Help with a system of inequalities with absolute values

$\{\begin{array}{c}|x-3|<2x\\ |2x+5|>3\end{array}$

The steps I'm taking are:

Finding the absolute values sings, so for $x-3\ge 0$ we have $x\ge 3$ therefore

$|x-3|=\{\begin{array}{cc}x-3& \text{for}x\ge 3\\ -x+3& \text{for}x3\end{array}$

and

$2x+5\ge 0$ we have $x\ge \frac{-2}{5}$ therefore

$|2x+5|=\{\begin{array}{cc}2x+5& \text{for}x\ge \frac{-2}{5}\\ -2x-5& \text{for}x\frac{-2}{5}\end{array}$

So I build a few systems with the complete inequalities, for the first one we have:

$\{\begin{array}{c}x\ge 3\\ x-3<2x=x>-3\end{array}$

So the solution here would be $x>3$, then:

$\{\begin{array}{c}x<3\\ -x+3<2x=x>1\end{array}$

The solution would be $1<x<3$, then

$\{\begin{array}{c}x\ge \frac{-2}{5}\\ 2x+5>3=x>-1\end{array}$

So the solution of the system is $x>-1$, then

$\{\begin{array}{c}x<\frac{-2}{5}\\ -2x-5>3=x<-4\end{array}$

And the solution is $x<-4$

What am I doing wrong?