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.


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


The error is on the left.
You went from
$$|y|>1$$
to
$$\left|\frac{1}{y}\right|>1$$
This is incorrect. You should have divided both sides by $|y|$, leaving
$$1>\frac{1}{|y|}$$
This agrees with the right-hand result.
However, something happened with the absolute value. Try the case where $y=-2$. In this case, the final equation holds, but the original equation does not. So for $y<0$, we instead have
$$1>\frac{1}{|y|}$$
Why did this happen? It was the first step in each one, multiplying (or dividing) by $y$ to get $y^2$. This loses the distinction between the signs.
