Examkin

Question

Difficulty: HardPositive and Negative Number Properties

If aa, bb, and cc are non-zero real numbers such that a2b3c<0a^2 b^3 c < 0, ab5c2>0\frac{a}{b^5 c^2} > 0, and ac>bca c > b c, which of the following expressions MUST be true?

  1. A
    a>ba > b
  2. B
    ac>0a c > 0
  3. b−ac<0\frac{b - a}{c} < 0Answer
  4. D
    b+c>0b + c > 0
  5. E
    ac>0\frac{a}{c} > 0

Answer

The expression \(\frac{b - a}{c} < 0\) MUST be true.
The condition a2b3c<0a^2 b^3 c < 0 requires bc<0b c < 0 because a2>0a^2 > 0. The condition ab5c2>0\frac{a}{b^5 c^2} > 0 requires ab>0a b > 0 because c2>0c^2 > 0. Combining these shows that aa and bb share the same sign, whereas cc has the opposite sign. Consequently, ac<0a c < 0. From ac>bca c > b c, subtracting bcb c yields (a−b)c>0(a - b) c > 0. Multiplying by −1-1 gives (b−a)c<0(b - a) c < 0, and dividing by c2>0c^2 > 0 produces b−ac<0\frac{b - a}{c} < 0, which MUST be true in all cases.

Step-by-Step Solution

1
Determine the relative signs of bb and cc using a2b3c<0a^2 b^3 c < 0.
bc<0b c < 0, meaning bb and cc have opposite signs.
Since a≠0a \neq 0, a2>0a^2 > 0 always. Dividing a2b3c<0a^2 b^3 c < 0 by a2a^2 yields b3c<0b^3 c < 0. Because b3b^3 has the same sign as bb, bc<0b c < 0.
2
Determine the relative signs of aa and bb using ab5c2>0\frac{a}{b^5 c^2} > 0.
ab>0a b > 0, meaning aa and bb have the same sign.
Since c≠0c \neq 0, c2>0c^2 > 0 always. Multiplying by c2c^2 gives ab5>0\frac{a}{b^5} > 0, which implies aa and b5b^5 have the same sign. Thus ab>0a b > 0.
3
Determine the relationship between aa and cc.
ac<0a c < 0, meaning aa and cc have opposite signs.
Since aa and bb have the same sign (ab>0a b > 0) while bb and cc have opposite signs (bc<0b c < 0), aa and cc must have opposite signs.
4
Analyze the inequality ac>bca c > b c.
\(\frac{b - a}{c} < 0\)
Rearranging ac>bca c > b c gives ac−bc>0  ⟹  (a−b)c>0a c - b c > 0 \implies (a - b) c > 0. Multiplying both sides by −1-1 flips the inequality: (b−a)c<0(b - a) c < 0. Dividing by the strictly positive quantity c2c^2 yields (b−a)cc2<0  ⟹  b−ac<0\frac{(b - a) c}{c^2} < 0 \implies \frac{b - a}{c} < 0.

Key Concept

Deducing sign relationships of variables in inequalities and algebraic transformations without assuming positive signs.
Estimated Time:2m 0s
Rate this question

Topics

Positive and Negative Number Properties

More questions from this topic