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

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

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

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

(−2,6)

Answer 1

Let's define the variables:

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

Considering Kiani's budget of $60, we can formulate the following inequality:

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

Now we can analyze the constraints of the problem:

1. Non-negativity constraints: Kiani cannot buy a negative number of books, so: \[ x \geq 0 \] \[ y \geq 0 \]

2. Budget constraint: Already represented by the inequality \( 12x + 5y \leq 60 \).

Now let's evaluate the provided points to determine which one satisfies all the constraints:

1. Point (6, 6):

   • Calculate: \( 12(6) + 5(6) = 72 + 30 = 102 \), which does not satisfy \( 12x + 5y \leq 60 \).

2. Point (2, 4):

   • Calculate: \( 12(2) + 5(4) = 24 + 20 = 44 \), which satisfies \( 12x + 5y \leq 60 \), and \( x \geq 0 \) and \( y \geq 0 \).

3. Point (1, 4.5):

   • Calculate: \( 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \), which satisfies \( 12x + 5y \leq 60 \), and \( x \geq 0 \) but \( y \geq 0 \).

4. Point (−2, 6):

   • This point cannot satisfy the non-negativity constraint \( x \geq 0 \).

Based on this evaluation, the only viable solution from the provided points is:

(2, 4).