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

Answer 1

Let's define the variables:

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

Given the costs:
- Each hardback costs $12.
- Each paperback costs $5.

Kiani's budget constraint can be expressed with the inequality:

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

In addition to this inequality, we also need to acknowledge that the number of books cannot be negative, giving us the following constraints:

1. \( x \geq 0 \) (cannot purchase a negative number of hardbacks)
2. \( y \geq 0 \) (cannot purchase a negative number of paperbacks)

So, the complete set of constraints is:

1. \( 12x + 5y \leq 60 \)
2. \( x \geq 0 \)
3. \( y \geq 0 \)

Now we will test each point to see if it satisfies the inequality \( 12x + 5y \leq 60 \) and the non-negativity constraints \( x \geq 0 \), \( y \geq 0 \).

**A. (6,6)**
\[
12(6) + 5(6) = 72 + 30 = 102 \quad (\text{not } \leq 60)
\]
This point does not satisfy the inequality.

**B. (1,4.5)**
\[
12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad (\text{is } \leq 60)
\]
This point satisfies the inequality. Also, \( 1 \geq 0 \) and \( 4.5 \geq 0 \).

**C. (2,4)**
\[
12(2) + 5(4) = 24 + 20 = 44 \quad (\text{is } \leq 60)
\]
This point satisfies the inequality. Also, \( 2 \geq 0 \) and \( 4 \geq 0 \).

**D. (-2,6)**
\[
12(-2) + 5(6) = -24 + 30 = 6 \quad (\text{is } \leq 60)
\]
While this point satisfies the inequality, \( -2 \) is not greater than or equal to 0, so it does not satisfy the non-negativity constraint.

**Viable Solutions:**
- Points B (1,4.5) and C (2,4) satisfy both the budget constraint and the non-negativity constraints.

Thus, the viable solutions are **B. (1,4.5)** and **C. (2,4)**.