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:
- \( x \geq 0 \) (since Kiani cannot buy a negative number of hardbacks)
- \( y \geq 0 \) (since Kiani cannot buy a negative number of paperbacks)
Now we have the set of inequalities:
- \( 12x + 5y \leq 60 \)
- \( x \geq 0 \)
- \( y \geq 0 \)
Next, let's evaluate the given points to see which one represents a viable solution within these constraints:
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Point (2, 4): \[ 12(2) + 5(4) = 24 + 20 = 44 \quad (44 \leq 60) \text{ (valid)} \]
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Point (6, 6): \[ 12(6) + 5(6) = 72 + 30 = 102 \quad (102 \leq 60) \text{ (invalid)} \]
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Point (−2, 6): Since \( x \) cannot be negative, this point is invalid.
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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).