (6, 6)**:
\[
12(6) + 5(6) = 72 + 30 = 102 \quad (\text{not valid since } 102 \nleq 60)
\]

2. **Point (-2, 6)**:
\[
12(-2) + 5(6) = -24 + 30 = 6 \quad (\text{not valid since } -2 < 0)
\]

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

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

From the above calculations, the points (2, 4) and (1, 4.5) are viable solutions. However, since you only need to identify one viable solution among the options given, both (2, 4) and (1, 4.5) are correct answers, while (6, 6) and (-2, 6) do not satisfy the constraints.

In conclusion, the viable solutions from the options are:
- (2, 4)
- (1, 4.5)

The point (2, 4) is more straightforward as it represents whole numbers of books, making it a better solution in the context of shopping for books, assuming Kiani prefers to buy whole items only.