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?(1 point) Responses (2,4) left parenthesis 2 comma 4 right parenthesis (6,6) left parenthesis 6 comma 6 right parenthesis (−2,6) left parenthesis negative 2 comma 6 right parenthesis (1,4.5) left parenthesis 1 comma 4.5 right parenthesis

Answer 1

To create a two-variable inequality for Kiani's budget when buying hardbacks and paperbacks, we will let:

- \( x \) = number of hardbacks
- \( y \) = number of paperbacks

The cost for hardbacks is $12 each and for paperbacks is $5 each. Kiani's total spending should not exceed her budget of $60. Therefore, we can express this as:

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

**Constraints of the problem:**
1. \( x \geq 0 \) (since Kiani cannot buy a negative number of hardbacks)
2. \( y \geq 0 \) (since Kiani cannot buy a negative number of paperbacks)

Now we have the set of inequalities:

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

Next, let's evaluate the given points to see which one represents a viable solution within these constraints:

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

- **Point (6, 6)**:
\[
12(6) + 5(6) = 72 + 30 = 102 \quad (102 \leq 60) \text{ (invalid)}
\]

- **Point (−2, 6)**:
Since \( x \) cannot be negative, this point is invalid.

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

**Valid Points**:
- (2, 4) is valid
- (1, 4.5) is valid

**Conclusion**: The viable solutions are (2, 4) and (1, 4.5). Thus, one viable solution point from the given options is:
- (2, 4) (this one specifically is a viable solution).