HumptyDumpty wrote:

Could someone please explain how to get from the marked 1. to 2.?

Nothing new to add. Just another way of arriving at the solution.

x(x-1)(x+1)<0.

A product of three factors can only be negative if

1. All three of them are negative --> x<0 , x<1 , x<-1. Now, if x<-1, then automatically, the other two conditions are also fulfilled. Thus, one solution is at x<-1.

2.Any two of the factors are positive and only one is negative --> Say x>0 AND x>1 AND x<-1. Is this possible simultaneously? NO. Now consider x>1 AND x>-1 AND x<0. Again not possible. Finally consider x>0 AND x>-1 AND x<1. This is possible if x>0 AND x<1. Thus, the second solution is for 0<x<1.

But again this method is only for clearing the concept involved. The normal method to these problems is discussed above.

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All that is equal and not-Deep Dive In-equality

Hit and Trial for Integral Solutions