To formulate a two-variable inequality based on Kiani's budget for buying hardbacks and paperbacks, we can let:
- \( x \) = the number of hardback books
- \( y \) = the number of paperback books
The cost of the hardback books is \( 12x \) dollars (since each hardback costs $12), and the cost of the paperback books is \( 5y \) dollars (since each paperback costs $5). Kiani's total expenditure must not exceed the budget of $60. Thus, the inequality representing this situation is:
\[ 12x + 5y \leq 60 \]
Now, we need to determine the constraints of the problem. Kiani cannot buy a negative number of books, so we have the following non-negative constraints:
\[ x \geq 0 \quad \text{(hardbacks)} \] \[ y \geq 0 \quad \text{(paperbacks)} \]
Therefore, the complete set of constraints is:
- \( 12x + 5y \leq 60 \)
- \( x \geq 0 \)
- \( y \geq 0 \)
Next, we will evaluate the given points to see which ones satisfy these constraints.
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\( (2, 4) \):
- \( 12(2) + 5(4) = 24 + 20 = 44 \) (which is ≤ 60)
- \( 2 \geq 0 \) (true)
- \( 4 \geq 0 \) (true)
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\( (6, 6) \):
- \( 12(6) + 5(6) = 72 + 30 = 102 \) (which is not ≤ 60)
- Not a viable solution.
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\( (-2, 6) \):
- \( -2 \) is not allowed since \( x \) must be ≥ 0.
- Not a viable solution.
-
\( (1, 4.5) \):
- \( 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \) (which is ≤ 60)
- \( 1 \geq 0 \) (true)
- \( 4.5 \geq 0 \) (true)
From this analysis, the viable solutions from the given points are \( (2, 4) \) and \( (1, 4.5) \). However, only one point can be chosen in your response.
The two points \( (2, 4) \) and \( (1, 4.5) \) both meet the constraints defined, but according to your responses, giving preference to the first viable solution yielded from the list:
The final answer as a viable solution is: (2, 4)
2 answers