Inequality with absolute value in denominator

Question

Is option 1 or 2 correct to solve the inequality?

Option 1:

for $x>0$

$\dfrac{x}{|x|}<x$ $\Rightarrow$ $\dfrac{x}{x}<x$ $\Rightarrow$ $1<x$

for $x<0$

$\dfrac{x}{|x|}<x$ $\Rightarrow$ $\dfrac{x}{-x}<x$ $\Rightarrow$ $-1<x$

Option 2:

for $x>0$

$\dfrac{x}{|x|}<x$ $\Rightarrow$ ${x}<x|x|$ $\Rightarrow$ ${x}<x^2$ $\Rightarrow$ $1<x$

for $x<0$

$\dfrac{x}{|x|}<x$ $\Rightarrow$ ${x}<x|x|$ $\Rightarrow$ ${x}<-x^2$ $\Rightarrow$ $-1>x$

Answer 1

The first one is correct. In the second answer the last step is wrong. $x <-x^{2}$ and $x <0$ gives $-1 <x$, not $-1 >x$ becasue when you divide by a negative number the inequality sign changes.