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

If xx is a real number, is ∣x2−4∣<5|x^2 - 4| < 5?

(1) ∣x−1∣<2|x - 1| < 2
(2) ∣x+1∣<3|x + 1| < 3

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 the question target ∣x2−4∣<5|x^2 - 4| < 5 gives −5<x2−4<5  ⟹  −1<x2<9-5 < x^2 - 4 < 5 \implies -1 < x^2 < 9. Since x2≥0x^2 \ge 0 for all real numbers, the condition reduces to x2<9x^2 < 9, or −3<x<3-3 < x < 3.

Statement (1) states ∣x−1∣<2|x - 1| < 2, which translates to −2<x−1<2  ⟹  −1<x<3-2 < x - 1 < 2 \implies -1 < x < 3. Any number xx in (−1,3)(-1, 3) automatically satisfies −3<x<3-3 < x < 3, giving a definitive 'Yes'. Hence, Statement (1) alone is sufficient.

Statement (2) states ∣x+1∣<3|x + 1| < 3, which translates to −3<x+1<3  ⟹  −4<x<2-3 < x + 1 < 3 \implies -4 < x < 2. If x=0x = 0, xx is in (−3,3)(-3, 3) ('Yes'); if x=−3.5x = -3.5, xx is not in (−3,3)(-3, 3) ('No'). Because both 'Yes' and 'No' are possible, Statement (2) alone is not sufficient.

Therefore, the statement indicating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient, is the correct choice.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
The target inequality ∣x2−4∣<5|x^2 - 4| < 5 is equivalent to −5<x2−4<5-5 < x^2 - 4 < 5. Adding 4 yields −1<x2<9-1 < x^2 < 9. Since x2≥0x^2 \ge 0 for all real numbers, −1<x2-1 < x^2 is always true. Thus, the target question simplifies to whether x2<9x^2 < 9, which means −3<x<3-3 < x < 3.
Simplifying the question stem simplifies evaluating the statement ranges.
2
Evaluate Statement (1): ∣x−1∣<2|x - 1| < 2.
Expanding the absolute value gives −2<x−1<2-2 < x - 1 < 2, which simplifies to −1<x<3-1 < x < 3. Every value of xx in the interval (−1,3)(-1, 3) is strictly inside (−3,3)(-3, 3). Thus, Statement (1) guarantees a definitive 'Yes'.
Since Statement (1) provides a single definitive answer ('Yes'), Statement (1) alone is sufficient.
3
Evaluate Statement (2): ∣x+1∣<3|x + 1| < 3.
Expanding the absolute value gives −3<x+1<3-3 < x + 1 < 3, which simplifies to −4<x<2-4 < x < 2. If x=0x = 0, then −3<0<3-3 < 0 < 3 is true ('Yes'). If x=−3.5x = -3.5, then ∣(−3.5)2−4∣=∣12.25−4∣=8.25<5|(-3.5)^2 - 4| = |12.25 - 4| = 8.25 < 5 is false ('No').
Since Statement (2) allows both 'Yes' and 'No' answers, Statement (2) alone is not sufficient.

Key Concept

Rephrasing absolute value inequality bounds in Data Sufficiency questions
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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