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)

Question

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).