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

(1,4.5)
left parenthesis 1 comma 4.5 right parenthesis

(6,6)
left parenthesis 6 comma 6 right parenthesis

(2,4)
left parenthesis 2 comma 4 right parenthesis

(−2,6)

Answer 1

Let's define the variables for the problem:

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

The cost of hardbacks is $12 each, and the cost of paperbacks is $5 each. Kiani has a budget of $60. This can be expressed as the following inequality:

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

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

1. Non-negativity constraints: Since Kiani cannot purchase a negative number of books, we have:
- \( x \geq 0 \)
- \( y \geq 0 \)

Now we have the following system of inequalities:

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

Next, we can evaluate the proposed points to see which one satisfies all the inequalities.

1. **Point (1, 4.5)**:
\[
12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad (\text{True: } 34.5 \leq 60)
\]
Non-negativity: \( x = 1 \geq 0, \, y = 4.5 \geq 0 \quad \text{(True)}\)

2. **Point (6, 6)**:
\[
12(6) + 5(6) = 72 + 30 = 102 \quad (\text{False: } 102 \leq 60)
\]
Non-negativity: \( x = 6 \geq 0, \, y = 6 \geq 0 \quad \text{(True)}\)

3. **Point (2, 4)**:
\[
12(2) + 5(4) = 24 + 20 = 44 \quad (\text{True: } 44 \leq 60)
\]
Non-negativity: \( x = 2 \geq 0, \, y = 4 \geq 0 \quad \text{(True)}\)

4. **Point (−2, 6)**:
\[
12(-2) + 5(6) = -24 + 30 = 6 \quad (\text{True: } 6 \leq 60)
\]
Non-negativity: \( x = -2 < 0 \quad \text{(False)}\)

After evaluating all the points, the viable solutions are:

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

The only point listed as an option that represents a viable solution is:

**(2, 4)**.