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

If rr and ss are real numbers, is r∣s∣<s∣r∣r|s| < s|r|?

(1) rs<0rs < 0
(2) r+s>∣r−s∣r + s > |r - s|

Which of the following options correctly describes the sufficiency of the statements?

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Answer
  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 (2) ALONE is sufficient, but statement (1) alone is not sufficient.
The choice stating that Statement (2) ALONE is sufficient while Statement (1) alone is not sufficient is correct. Rephrased, r∣s∣<s∣r∣r|s| < s|r| is true only when r<0r < 0 and s>0s > 0. Statement (1) allows rr to be positive while ss is negative, yielding a 'No', as well as rr negative while ss is positive, yielding a 'Yes', so it is insufficient. Statement (2) simplifies algebraically to rs>0rs > 0 and r+s>0r + s > 0, which guarantees both numbers are positive. When both numbers are positive, r∣s∣<s∣r∣r|s| < s|r| reduces to rs<rsrs < rs, which is false. Because Statement (2) yields a definitive 'No', it is sufficient on its own.

Step-by-Step Solution

1
Rephrase the question stem target
The inequality r∣s∣<s∣r∣r|s| < s|r| holds if and only if r<0r < 0 and s>0s > 0.
If rr and ss have the same sign or either is zero, r∣s∣=s∣r∣r|s| = s|r|, making the strict inequality false. If r>0r > 0 and s<0s < 0, r∣s∣=−rsr|s| = -rs and s∣r∣=−rss|r| = -rs, making both sides equal. Only when r<0r < 0 and s>0s > 0 does r∣s∣=rs<0r|s| = rs < 0 and s∣r∣=−rs>0s|r| = -rs > 0, which makes the inequality true.
2
Evaluate Statement (1): rs<0rs < 0
Statement (1) is NOT sufficient.
This condition means rr and ss have opposite signs. Case 1: If r=−2r = -2 and s=3s = 3, then r<0r < 0 and s>0s > 0, so the answer is YES. Case 2: If r=3r = 3 and s=−2s = -2, then r>0r > 0 and s<0s < 0, so the answer is NO. Since both YES and NO are possible, Statement (1) is insufficient.
3
Evaluate Statement (2): r+s>∣r−s∣r + s > |r - s|
Statement (2) ALONE is sufficient.
Since ∣r−s∣≥0|r - s| \ge 0, we have r+s>0r + s > 0. Squaring both positive sides gives (r+s)2>(r−s)2  ⟹  r2+2rs+s2>r2−2rs+s2  ⟹  4rs>0  ⟹  rs>0(r + s)^2 > (r - s)^2 \implies r^2 + 2rs + s^2 > r^2 - 2rs + s^2 \implies 4rs > 0 \implies rs > 0. Since rs>0rs > 0 and r+s>0r + s > 0, both rr and ss must be positive. If r>0r > 0 and s>0s > 0, then r∣s∣=rsr|s| = rs and s∣r∣=rss|r| = rs, so r∣s∣<s∣r∣r|s| < s|r| simplifies to rs<rsrs < rs, which is FALSE. This gives a definitive NO to the question, making Statement (2) sufficient.

Key Concept

Data Sufficiency Yes/No Decision Logic with Absolute Value Inequalities
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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