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

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

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

Which of the following choices correctly describes the sufficiency of the statements?

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.Cevap
  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 (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Rephrasing the stem ∣x−3∣<2x|x - 3| < 2x shows it is equivalent to asking whether x>1x > 1. Statement (2) restricts xx to the range 1<x<31 < x < 3, where every value of xx satisfies x>1x > 1, providing a definitive YES answer. Statement (1) permits x<1x < 1 (giving a NO) and x>3x > 3 (giving a YES), making Statement (1) insufficient.

Adım Adım Çözüm

1
Rephrase the target question stem algebraically.
The target inequality is ∣x−3∣<2x|x - 3| < 2x. Since absolute values are non-negative (∣x−3∣≥0|x - 3| \ge 0), 2x2x must be strictly positive, so x>0x > 0.
- If x≥3x \ge 3, ∣x−3∣=x−3<2x  ⟹  x>−3|x - 3| = x - 3 < 2x \implies x > -3, which holds for all x≥3x \ge 3.
- If 0<x<30 < x < 3, ∣x−3∣=3−x<2x  ⟹  3x>3  ⟹  x>1|x - 3| = 3 - x < 2x \implies 3x > 3 \implies x > 1.
Thus, the inequality ∣x−3∣<2x|x - 3| < 2x holds if and only if x>1x > 1. The target question simplifies to: 'Is x>1x > 1?'
Simplifying the question stem before analyzing statements prevents misinterpreting the range of valid solutions.
2
Evaluate Statement (1): x2−4x+3>0x^2 - 4x + 3 > 0.
Factoring the quadratic yields (x−1)(x−3)>0(x - 1)(x - 3) > 0, so x<1x < 1 or x>3x > 3.
- If x=0x = 0 (which is <1< 1), ∣0−3∣=3|0 - 3| = 3 and 2(0)=02(0) = 0. Is 3<03 < 0? NO.
- If x=4x = 4 (which is >3> 3), ∣4−3∣=1|4 - 3| = 1 and 2(4)=82(4) = 8. Is 1<81 < 8? YES.
Since Statement (1) allows for both YES and NO outcomes, it is NOT sufficient.
Testing specific values from the disjoint solution set demonstrates that a single definitive answer cannot be reached.
3
Evaluate Statement (2): ∣x−2∣<1|x - 2| < 1.
Solving the absolute value inequality gives −1<x−2<1  ⟹  1<x<3-1 < x - 2 < 1 \implies 1 < x < 3.
For every value of xx in the interval (1,3)(1, 3), xx is strictly greater than 11. Therefore, the answer to 'Is x>1x > 1?' is a definitive YES for all possible values of xx in this statement.
A statement is sufficient if every allowed value of xx yields a consistent YES answer to the rephrased question stem.

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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