$\displaystyle 2x+3\ \ge\ x-5\Rightarrow\ x+x-5+8\ \ge\ x-5$
$\displaystyle \Rightarrow\ (x+8)+x-5\ \ge\ x-5$
Both sides are equal if $\displaystyle x+8=0$
The left-hand side is greater than the right-hand side if $\displaystyle x+8>0$
Therefore the solution is $\displaystyle x+8\ \ge\ 0\Rightarrow\ x\ \ge\ -8$