Examkin

Soru

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

If xx is a real number, is x2−4x+3<0x^2 - 4x + 3 < 0?

(1) (x−4)2≤0(x - 4)^2 \le 0
(2) ∣x−2∣<2|x - 2| < 2

Which of the following statements correctly describes the sufficiency of the given statements to answer the question?

  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 choice stating that Statement (1) alone is sufficient, but Statement (2) alone is not sufficient is correct. Statement (1) forces x=4x = 4. Evaluating x=4x = 4 against the condition x2−4x+3<0x^2 - 4x + 3 < 0 gives 42−4(4)+3=3<04^2 - 4(4) + 3 = 3 < 0, which is conclusively false. A statement that yields a definitive 'No' to a Yes/No question is sufficient. Statement (2) yields the range 0<x<40 < x < 4, which contains values of xx that make x2−4x+3<0x^2 - 4x + 3 < 0 true (such as x=2x = 2) and false (such as x=3.5x = 3.5), making Statement (2) insufficient.

Adım Adım Çözüm

1
Rephrase the question stem target inequality.
The inequality x2−4x+3<0x^2 - 4x + 3 < 0 factors into (x−1)(x−3)<0(x - 1)(x - 3) < 0, which holds true if and only if 1<x<31 < x < 3. Thus, the question asks: 'Is 1<x<31 < x < 3?'
Simplifying the question stem establishes the exact criteria needed for a 'Yes' vs. 'No' response.
2
Evaluate Statement (1): (x−4)2≤0(x - 4)^2 \le 0.
Since any square of a real number is non-negative, (x−4)2≥0(x - 4)^2 \ge 0. The only way (x−4)2≤0(x - 4)^2 \le 0 can hold is if (x−4)2=0(x - 4)^2 = 0, which means x=4x = 4. Substituting x=4x = 4 into 1<x<31 < x < 3 gives 1<4<31 < 4 < 3, which is false. Therefore, the answer to the stem's question is a definitive 'NO'. In Yes/No Data Sufficiency logic, a definitive 'No' means the statement IS SUFFICIENT.
A statement that yields a consistent and definitive answer—whether 'Yes' or 'No'—provides sufficient information.
3
Evaluate Statement (2): ∣x−2∣<2|x - 2| < 2.
The absolute value inequality ∣x−2∣<2|x - 2| < 2 unravels to −2<x−2<2-2 < x - 2 < 2, or 0<x<40 < x < 4.
- If x=2x = 2 (which is in 0<x<40 < x < 4), then 1<2<31 < 2 < 3 is true ('YES').
- If x=0.5x = 0.5 (which is in 0<x<40 < x < 4), then 1<0.5<31 < 0.5 < 3 is false ('NO').
Since Statement (2) allows both 'Yes' and 'No' answers depending on the value chosen for xx, Statement (2) is NOT SUFFICIENT.
If a statement leads to contradictory answers ('Yes' under some conditions, 'No' under others), it is insufficient.

Anahtar Kavram

Value vs. Yes/No Data Sufficiency Decision Logic
Tahmini Süre:2m 0s
Bu soruyu puanla

Konular

Value vs. Yes/No Data Sufficiency Decision Logic

Bu konudan daha fazla soru