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Zorluk: OrtaInequalities, 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.Cevap
  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.

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Data Sufficiency evaluation of absolute values and algebraic inequalities
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Konular

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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