solve |2x-1|=|3x+5|

$$\\|2x-1|=|3x+5|\\\\ 2x-1=3x+5\qquad or \qquad 2x-1=-(3x+5)\\\\ 2x=3x+6\qquad or \qquad 2x-1=-3x-5\\\\ -x=6\qquad or \qquad 2x=-3x-4\\\\ x=-6\qquad or \qquad 5x=-4\\\\ x=-6\qquad or \qquad x=\frac{-4}{5}\\\\ $check$\\\\ If\;\; x=-6\\\\ |-12-1|=|-18+5|\\\\ 13=13\qquad True\\\\$$

$$\\If\;\; x=\frac{-4}{5}\\\\ |\frac{2*-4}{5}-1|=|\frac{3*-4}{5}+5|\\\\ |\frac{2*-4-5}{5}|=|\frac{3*-4+25}{5}|\\\\ |\frac{-13}{5}|=|\frac{13}{5}|\\\\ \frac{13}{5}= \frac{13}{5} \qquad True\\\\ $therefore the solutions are $\\\\ x=-6\;\;and \;\; x=\frac{-4}{5}\\\\$$

Here is the graphical solutions

https://www.desmos.com/calculator/rvzeay4sfu