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

If xx is a real number, is ∣x−2∣<3|x - 2| < 3?

(1) x2−4x+3<0x^2 - 4x + 3 < 0
(2) ∣x∣<4|x| < 4

Which of the options 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.
The correct answer specifies that Statement (1) alone is sufficient while Statement (2) alone is not. Rephrasing the question stem shows that ∣x−2∣<3|x - 2| < 3 is equivalent to −1<x<5-1 < x < 5. Statement (1) yields 1<x<31 < x < 3, which lies entirely inside (−1,5)(-1, 5), guaranteeing a 'Yes' answer. Statement (2) yields −4<x<4-4 < x < 4, which allows values outside (−1,5)(-1, 5) (such as x=−2x = -2, giving 'No') as well as values inside (such as x=0x = 0, giving 'Yes'), so it is not sufficient.

Step-by-Step Solution

1
Rephrase the target question stem algebraically.
The inequality ∣x−2∣<3|x - 2| < 3 is equivalent to −3<x−2<3-3 < x - 2 < 3, which simplifies to −1<x<5-1 < x < 5. The question asks: Is xx strictly between −1-1 and 55?
Simplifying the target inequality establishes the exact boundary values needed for evaluation.
2
Evaluate Statement (1): x2−4x+3<0x^2 - 4x + 3 < 0.
Factoring the quadratic gives (x−1)(x−3)<0(x - 1)(x - 3) < 0, which holds when 1<x<31 < x < 3.
Since the interval (1,3)(1, 3) is entirely contained within the target range (−1,5)(-1, 5), every value of xx satisfying Statement (1) results in a definitive 'Yes' to the question. Thus, Statement (1) is sufficient.
3
Evaluate Statement (2): ∣x∣<4|x| < 4.
This absolute value inequality simplifies to −4<x<4-4 < x < 4.
If x=0x = 0, then −1<0<5-1 < 0 < 5 is True ('Yes'). If x=−2x = -2, then −1<−2<5-1 < -2 < 5 is False ('No'). Because Statement (2) allows both 'Yes' and 'No' outcomes, Statement (2) is not sufficient.

Key Concept

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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