Question 23673
This is really hard to be sure what you meant.  Let me take a shot at it.
{{{2/(5X +6)=1/(6X-5) }}}


If so, then remember that old rule {{{a/b= c/d}}} means that {{{ad= bc}}}.


In the same way, {{{2/(5X +6)=1/(6X-5) }}} means that {{{2(6x-5) = 1(5x+6)}}}
{{{12x - 10 = 5x+6}}}
{{{7x = 16}}}
{{{x = 16/7}}}


On the other hand, the problem could be interpreted to be:

{{{(2/5)*X + 6  = (1/6)*X - 5}}}


In this case, the LCD is 30, so multiply both sides by 30 to clear the fractions.

{{{30*(2/5)*X + 30*6  =30*(1/6)*X - 30*5}}}
{{{12X+ 180= 5X - 150}}}


Subtract 5x from each side:
{{{12x-5x + 180 = 5x-5x - 150}}}

{{{7x + 180 =-150}}}

Subtract 180 from each side:
{{{7x +180 -180 = -150-180}}}
{{{7x = -330}}}


Divide both sides by 7:

{{{(7x)/7 = -330/7}}}
{{{x= -330/7}}} another UGLY answer!  

R^2 at SCC