Let's define the variables:
- Let \( x \) be the number of hardbacks purchased.
- Let \( y \) be the number of paperbacks purchased.
Given the costs:
- Each hardback costs $12.
- Each paperback costs $5.
Kiani's budget constraint can be expressed with the inequality:
\[ 12x + 5y \leq 60 \]
In addition to this inequality, we also need to acknowledge that the number of books cannot be negative, giving us the following constraints:
- \( x \geq 0 \) (cannot purchase a negative number of hardbacks)
- \( y \geq 0 \) (cannot purchase a negative number of paperbacks)
So, the complete set of constraints is:
- \( 12x + 5y \leq 60 \)
- \( x \geq 0 \)
- \( y \geq 0 \)
Now we will test each point to see if it satisfies the inequality \( 12x + 5y \leq 60 \) and the non-negativity constraints \( x \geq 0 \), \( y \geq 0 \).
A. (6,6) \[ 12(6) + 5(6) = 72 + 30 = 102 \quad (\text{not } \leq 60) \] This point does not satisfy the inequality.
B. (1,4.5) \[ 12(1) + 5(4.5) = 12 + 22.5 = 34.5 \quad (\text{is } \leq 60) \] This point satisfies the inequality. Also, \( 1 \geq 0 \) and \( 4.5 \geq 0 \).
C. (2,4) \[ 12(2) + 5(4) = 24 + 20 = 44 \quad (\text{is } \leq 60) \] This point satisfies the inequality. Also, \( 2 \geq 0 \) and \( 4 \geq 0 \).
D. (-2,6) \[ 12(-2) + 5(6) = -24 + 30 = 6 \quad (\text{is } \leq 60) \] While this point satisfies the inequality, \( -2 \) is not greater than or equal to 0, so it does not satisfy the non-negativity constraint.
Viable Solutions:
- Points B (1,4.5) and C (2,4) satisfy both the budget constraint and the non-negativity constraints.
Thus, the viable solutions are B. (1,4.5) and C. (2,4).
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