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Zorluk: ZorPositive and Negative Number Properties

If aa, bb, and cc are non-zero real numbers such that a3b2c<0a^3 b^2 c < 0, ab3>0a b^3 > 0, and ac<0\frac{a}{c} < 0, which of the following expressions MUST be positive?

  1. b−ca\frac{b - c}{a}Cevap
  2. B
    abc\frac{a b}{c}
  3. C
    c(a+b)c(a + b)
  4. D
    a+cb\frac{a + c}{b}
  5. E
    b(c−a)b(c - a)

Cevap

b−ca\frac{b - c}{a}
Deducing the signs of aa, bb, and cc reveals two possible cases: either a>0,b>0,c<0a > 0, b > 0, c < 0 or a<0,b<0,c>0a < 0, b < 0, c > 0. In Case 1, b−c>0b - c > 0 and a>0a > 0, so b−ca>0\frac{b - c}{a} > 0. In Case 2, b−c<0b - c < 0 and a<0a < 0, so b−ca>0\frac{b - c}{a} > 0. Therefore, b−ca\frac{b - c}{a} is strictly positive in all cases.

Adım Adım Çözüm

1
Analyze the sign conditions from the given inequalities.
From ab3>0a b^3 > 0, aa and bb must have the same sign. From ac<0\frac{a}{c} < 0, aa and cc must have opposite signs. From a3b2c<0a^3 b^2 c < 0, since b2>0b^2 > 0 for non-zero bb, we have a3c<0a^3 c < 0, which confirms aa and cc have opposite signs.
Odd powers preserve the sign of a variable, whereas even powers are strictly positive for non-zero real numbers.
2
Determine the two possible sign scenarios for (a,b,c)(a, b, c).
Case 1: a>0,b>0,c<0a > 0, b > 0, c < 0.
Case 2: a<0,b<0,c>0a < 0, b < 0, c > 0.
Since aa and bb share the same sign and cc has the opposite sign, these are the only two valid assignments.
3
Evaluate the sign of the numerator and denominator of b−ca\frac{b - c}{a} in Case 1.
In Case 1 (a>0,b>0,c<0a > 0, b > 0, c < 0): b−c=positive−negative=positiveb - c = \text{positive} - \text{negative} = \text{positive}. Denominator a>0a > 0. Ratio positivepositive>0\frac{\text{positive}}{\text{positive}} > 0.
Subtracting a negative number from a positive number yields a positive result.
4
Evaluate the sign of the numerator and denominator of b−ca\frac{b - c}{a} in Case 2.
In Case 2 (a<0,b<0,c>0a < 0, b < 0, c > 0): b−c=negative−positive=negativeb - c = \text{negative} - \text{positive} = \text{negative}. Denominator a<0a < 0. Ratio negativenegative>0\frac{\text{negative}}{\text{negative}} > 0.
Dividing two negative values produces a positive quotient.

Anahtar Kavram

Positive and Negative Number Properties in Inequalities
Tahmini Süre:2m 0s
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Positive and Negative Number Properties

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