I am trying to simplify the function $D(\alpha,\beta)$ shown below (with $\alpha,\beta>0):$
$$ D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{|\alpha-1|}{2} - 2 \left(\frac{\frac{1}{2}+\frac{|2\alpha-1|}{2}+\beta}{2}+\frac{|\beta-\frac{1}{2}-\frac{|2\alpha-1|}{2}|}{2}\right)-2 \left(\frac{\alpha+\beta}{2}-\frac{|\beta-\alpha|}{2}\right)=0$$
This is how I proceed
${\it Case 1: }$ $$\alpha-1>0 \hspace{1cm}\text{and}\hspace{1cm} \beta-\alpha>0 $$ $$D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{\alpha-1}{2} - 2 \left(\frac{\frac{1}{2}+\frac{2\alpha-1|}{2}+\beta}{2}+\frac{\beta-\frac{1}{2}-\frac{2\alpha-1}{2}}{2}\right)-2 \left(\frac{\alpha+\beta}{2}-\frac{\beta-\alpha}{2}\right) $$
$$\rightarrow \alpha=\beta$$
${\it Case 2: }$ $$\alpha-1<0 \hspace{1cm}\text{and}\hspace{1cm} 2\alpha-1<0\hspace{1cm}\text{and} \hspace{1cm} \beta-\alpha<0 $$ $$ D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{1-\alpha}{2} - 2 \left(\frac{\frac{1}{2}+\frac{1-2\alpha}{2}+\beta}{2}+\frac{|\beta-\frac{1}{2}-\frac{1-2\alpha}{2}|}{2}\right)-2 \left(\frac{\alpha+\beta}{2}-\frac{-\beta+\alpha}{2}\right)=0$$
${\it Case 3: }$ $$\alpha-1<0 \hspace{1cm}\text{and}\hspace{1cm} 2\alpha-1>0\hspace{1cm}\text{and} \hspace{1cm} \beta-\alpha<0 $$
and so on..
As you can see its quite tedious, I want to ask, do you think my approach is correct or am I doing stuff wrong? Is this the only way for me?