Examkin

Question

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

If kk is a real number, is ∣k∣<4|k| < 4?

(1) k>−3k > -3
(2) k<3k < 3

Which of the following describes the sufficiency of the statements?

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Answer
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Rephrasing ∣k∣<4|k| < 4 gives −4<k<4-4 < k < 4. Neither statement alone provides both an upper and lower bound. However, combining Statement (1) (k>−3k > -3) and Statement (2) (k<3k < 3) establishes −3<k<3-3 < k < 3. Because the interval (−3,3)(-3, 3) is entirely contained inside (−4,4)(-4, 4), any value of kk satisfying both statements guarantees ∣k∣<4|k| < 4. Thus, both statements together are sufficient.

Step-by-Step Solution

1
Rephrase the question stem target
The target inequality ∣k∣<4|k| < 4 is equivalent to −4<k<4-4 < k < 4. The question asks whether kk lies strictly inside the range (−4,4)(-4, 4).
Simplifying an absolute value inequality into a compound inequality clarifies the exact target range.
2
Evaluate Statement (1) independently
Statement (1) specifies k>−3k > -3. If k=0k = 0, then ∣0∣=0<4|0| = 0 < 4 (Yes). If k=5k = 5, then ∣5∣=5≮4|5| = 5 \not< 4 (No).
Since Statement (1) yields both 'Yes' and 'No' answers, it is insufficient.
3
Evaluate Statement (2) independently
Statement (2) specifies k<3k < 3. If k=0k = 0, then ∣0∣=0<4|0| = 0 < 4 (Yes). If k=−5k = -5, then ∣−5∣=5≮4|-5| = 5 \not< 4 (No).
Since Statement (2) yields both 'Yes' and 'No' answers, it is insufficient.
4
Evaluate Statements (1) and (2) combined
Combining k>−3k > -3 and k<3k < 3 gives −3<k<3-3 < k < 3. Any value of kk in (−3,3)(-3, 3) automatically falls within (−4,4)(-4, 4), yielding a definitive 'Yes' to ∣k∣<4|k| < 4.
The interval (−3,3)(-3, 3) is a complete subset of (−4,4)(-4, 4), making the combined information sufficient.

Key Concept

Absolute Value Range Simplification and Subset Verification in Data Sufficiency
Rate this question

Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

More questions from this topic