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?
A. (2, 4)
B. (-2, 6)
C. (6, 6)
D. (1, 4.5)

Answer 1

Let \( x \) represent the number of hardbacks that Kiani buys, and \( y \) represent the number of paperbacks. The cost of hardbacks is $12 each, while paperbacks cost $5 each. Given Kiani's budget of $60, the inequality that represents this situation is:

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

**Constraints of the problem:**

1. **Non-negativity constraint:** Since Kiani cannot buy a negative number of books:
\[
x \geq 0
\]
\[
y \geq 0
\]

2. **Budget constraint:** We already established this with:
\[
12x + 5y \leq 60
\]

Now, we will check each of the given points to see if they satisfy these constraints:

**A. (2, 4)**
\[
x = 2, \quad y = 4
\]
Substituting into the budget equation:
\[
12(2) + 5(4) = 24 + 20 = 44 \leq 60 \quad \text{(satisfies the budget)}
\]
Both \( x \) and \( y \) are non-negative.
**Point A is viable.**

**B. (-2, 6)**
\[
x = -2, \quad y = 6
\]
\( x \) is negative, which violates the non-negativity constraint.
**Point B is not viable.**

**C. (6, 6)**
\[
x = 6, \quad y = 6
\]
Substituting into the budget equation:
\[
12(6) + 5(6) = 72 + 30 = 102 \nleq 60 \quad \text{(does not satisfy the budget)}
\]
Both \( x \) and \( y \) are non-negative.
**Point C is not viable.**

**D. (1, 4.5)**
\[
x = 1, \quad y = 4.5
\]
Substituting into the budget equation:
\[
12(1) + 5(4.5) = 12 + 22.5 = 34.5 \leq 60 \quad \text{(satisfies the budget)}
\]
Both \( x \) and \( y \) are non-negative.
**Point D is viable.**

**Conclusion:** The viable solutions from the given points are **A (2, 4)** and **D (1, 4.5)**.