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Zorluk: Çok zorInequalities, Absolute Values, and Number Ranges in Data Sufficiency

If mm is a real number, is ∣m2−2m−3∣<5|m^2 - 2m - 3| < 5?

(1) ∣m−2∣<3|m - 2| < 3
(2) ∣m+1∣<4|m + 1| < 4

  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 question stem requires solving ∣m2−2m−3∣<5|m^2 - 2m - 3| < 5. This breaks down into −5<m2−2m−3<5-5 < m^2 - 2m - 3 < 5. The left inequality m2−2m+2>0m^2 - 2m + 2 > 0 holds for all real numbers because its discriminant is negative and the quadratic opens upward. The right inequality m2−2m−8<0m^2 - 2m - 8 < 0 factors as (m−4)(m+2)<0(m - 4)(m + 2) < 0, giving the target range −2<m<4-2 < m < 4.

Evaluating the first statement gives ∣m−2∣<3  ⟹  −1<m<5|m - 2| < 3 \implies -1 < m < 5. This interval extends past 4 (e.g., m=4.5m = 4.5 yields a 'No', while m=1m = 1 yields a 'Yes'), so the first statement alone is not sufficient.

Evaluating the second statement gives ∣m+1∣<4  ⟹  −5<m<3|m + 1| < 4 \implies -5 < m < 3. This interval extends past −2-2 (e.g., m=−4m = -4 yields a 'No', while m=1m = 1 yields a 'Yes'), so the second statement alone is not sufficient.

Combining both statements requires taking the intersection of −1<m<5-1 < m < 5 and −5<m<3-5 < m < 3, which gives −1<m<3-1 < m < 3. Since every number in the range (−1,3)(-1, 3) lies strictly between −2-2 and 44, the target condition is guaranteed to be true. Therefore, both statements together are sufficient, but neither statement alone is sufficient.

Adım Adım Çözüm

1
Rephrase the target question stem algebraically.
The target inequality ∣m2−2m−3∣<5|m^2 - 2m - 3| < 5 is equivalent to −5<m2−2m−3<5-5 < m^2 - 2m - 3 < 5. Solving the upper bound gives m2−2m−8<0  ⟹  (m−4)(m+2)<0  ⟹  −2<m<4m^2 - 2m - 8 < 0 \implies (m - 4)(m + 2) < 0 \implies -2 < m < 4. The lower bound m2−2m−3>−5  ⟹  m2−2m+2>0  ⟹  (m−1)2+1>0m^2 - 2m - 3 > -5 \implies m^2 - 2m + 2 > 0 \implies (m - 1)^2 + 1 > 0 is true for all real mm. Thus, the question simplifies to: 'Is −2<m<4-2 < m < 4?'
Simplifying the target stem establishes the exact numerical interval required to yield a definitive 'Yes' or 'No' response.
2
Evaluate Statement (1) independently.
Statement (1) states ∣m−2∣<3  ⟹  −3<m−2<3  ⟹  −1<m<5|m - 2| < 3 \implies -3 < m - 2 < 3 \implies -1 < m < 5. If m=1m = 1, −2<1<4-2 < 1 < 4 (Yes). However, if m=4.5m = 4.5, mm is not in (−2,4)(-2, 4) (No). Since Statement (1) allows both 'Yes' and 'No' outcomes, it is NOT sufficient.
A statement is sufficient only if every value in its allowed range produces the same answer to the rephrased question.
3
Evaluate Statement (2) independently.
Statement (2) states ∣m+1∣<4  ⟹  −4<m+1<4  ⟹  −5<m<3|m + 1| < 4 \implies -4 < m + 1 < 4 \implies -5 < m < 3. If m=1m = 1, −2<1<4-2 < 1 < 4 (Yes). However, if m=−4m = -4, mm is not in (−2,4)(-2, 4) (No). Thus, Statement (2) is NOT sufficient.
Checking boundary values reveals that Statement (2) allows values outside the required interval.
4
Evaluate Statement (1) and Statement (2) together.
Combining Statement (1) range (−1<m<5)(-1 < m < 5) and Statement (2) range (−5<m<3)(-5 < m < 3) requires taking their intersection: −1<m<3-1 < m < 3. Since every value in (−1,3)(-1, 3) satisfies −2<m<4-2 < m < 4, the answer to the question is a definitive 'Yes'.
The intersection of the two ranges falls strictly inside the target range, guaranteeing sufficiency.

Anahtar Kavram

Data Sufficiency range evaluation for quadratic absolute value inequalities
Tahmini Süre:2m 0s
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Konular

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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