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Zorluk: OrtaPositive and Negative Number Properties

Let aa, bb, and cc be integers such that −5≤a≤3-5 \le a \le 3, −4≤b≤6-4 \le b \le 6, and −3≤c≤4-3 \le c \le 4. If ab<0a b < 0 and bc2>0b c^2 > 0, what is the maximum possible value of the expression a−cb\frac{a - c}{b}?

Cevap: 2

Cevap

The maximum possible value of the expression is 2.
From bc2>0b c^2 > 0, since c2>0c^2 > 0 for any non-zero real number cc, it follows that b>0b > 0 and c≠0c \neq 0. Given ab<0a b < 0 and b>0b > 0, aa must be strictly negative (a<0a < 0). To maximize a−cb\frac{a - c}{b} where b>0b > 0, we maximize the numerator (a−c)(a - c) and minimize the denominator bb. The maximum negative integer for aa is −1-1, and the minimum integer for cc is −3-3, giving a maximum numerator of −1−(−3)=2-1 - (-3) = 2. The minimum positive integer for bb is 11. Hence, the maximum value is 21=2\frac{2}{1} = 2.

Adım Adım Çözüm

1
Analyze the sign of bc2>0b c^2 > 0
b>0b > 0 and c≠0c \neq 0
The square of any non-zero real number is strictly positive (c2>0c^2 > 0), so bb must be positive for the product bc2b c^2 to be positive.
2
Analyze the sign of ab<0a b < 0
a<0a < 0
Since b>0b > 0, aa must be negative for their product to be less than zero.
3
Determine candidate integer sets
a∈{−5,−4,−3,−2,−1}a \in \{-5, -4, -3, -2, -1\}, b∈{1,2,3,4,5,6}b \in \{1, 2, 3, 4, 5, 6\}, c∈{−3,−2,−1,1,2,3,4}c \in \{-3, -2, -1, 1, 2, 3, 4\}
Filter the given intervals by the sign restrictions (a<0a < 0, b>0b > 0, c≠0c \neq 0).
4
Optimize numerator (a−c)(a - c) and denominator bb
a=−1a = -1, c=−3c = -3, b=1b = 1
To maximize a quotient with a positive denominator, make the numerator as large as possible (a−c=−1−(−3)=2a - c = -1 - (-3) = 2) and the denominator as small as possible (b=1b = 1).
5
Evaluate the maximum value
2
−1−(−3)1=21=2\frac{-1 - (-3)}{1} = \frac{2}{1} = 2.

Anahtar Kavram

Deducing sign constraints from products and powers of variables and optimizing algebraic quotients under sign and domain restrictions.
Tahmini Süre:1m 30s
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Konular

Positive and Negative Number Properties

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