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