I am having trouble finding the correct method for solving algebraic fractions. I have tried the sequence of operations below and it appears to fit, but would appreciate some help / guidance on a robust process for working through to solve these equations. Thanks

$\displaystyle Solve for x$

$\displaystyle x+5/x - 8/2x-3 = 0$

can be rewritten as $\displaystyle x+5/x = 8/2x-3$

which cancels out to $\displaystyle 5 = 8/2x-3$

which can be rewritten as $\displaystyle 8/5 = 2x-3 => 1 $ $\displaystyle 3/5 = 2x-3$

and therefore $\displaystyle 2x = 1 3/5 + 3$

and $\displaystyle x = 1/2(4 $ $\displaystyle 3/5)$

==> $\displaystyle x = 2 $ $\displaystyle 3/10$