Basic question about inequality let $x>1$
Obviously $x^2>x$
Then $x^2>x>1$
Taking $x^2>1$, we can assert that this holds true for all the values for $x>1$ and $x<-1$
But if I take $-5$ such that $x<-1$, then $x^2>x$ holds but $x>1$ doesn't. Why is it so? Doesn't $x^2>x>1$ mean that all of the three must be true?

When you deduced that $^2>>1$, you specifically had the constraint that $x > 1$. Of course, this inequality is not then applicable since now you have $x > -5$ which is a larger domain than $x > -1$. 
More specifically, $x > 1 \Rightarrow x^2 > x > 1$ is a true statement, but $x^2 > x \iff x> 1$ is clearly not a true statement as you have demonstrated. 
