Examkin

Question

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

If mm and nn are real numbers, is m2<n2m^2 < n^2?

(1) ∣m∣+n<0|m| + n < 0
(2) m+n>0m + n > 0

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  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. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Rephrased, the question asks whether ∣m∣<∣n∣|m| < |n|. Evaluating the first condition gives n<−∣m∣n < -|m|, which shows that nn must be negative and its absolute value ∣n∣=−n|n| = -n must be strictly greater than ∣m∣|m|. Hence, m2<n2m^2 < n^2 is definitely true. Evaluating the second condition allows both m=2,n=5m=2, n=5 (where m2<n2m^2 < n^2) and m=5,n=2m=5, n=2 (where m2>n2m^2 > n^2), making it insufficient. Therefore, the first condition alone is sufficient while the second condition alone is not.

Step-by-Step Solution

1
Rephrase the target question stem m2<n2m^2 < n^2.
The target condition m2<n2m^2 < n^2 is equivalent to ∣m∣2<∣n∣2|m|^2 < |n|^2, which is true if and only if ∣m∣<∣n∣|m| < |n|.
Since both m2m^2 and n2n^2 are non-negative, taking the square root of both sides preserves the inequality order for non-negative magnitudes.
2
Evaluate Statement (1): ∣m∣+n<0|m| + n < 0.
Rearranging gives n<−∣m∣n < -|m|. Since ∣m∣≥0|m| \ge 0, this implies nn is strictly negative. Taking absolute values of both sides of n<−∣m∣n < -|m| gives ∣n∣=−n>∣m∣|n| = -n > |m|. Therefore, ∣n∣>∣m∣|n| > |m|, which means n2>m2n^2 > m^2 or m2<n2m^2 < n^2.
Statement (1) yields a definitive 'Yes' to the target question. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): m+n>0m + n > 0.
Case 1: Let m=2m = 2 and n=5n = 5. Then 2+5=7>02 + 5 = 7 > 0, and 22=4<25=522^2 = 4 < 25 = 5^2 (Yes). Case 2: Let m=5m = 5 and n=2n = 2. Then 5+2=7>05 + 2 = 7 > 0, but 52=25>4=225^2 = 25 > 4 = 2^2 (No).
Because Statement (2) allows both 'Yes' and 'No' outcomes, Statement (2) alone is not sufficient.

Key Concept

Data Sufficiency evaluation of absolute values and algebraic inequalities
Rate this question

Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

More questions from this topic