Value of x

Question

If x^3<x<1/x, then x could be equal to?

Answer 1

x³<x<1/x; x³<x, x³-x<0, x(x²-1)<0 so EITHER (1) x<0 and x²-1>0, that is, (x-1)(x+1)>0, so x-1>0 and x+1>0⇒x>1. But x cannot be both negative and greater than 1 at the same time, so we reject this condition; therefore if x<0 x-1<0 and x+1<0⇒x<-1. Conclusion: x<-1 satisfies x<0 and x²-1>0. Example: x=-2→x³-x=-8+2=-6<0;

OR (2) x>0 and x²-1<0⇒(x+1)(x-1)<0, so x+1<0, x<-1 and x-1>0, x>1; or x+1>0, x>-1 and x-1<0, x<1. Conclusion: 0<x<1. Example: x=½→x³-x=⅛-½=-⅜<0.

But we now have to look at x<1/x. If x>0, then x²-1<0, (x-1)(x+1)<0, so 0<x<1, which concurs with (2). When x=½, ½<2✔️

If x<0, what can we say about x<1/x? x²>0, so x²<x(1/x), x²<1, (x-1)(x+1)<0⇒x<-1. Conclusion: x<-1 satisfies x<0. Example: x=-2, -2<-½✔️

SOLUTION: x³<x<1/x⇒x<-1 or 0<x<1.