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Zorluk: OrtaValue vs. Yes/No Data Sufficiency Decision Logic

If xx is a positive integer, is x2−5x+6>0x^2 - 5x + 6 > 0?

(1) (x−2)(x−3)≤0(x - 2)(x - 3) \le 0
(2) x<5x < 5

  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 option stating that Statement (1) alone is sufficient while Statement (2) alone is not sufficient is correct. Statement (1) restricts the positive integer xx to {2,3}\{2, 3\}, for both of which x2−5x+6=0x^2 - 5x + 6 = 0. This yields a definitive 'No' to the question of whether the expression is strictly greater than 0, making Statement (1) sufficient. Statement (2) permits values such as x=1x = 1 (which yields 'Yes') and x=2x = 2 (which yields 'No'), making it insufficient.

Adım Adım Çözüm

1
Rephrase the question stem target algebraically.
The target expression x2−5x+6x^2 - 5x + 6 factors into (x−2)(x−3)(x - 2)(x - 3). The question asks: 'Is (x−2)(x−3)>0(x - 2)(x - 3) > 0?'
Factoring simplifies evaluating the sign of the expression for specific values of xx.
2
Evaluate Statement (1) independently.
Statement (1) gives (x−2)(x−3)≤0(x - 2)(x - 3) \le 0. Since xx is a positive integer, the integer solutions to this inequality are x=2x = 2 and x=3x = 3. For x=2x = 2, (2−2)(2−3)=0(2-2)(2-3) = 0. For x=3x = 3, (3−2)(3−3)=0(3-2)(3-3) = 0. In both cases, x2−5x+6=0x^2 - 5x + 6 = 0, which is NOT >0> 0. Thus, the answer to the stem question is a definitive 'NO'.
In a Yes/No Data Sufficiency question, any statement that yields a definitive 'No' for all possible values of the variable is SUFFICIENT.
3
Evaluate Statement (2) independently.
Statement (2) gives x<5x < 5. Since xx is a positive integer, xx can be 1, 2, 3, or 4. If x=1x = 1, 12−5(1)+6=2>01^2 - 5(1) + 6 = 2 > 0 (Answer: YES). If x=2x = 2, 22−5(2)+6=02^2 - 5(2) + 6 = 0, which is not >0> 0 (Answer: NO). Since Statement (2) allows both 'YES' and 'NO' outcomes, it is NOT SUFFICIENT.
A statement that leads to conflicting answers ('Maybe') is insufficient.

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Value vs. Yes/No Data Sufficiency Decision Logic
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Value vs. Yes/No Data Sufficiency Decision Logic

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