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Difficulty: HardInequalities, Absolute Values, and Number Ranges in Data Sufficiency

If mm and nn are non-zero real numbers, is ∣m+n∣m>1\frac{|m + n|}{m} > 1?

(1) m<−∣n∣m < -|n|
(2) mn>0mn > 0 and ∣m+n∣=m+n|m + n| = m + n

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. EACH statement ALONE is sufficient.Answer
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

EACH statement ALONE is sufficient.
The correct choice is that EACH statement ALONE is sufficient. Statement (1) restricts mm to negative values, guaranteeing that the numerator is non-negative while the denominator is negative, so the ratio can never exceed 1 (yielding a definitive 'No'). Statement (2) forces both mm and nn to be strictly positive, which simplifies the ratio to 1+nm>11 + \frac{n}{m} > 1 (yielding a definitive 'Yes'). Since each statement provides a definitive answer independently, each statement alone is sufficient.

Step-by-Step Solution

1
Analyze the question stem target ∣m+n∣m>1\frac{|m + n|}{m} > 1.
Since absolute value ∣m+n∣|m + n| is always non-negative (∣m+n∣≥0|m + n| \ge 0), if m<0m < 0, the ratio ∣m+n∣m≤0\frac{|m + n|}{m} \le 0, which can never be greater than 1. Thus, if m<0m < 0, the answer is a definitive 'No'. If m>0m > 0, the expression simplifies to ∣m+n∣>m|m + n| > m.
Establishing the behavior of the expression based on the sign of mm simplifies statement evaluation.
2
Evaluate Statement (1): m<−∣n∣m < -|n|.
Since n≠0n \neq 0, ∣n∣>0|n| > 0, which implies −∣n∣<0-|n| < 0. Therefore, m<0m < 0. Because m<0m < 0 and ∣m+n∣≥0|m + n| \ge 0, the fraction ∣m+n∣m\frac{|m + n|}{m} is non-positive (≤0\le 0). Thus, ∣m+n∣m\frac{|m + n|}{m} cannot be greater than 1, yielding a definitive 'No'. Statement (1) ALONE is sufficient.
A statement that yields a definitive 'No' to a Yes/No Data Sufficiency question is sufficient.
3
Evaluate Statement (2): mn>0mn > 0 and ∣m+n∣=m+n|m + n| = m + n.
From mn>0mn > 0, mm and nn must have the same sign (both positive or both negative). The condition ∣m+n∣=m+n|m + n| = m + n requires m+n≥0m + n \ge 0. If both mm and nn were negative, m+nm + n would be negative, contradicting ∣m+n∣≥0|m + n| \ge 0. Hence, both mm and nn must be positive (m>0m > 0 and n>0n > 0). Substituting into the expression: ∣m+n∣m=m+nm=1+nm\frac{|m + n|}{m} = \frac{m + n}{m} = 1 + \frac{n}{m}. Since n>0n > 0 and m>0m > 0, nm>0\frac{n}{m} > 0, so 1+nm>11 + \frac{n}{m} > 1. This yields a definitive 'Yes'. Statement (2) ALONE is sufficient.
Determining that mm and nn must both be positive allows exact evaluation of the inequality.

Key Concept

Data Sufficiency logic for Yes/No questions involving absolute value properties and variable sign analysis.
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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