Is n negative?
1. n^5(1 – n^4) < 0
2. n^4 – 1 < 0
Go step by step.
Question: Is n negative?
Statement 1: \(n^5(1 - n^4) < 0\)
Re-write as: \(n^5(n^4 - 1) > 0\) (I do this only for convenience. It is easier to handle >0 because either both are positive or both negative so less confusion.)
This tells me that either both \(n^5\) and \((n^4 - 1)\) are positive (which means n is positive) or both are negative (which means n is negative). n can be positive or negative so not sufficient.
Statement 2: \(n^4 - 1 < 0\)
This implies that n is between -1 and 1.
\(n^4 - 1 = (n^2 + 1)(n^2 - 1) = (n^2 + 1)(n + 1)(n - 1)\)
\(n^2 + 1\) is always positive so we can ignore it.
We get, \((n+1)(n - 1) < 0\)
-1 < n < 1 (check out: inequalities-trick-91482.html#p804990
Since n can be positive or negative, not sufficient alone.
Using both statements together,
\(n^4 - 1\) is negative, so from the analysis of statement 1, \(n^5\) must be negative too. This means n must be negative. Sufficient.
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