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

If kk is a real number, is ∣k2−4∣<3k|k^2 - 4| < 3k?

(1) k2−5k+4<0k^2 - 5k + 4 < 0
(2) ∣k−1∣<2|k - 1| < 2

Which statement(s) provide sufficient information to determine whether the inequality holds?

  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.
The stem inequality ∣k2−4∣<3k|k^2 - 4| < 3k simplifies to 1<k<41 < k < 4. Statement (1) directly yields 1<k<41 < k < 4, providing a definitive Yes answer. Statement (2) yields −1<k<3-1 < k < 3, which includes values inside (1,4)(1, 4) such as k=2k = 2 (Yes) and values outside (1,4)(1, 4) such as k=0.5k = 0.5 (No), making Statement (2) insufficient.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
The target inequality ∣k2−4∣<3k|k^2 - 4| < 3k requires 3k>03k > 0 (so k>0k > 0) and translates to −3k<k2−4<3k-3k < k^2 - 4 < 3k. Solving k2−4<3kk^2 - 4 < 3k yields (k−4)(k+1)<0  ⟹  −1<k<4(k - 4)(k + 1) < 0 \implies -1 < k < 4. Solving −3k<k2−4-3k < k^2 - 4 for k>0k > 0 yields k2+3k−4>0  ⟹  (k+4)(k−1)>0  ⟹  k>1k^2 + 3k - 4 > 0 \implies (k + 4)(k - 1) > 0 \implies k > 1. Combining these gives the rephrased question: 'Is 1<k<41 < k < 4?'
Simplifying the question stem before evaluating the statements prevents misinterpretation of absolute value boundary conditions.
2
Evaluate Statement (1): k2−5k+4<0k^2 - 5k + 4 < 0.
Factoring the quadratic gives (k−1)(k−4)<0(k - 1)(k - 4) < 0, which means 1<k<41 < k < 4.
Since Statement (1) guarantees that kk lies strictly between 11 and 44, the answer to the rephrased question is a definitive Yes. Statement (1) ALONE is sufficient.
3
Evaluate Statement (2): ∣k−1∣<2|k - 1| < 2.
Removing the absolute value gives −2<k−1<2-2 < k - 1 < 2, or −1<k<3-1 < k < 3.
If k=2k = 2, then 1<2<41 < 2 < 4, giving a Yes answer. If k=0.5k = 0.5, then 0.50.5 is in (−1,3)(-1, 3) but not in (1,4)(1, 4), giving a No answer (since ∣0.52−4∣=3.75>3(0.5)=1.5|0.5^2 - 4| = 3.75 > 3(0.5) = 1.5). Because Statement (2) allows both Yes and No outcomes, it is NOT sufficient.

Key Concept

Absolute Value Inequalities and Question Stem Rephrasing

Alternative Method

Instead of solving the quadratic inequality −3k<k2−4<3k-3k < k^2 - 4 < 3k algebraically, test the boundary points k=1k = 1 and k=4k = 4 in ∣k2−4∣=3k|k^2 - 4| = 3k: at k=1k = 1, ∣1−4∣=3=3(1)|1 - 4| = 3 = 3(1); at k=4k = 4, ∣16−4∣=12=3(4)|16 - 4| = 12 = 3(4). Since 3k3k grows linearly while k2−4k^2 - 4 grows quadratically, the inequality holds strictly between these two positive roots 1<k<41 < k < 4.
Estimated Time:2m 0s
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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