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Zorluk: OrtaInequalities, Absolute Values, and Number Ranges in Data Sufficiency

If xx is a real number, is ∣x−5∣<x+1|x - 5| < x + 1?

(1) x2−6x+8<0x^2 - 6x + 8 < 0
(2) ∣x−3∣<2|x - 3| < 2

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.Cevap
  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.

Cevap

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
The question target ∣x−5∣<x+1|x - 5| < x + 1 simplifies directly to x>2x > 2. Statement (1) bounds xx to 2<x<42 < x < 4, which guarantees that x>2x > 2 is always true, providing a definitive 'Yes'. Statement (2) bounds xx to 1<x<51 < x < 5, allowing values such as 1.51.5 (where x>2x > 2 is false) and 33 (where x>2x > 2 is true), which is insufficient. Therefore, Statement (1) alone is sufficient, but Statement (2) alone is not sufficient.

Adım Adım Çözüm

1
Rephrase the question stem by simplifying the absolute value inequality ∣x−5∣<x+1|x - 5| < x + 1.
The inequality holds if and only if −(x+1)<x−5<x+1-(x + 1) < x - 5 < x + 1. Solving −x−1<x−5-x - 1 < x - 5 yields 2x>42x > 4, or x>2x > 2. Solving x−5<x+1x - 5 < x + 1 yields −5<1-5 < 1, which is always true. Thus, the question rephrases to: 'Is x>2x > 2?'
Simplifying absolute value conditions in the stem reduces complex expressions to a single target range.
2
Evaluate Statement (1): x2−6x+8<0x^2 - 6x + 8 < 0.
Factoring gives (x−2)(x−4)<0(x - 2)(x - 4) < 0, which yields the range 2<x<42 < x < 4. For any value of xx in (2,4)(2, 4), xx is strictly greater than 22. This gives a definitive 'Yes' to the rephrased question.
Since every value satisfying Statement (1) satisfies x>2x > 2, Statement (1) alone is sufficient.
3
Evaluate Statement (2): ∣x−3∣<2|x - 3| < 2.
Unfolding the absolute value gives −2<x−3<2-2 < x - 3 < 2, or 1<x<51 < x < 5. If x=1.5x = 1.5, then x>2x > 2 is false ('No'). If x=3x = 3, then x>2x > 2 is true ('Yes'). Because both 'Yes' and 'No' answers are possible, Statement (2) is not sufficient.
Statement (2) permits values both above and below the threshold of 2.

Anahtar Kavram

Question Stem Rephrasing for Absolute Value Inequalities
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Konular

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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