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Difficulty: Very hardInequalities, Absolute Values, and Number Ranges in Data Sufficiency

If xx is a real number, is ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8?

(1) ∣x−1∣≤4|x - 1| \le 4
(2) x2−x−6≤0x^2 - x - 6 \le 0

  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. C
    BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
  4. EACH statement ALONE is sufficient.Answer
  5. E
    Statements (1) and (2) TOGETHER are NOT sufficient.

Answer

Each statement alone is sufficient.
Rephrasing the question stem shows that ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8 is equivalent to −3≤x≤5-3 \le x \le 5. Statement (1) gives −3≤x≤5-3 \le x \le 5, which directly yields a definitive 'Yes'. Statement (2) gives −2≤x≤3-2 \le x \le 3, which is a subset of [−3,5][-3, 5], so any value of xx satisfying Statement (2) must also satisfy −3≤x≤5-3 \le x \le 5, also yielding a definitive 'Yes'. Thus, each statement alone is sufficient.

Step-by-Step Solution

1
Rephrase the question stem target by analyzing critical points x=−2x = -2 and x=4x = 4 for ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8.
The target inequality ∣x+2∣+∣x−4∣≤8|x + 2| + |x - 4| \le 8 simplifies to the range −3≤x≤5-3 \le x \le 5.
For x≥4x \ge 4: (x+2)+(x−4)≤8  ⟹  2x−2≤8  ⟹  x≤5(x + 2) + (x - 4) \le 8 \implies 2x - 2 \le 8 \implies x \le 5, giving [4,5][4, 5]. For −2≤x<4-2 \le x < 4: (x+2)+(4−x)=6≤8(x + 2) + (4 - x) = 6 \le 8, which is true for all x∈[−2,4)x \in [-2, 4). For x<−2x < -2: (−x−2)+(4−x)≤8  ⟹  2−2x≤8  ⟹  −2x≤6  ⟹  x≥−3(-x - 2) + (4 - x) \le 8 \implies 2 - 2x \le 8 \implies -2x \le 6 \implies x \ge -3, giving [−3,−2)[-3, -2). Combining all intervals gives −3≤x≤5-3 \le x \le 5.
2
Evaluate Statement (1): ∣x−1∣≤4|x - 1| \le 4.
−4≤x−1≤4  ⟹  −3≤x≤5-4 \le x - 1 \le 4 \implies -3 \le x \le 5.
Statement (1) states that xx is precisely in the range [−3,5][-3, 5]. This provides a definitive 'Yes' answer to the question 'Is −3≤x≤5-3 \le x \le 5?'. Thus, Statement (1) alone is sufficient.
3
Evaluate Statement (2): x2−x−6≤0x^2 - x - 6 \le 0.
(x−3)(x+2)≤0  ⟹  −2≤x≤3(x - 3)(x + 2) \le 0 \implies -2 \le x \le 3.
Since [−2,3][-2, 3] is entirely contained within [−3,5][-3, 5], every value of xx satisfying Statement (2) automatically satisfies −3≤x≤5-3 \le x \le 5. This yields a definitive 'Yes' to the question. Thus, Statement (2) alone is sufficient.
4
Combine evaluations of Statement (1) and Statement (2).
Each statement alone is sufficient.
Because both statements independently provide enough information to give a definitive 'Yes' answer, the correct answer choice is that each statement alone is sufficient.

Key Concept

Absolute Value Distance Interpretation and Subset Range Sufficiency
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Topics

Inequalities, Absolute Values, and Number Ranges in Data Sufficiency

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