Question

Kiana is shopping at a bookstore with a budget of $60. Hardbacks cost $12 and paperbacks are $5 write a two-variable inequality and determine the constraints of the problem
(1, 4.5)
(6,6)
(2,4)
(-2,6)

Answer 1

To write a two-variable inequality for Kiana's situation at the bookstore, let's define our variables:

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

The cost of the hardbacks is $12 each, and the cost of the paperbacks is $5 each. Given that Kiana has a budget of $60, we can express this situation with the following inequality:

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

This inequality represents the constraint that the total cost of the books cannot exceed her budget of $60.

### Constraints of the Problem

1. **Non-negativity Constraints**: Since Kiana cannot buy a negative number of books, we have:
- \( x \geq 0 \) (she cannot buy a negative number of hardbacks)
- \( y \geq 0 \) (she cannot buy a negative number of paperbacks)

Putting it all together, the constraints of the problem can be summarized as:
\[
12x + 5y \leq 60
\]
\[
x \geq 0
\]
\[
y \geq 0
\]

### Evaluating the Given Points

Now, let's evaluate the given points to determine if they satisfy the inequality \( 12x + 5y \leq 60 \):

1. **Point (1, 4.5)**:
\[
12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad (\text{Satisfies } 34.5 \leq 60)
\]

2. **Point (6, 6)**:
\[
12(6) + 5(6) = 72 + 30 = 102 \quad (\text{Does not satisfy } 102 \leq 60)
\]

3. **Point (2, 4)**:
\[
12(2) + 5(4) = 24 + 20 = 44 \quad (\text{Satisfies } 44 \leq 60)
\]

4. **Point (-2, 6)**:
\[
\text{(Not applicable since } x \geq 0)
\]

### Summary

- The valid points that satisfy the budget constraint \( 12x + 5y \leq 60 \) are **(1, 4.5)** and **(2, 4)**.
- The point **(6, 6)** does not satisfy the budget constraint.
- The point **(-2, 6)** is invalid since the number of hardbacks cannot be negative.

This completes the evaluation and determination of the constraints for Kiana's bookstore shopping situation.