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Is \(\mathrm{x} > 1\)?
We need to determine: Is \(\mathrm{x > 1}\)?
This is a yes/no question. To be sufficient, a statement must allow us to definitively answer either "yes, x is always greater than 1" or "no, x is not always greater than 1" (meaning \(\mathrm{x ≤ 1}\) for at least some values).
Given Information: No constraints are provided in the question itself.
Key Insight: Since we're dealing with inequalities involving absolute values, we should focus on where the expressions change sign rather than testing every possible region.
Statement 1: \(\mathrm{(x + 1)(|x| - 1) > 0}\)
For a product to be positive, both factors must have the same sign (both positive or both negative).
Let's identify the critical points where each factor changes sign:
- \(\mathrm{(x + 1) = 0}\) when \(\mathrm{x = -1}\)
- Negative when \(\mathrm{x < -1}\)
- Positive when \(\mathrm{x > -1}\)
- \(\mathrm{|x| - 1 = 0}\) when \(\mathrm{|x| = 1}\), which means \(\mathrm{x = -1}\) or \(\mathrm{x = 1}\)
- When \(\mathrm{-1 < x < 1}\): \(\mathrm{|x| < 1}\), so \(\mathrm{|x| - 1 < 0}\) (negative)
- When \(\mathrm{x < -1}\) or \(\mathrm{x > 1}\): \(\mathrm{|x| > 1}\), so \(\mathrm{|x| - 1 > 0}\) (positive)
Case 1: Both factors positive
- Need \(\mathrm{x + 1 > 0}\) → \(\mathrm{x > -1}\)
- Need \(\mathrm{|x| - 1 > 0}\) → \(\mathrm{x < -1}\) or \(\mathrm{x > 1}\)
- The overlap: Only when \(\mathrm{x > 1}\) ✓
Case 2: Both factors negative
- Need \(\mathrm{x + 1 < 0}\) → \(\mathrm{x < -1}\)
- Need \(\mathrm{|x| - 1 < 0}\) → \(\mathrm{-1 < x < 1}\)
- No overlap exists! These conditions contradict each other.
Therefore, \(\mathrm{(x + 1)(|x| - 1) > 0}\) holds only when \(\mathrm{x > 1}\).
Since Statement 1 forces \(\mathrm{x > 1}\), we can definitively answer "yes" to our question.
[STOP - Statement 1 is Sufficient!]
This eliminates choices B, C, and E.
Now we analyze Statement 2 independently, forgetting Statement 1 completely.
Statement 2: \(\mathrm{|x| < 5}\)
This means \(\mathrm{-5 < x < 5}\).
Within this range, x could be:
- \(\mathrm{x = 0}\) → Is \(\mathrm{0 > 1}\)? No
- \(\mathrm{x = 2}\) → Is \(\mathrm{2 > 1}\)? Yes
Since we get different answers ("no" for \(\mathrm{x = 0}\), "yes" for \(\mathrm{x = 2}\)), we cannot definitively determine whether \(\mathrm{x > 1}\).
Statement 2 is NOT sufficient.
This eliminates choices B and D.
- Statement 1 alone: Forces \(\mathrm{x > 1}\) → Can answer "yes" → SUFFICIENT
- Statement 2 alone: Allows both \(\mathrm{x ≤ 1}\) and \(\mathrm{x > 1}\) → Cannot answer definitively → NOT SUFFICIENT
Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."