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Difficulty: MediumPositive and Negative Number Properties

If aa, bb, and cc are non-zero real numbers such that ab2c3<0a b^2 c^3 < 0 and ab>0\frac{a}{b} > 0, which of the following expressions MUST be negative?

  1. a2bca^2 b cAnswer
  2. B
    abca b c
  3. C
    abc2a b c^2
  4. D
    abc2\frac{a}{b c^2}
  5. E
    acb\frac{a c}{b}

Answer

The expression a2bca^2 b c must be negative.
For any non-zero real number aa, the term a2a^2 is strictly positive. The given conditions show that b2>0b^2 > 0, which simplifies ab2c3<0a b^2 c^3 < 0 to ac<0a c < 0, proving aa and cc have opposite signs. Furthermore, ab>0\frac{a}{b} > 0 means aa and bb have the same sign. Combining these facts shows that bb and cc must always have opposite signs, so bc<0b c < 0. Therefore, a2bc=(a2)(bc)=(+)×(−)<0a^2 b c = (a^2)(b c) = (+) \times (-) < 0 in all valid cases.

Step-by-Step Solution

1
Analyze the sign implication of ab2c3<0a b^2 c^3 < 0.
ac<0a c < 0, meaning aa and cc have opposite signs.
Because b≠0b \neq 0, b2b^2 is strictly positive (b2>0b^2 > 0). Dividing ab2c3<0a b^2 c^3 < 0 by b2b^2 yields ac3<0a c^3 < 0. Since c3c^3 has the exact same sign as cc, it follows that ac<0a c < 0.
2
Analyze the sign implication of ab>0\frac{a}{b} > 0.
aa and bb must have the same sign (ab>0a b > 0).
A quotient of two non-zero numbers is positive if and only if both numbers have the same sign.
3
Combine sign conditions to evaluate the possible cases for a,b,ca, b, c.
Either Case 1: a>0,b>0,c<0a > 0, b > 0, c < 0, or Case 2: a<0,b<0,c>0a < 0, b < 0, c > 0.
If a>0a > 0, then b>0b > 0 (from step 2) and c<0c < 0 (from step 1). If a<0a < 0, then b<0b < 0 (from step 2) and c>0c > 0 (from step 1).
4
Test the expression a2bca^2 b c under both cases.
In Case 1: (+)2⋅(+)⋅(−)=(−)(+)^2 \cdot (+) \cdot (-) = (-). In Case 2: (−)2⋅(−)⋅(+)=(−)(-)^2 \cdot (-) \cdot (+) = (-). Thus a2bc<0a^2 b c < 0 always.
Since a2>0a^2 > 0 for all non-zero real numbers aa, and bb and cc have opposite signs in every valid scenario (bc<0b c < 0), the product a2(bc)a^2 (b c) is always strictly negative.

Key Concept

Deducing signs of variables using exponent rules and inequalities
Estimated Time:1m 30s
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Topics

Positive and Negative Number Properties

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