Evaluate x in
1/x<x
1/x < x
or x > 1/x
x^2 > 1
critical values are -1 and 1 , clearly x ≠ 0
We have 4 choices for x:
using x > 1/x
1. x < -1
test for some value, say x = -5
is -5 > -1/5 ? , NO
2. -1 < x < 0
test for some value, say x = -.5
is -.5 > 1/-.5 ? , YES
3. 0 < x < 1
pick x = .25
is .25 > 1/.25 ? NO
4. x > 1
pick x = 10
is 10 > 1/10 > YES
so x > 1 OR -1<x<0
check with Wolfram
http://www.wolframalpha.com/input/?i=y+%3D+x+-+1%2Fx
Where is the graph above the x-axis ?