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
- 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 \( 12x + 5y \leq 60 \):
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Point (1, 4.5): \[ 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad (\text{Satisfies } 34.5 \leq 60) \]
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Point (6, 6): \[ 12(6) + 5(6) = 72 + 30 = 102 \quad (\text{Does not satisfy } 102 \leq 60) \]
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Point (2, 4): \[ 12(2) + 5(4) = 24 + 20 = 44 \quad (\text{Satisfies } 44 \leq 60) \]
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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.
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