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Difficulty: HardInequalities, Absolute Values, and Number Ranges in Data Sufficiency

If xx is a real number, is ∣x−3∣+∣x+2∣≤7|x - 3| + |x + 2| \le 7?

(1) x2−2x−8≤0x^2 - 2x - 8 \le 0
(2) x(x−5)≤0x(x - 5) \le 0

  1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.Answer
  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.

Answer

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Rephrasing the question target ∣x−3∣+∣x+2∣≤7|x - 3| + |x + 2| \le 7 shows that it is satisfied if and only if −3≤x≤4-3 \le x \le 4. Statement (1) yields the interval [−2,4][-2, 4], which is completely contained within [−3,4][-3, 4], guaranteeing a definitive 'Yes'. Statement (2) yields [0,5][0, 5], which includes values like 4.54.5 that result in a 'No'. Thus, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Step-by-Step Solution

1
Rephrase the question stem inequality ∣x−3∣+∣x+2∣≤7|x - 3| + |x + 2| \le 7.
The target inequality simplifies to −3≤x≤4-3 \le x \le 4.
Interpret ∣x−3∣+∣x+2∣|x - 3| + |x + 2| as the sum of distances from xx to 33 and −2-2 on the real number line. The distance between −2-2 and 33 is 55. For −2≤x≤3-2 \le x \le 3, the sum of distances is constantly 5≤75 \le 7. For x>3x > 3, the sum is 2x−1≤7  ⟹  x≤42x - 1 \le 7 \implies x \le 4. For x<−2x < -2, the sum is 1−2x≤7  ⟹  x≥−31 - 2x \le 7 \implies x \ge -3. Thus, the inequality holds if and only if x∈[−3,4]x \in [-3, 4].
2
Evaluate Statement (1): x2−2x−8≤0x^2 - 2x - 8 \le 0.
Statement (1) gives −2≤x≤4-2 \le x \le 4.
Factoring gives (x−4)(x+2)≤0(x - 4)(x + 2) \le 0, which holds when x∈[−2,4]x \in [-2, 4]. Since [−2,4]⊆[−3,4][-2, 4] \subseteq [-3, 4], every value of xx satisfying Statement (1) guarantees that ∣x−3∣+∣x+2∣≤7|x - 3| + |x + 2| \le 7 is true. This yields a definitive 'Yes'. Statement (1) alone is sufficient.
3
Evaluate Statement (2): x(x−5)≤0x(x - 5) \le 0.
Statement (2) gives 0≤x≤50 \le x \le 5.
The solution set is x∈[0,5]x \in [0, 5]. If x=1x = 1, −3≤1≤4-3 \le 1 \le 4 is true (Yes). If x=4.5x = 4.5, ∣4.5−3∣+∣4.5+2∣=1.5+6.5=8≤7|4.5 - 3| + |4.5 + 2| = 1.5 + 6.5 = 8 \le 7 is false (No). Since Statement (2) allows both 'Yes' and 'No' outcomes, Statement (2) alone is not sufficient.

Key Concept

Question stem rephrasing for distance-based absolute values and set-inclusion sufficiency evaluation
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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