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

If rr is a real number, is r3<rr^3 < r?

(1) ∣r−0.5∣<0.5|r - 0.5| < 0.5
(2) ∣r∣<1|r| < 1

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

  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.
Rephrasing r3<rr^3 < r shows that the inequality holds true if and only if r<−1r < -1 or 0<r<10 < r < 1. Statement (1) bounds rr precisely within the interval (0,1)(0, 1), ensuring a definitive 'Yes' answer. Statement (2) defines the range (−1,1)(-1, 1), which includes values where the inequality holds (such as 0.50.5) and values where it fails (such as −0.5-0.5), making it insufficient.

Step-by-Step Solution

1
Rephrase the target inequality algebraically.
The inequality r3<rr^3 < r rearranges to r3−r<0r^3 - r < 0, which factors as r(r−1)(r+1)<0r(r - 1)(r + 1) < 0.
Simplifying the stem isolates the specific intervals on the number line where the statement holds true.
2
Determine the valid number ranges for the rephrased question.
The expression r(r−1)(r+1)r(r - 1)(r + 1) is strictly negative when r<−1r < -1 or when 0<r<10 < r < 1.
Testing sign changes across key boundary points (r=−1r = -1, r=0r = 0, r=1r = 1) identifies the target ranges.
3
Evaluate Statement (1): ∣r−0.5∣<0.5|r - 0.5| < 0.5.
This absolute value inequality expands to −0.5<r−0.5<0.5-0.5 < r - 0.5 < 0.5, which simplifies to 0<r<10 < r < 1.
Since every value in the range 0<r<10 < r < 1 satisfies r3<rr^3 < r, Statement (1) yields a definitive 'Yes'. Thus, Statement (1) alone is sufficient.
4
Evaluate Statement (2): ∣r∣<1|r| < 1.
This absolute value inequality simplifies to −1<r<1-1 < r < 1.
If r=0.5r = 0.5, then r3=0.125<0.5r^3 = 0.125 < 0.5 (Yes). However, if r=−0.5r = -0.5, then r3=−0.125>−0.5r^3 = -0.125 > -0.5 (No). Because Statement (2) allows both 'Yes' and 'No' outcomes, it is not sufficient.

Key Concept

Data Sufficiency evaluation of cubic inequalities and absolute value range constraints
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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