reto wrote:
Is \(x^{12}\)– \(2x^{11}\)negative?
(1) \(x^2\) < |x|
(2) \(x^{-1}\) < -1
The question asks whether , \(x^{12} – 2*x^{11}<0\) ---> \(x^{11}*(x-2)<0\). Remember that \(x^{odd} <0\) for \(x<0\)and \(x^{odd}>0\) for \(x>0\)
Per statement 1, \(x^2< |x|\) ---> carefully note this statement tells us that a squared positive quantity (x^2) < another positive quantity (|x|). This is only possible when -1<x<0 or 0<x<1. For these 2 ranges, you will get both "no" for -1<x<0 and "yes" for 0<x<1, making this statement NOT sufficient.
Per statement 2, \(x^{-1} < -1\) ---> (DO NOT MULTIPLY BY x as you do not know the sign of x) \(x^{-1} < -1 = x^{-1}+1 < 0\) ---> (x+1)/x < 0 ---> as the result of this fraction <0 ---> (x+1) and x are of OPPOSITE SIGNS.
2 cases :
Case 1: x+1 > 0 and x < 0 ---> x>-1 and x<0 ---> -1<x<0 and
Case 2: x+1<0 --> x<-1 and x>0 ---> not possible.
Thus, the only range possible -->-1<x<0 and for this range you get a NO for the question asked. Thus statement 2 is sufficient. B is thus the correct answer.
Hope this helps.