Examkin

Soru

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

If xx is a real number, is x2<4x^2 < 4?

(1) ∣x−1∣<2|x - 1| < 2
(2) ∣x+1∣<3|x + 1| < 3

  1. A
    Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
  2. B
    Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
  3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.Cevap
  4. D
    EACH statement ALONE is sufficient.
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Cevap

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Rephrasing the target question x2<4x^2 < 4 shows that we need to determine if −2<x<2-2 < x < 2. Statement (1) gives −1<x<3-1 < x < 3, which allows values outside (−2,2)(-2, 2) like x=2.5x = 2.5, so it is insufficient alone. Statement (2) gives −4<x<2-4 < x < 2, which allows values outside (−2,2)(-2, 2) like x=−3x = -3, so it is insufficient alone. Taking the intersection of both statements gives −1<x<2-1 < x < 2. Since every number in (−1,2)(-1, 2) satisfies −2<x<2-2 < x < 2, the combined statements definitively answer YES.

Adım Adım Çözüm

1
Rephrase the target question
The target question asking whether x2<4x^2 < 4 is equivalent to asking whether −2<x<2-2 < x < 2.
Taking the square root of both sides of x2<4x^2 < 4 yields ∣x∣<2|x| < 2, which expands to −2<x<2-2 < x < 2.
2
Evaluate Statement (1) independently
Statement (1) states ∣x−1∣<2|x - 1| < 2, which expands to −2<x−1<2-2 < x - 1 < 2, or −1<x<3-1 < x < 3.
If x=0x = 0, then −1<0<3-1 < 0 < 3 is true and 02<40^2 < 4 (Yes). If x=2.5x = 2.5, then −1<2.5<3-1 < 2.5 < 3 is true, but 2.52=6.25≮42.5^2 = 6.25 \not< 4 (No). Because we get both Yes and No answers, Statement (1) alone is insufficient.
3
Evaluate Statement (2) independently
Statement (2) states ∣x+1∣<3|x + 1| < 3, which expands to −3<x+1<3-3 < x + 1 < 3, or −4<x<2-4 < x < 2.
If x=0x = 0, then −4<0<2-4 < 0 < 2 is true and 02<40^2 < 4 (Yes). If x=−3x = -3, then −4<−3<2-4 < -3 < 2 is true, but (−3)2=9≮4(-3)^2 = 9 \not< 4 (No). Because we get both Yes and No answers, Statement (2) alone is insufficient.
4
Evaluate Statements (1) and (2) together
Combining −1<x<3-1 < x < 3 and −4<x<2-4 < x < 2 requires xx to satisfy both inequalities simultaneously, yielding the intersection −1<x<2-1 < x < 2.
Every value of xx in the interval (−1,2)(-1, 2) lies strictly inside the required target interval (−2,2)(-2, 2). Thus, x2<4x^2 < 4 is definitively YES. Both statements together are sufficient.

Anahtar Kavram

Inequality range intersection and absolute value distance expansion in Data Sufficiency
Tahmini Süre:2m 0s
Bu soruyu puanla

Konular

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

Bu konudan daha fazla soru