Basic algebra inequality mistake (I probably didn't break algebra) I'm not sure what I have done here. I'm guessing it's something to do with the absolute values but I really have no idea what it is.


$$\begin{array}{ccc}
    y>\frac1y &  & y>\frac1y \\
    y^2>1 &  & 1>\frac1{y^2}  \\
    |y|>1 &  & 1>\frac1{|y|}  \\    
    \frac1{|y|}>1 &  & \downarrow \\        
    \searrow &  & \swarrow  \\            
    & 1 < \left|\dfrac1y\right| < 1 & 
  \end{array}
$$


Check out this graph here:
http://m.wolframalpha.com/input/?i=1%2Fx+%2C+x&x=10&y=14
The values you are looking for are the ones where the hyperbola is below the straight line. 
Values between -1 and zero (not inclusive)
 And values between one and infinity (also not inclusive)
You have to be carefull with inequalities, you can't use them in the same way you treat equality.
For example $y^2 > 1$ is not correct. Since these include values between -1 and infinity. Which is not what was shown on the graph above. 
Best to draw the graph before drawing conclusions. 
