Kiana 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?

Question

(2, 4)
(1, 4.5)
(6, 6)
(-2, 6)

Answer 1

To formulate the two-variable inequality representing Kiana's budget situation, let's define the variables:

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

The cost of the hardbacks is $12 each, and the cost of the paperbacks is $5 each. Kiana's budget constraint can then be expressed by the following inequality:

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

Next, we should define the constraints of the problem further:

1. Non-negativity Constraints: Since Kiana can't buy a negative number of books, we have: \[ x \geq 0 \] \[ y \geq 0 \]

2. Cost Constraint: As indicated above: \[ 12x + 5y \leq 60 \]

Now we need to evaluate the given points to check if they meet these constraints. We'll plug each point into the budget inequality \( 12x + 5y \leq 60 \) and also check for non-negativity.

1. Point (2, 4): \[ 12(2) + 5(4) = 24 + 20 = 44 \quad (44 \leq 60) \text{ (Valid)} \] \(\Rightarrow x=2 \geq 0\) and \(y=4 \geq 0\)

2. Point (1, 4.5): \[ 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad (34.5 \leq 60) \text{ (Valid)} \] \(\Rightarrow x=1 \geq 0\) and \(y=4.5 \geq 0\)

3. Point (6, 6): \[ 12(6) + 5(6) = 72 + 30 = 102 \quad (102 \leq 60) \text{ (Invalid)} \] \(\Rightarrow x=6 \geq 0\) and \(y=6 \geq 0\)

4. Point (-2, 6): \[ 12(-2) + 5(6) = -24 + 30 = 6 \quad (6 \leq 60) \text{ (Valid for cost)} \] However, \(-2 < 0\) is not a valid solution for \(x\).

From this analysis, the viable solutions are (2, 4) and (1, 4.5).

The only points that represent viable solutions to Kiana's problem are (2, 4) and (1, 4.5).