Question 88323
Ok, your first move was good! So you have:
{{{9/(x-5) - 8/(x+5) - 1 = 0}}} Now add 1 (move the -1 to the right side) to both sides.
{{{9/(x-5) - 8/(x+5) = 1}}} Now add the two fractions on the left side.  The LCD is {{{(x-5)(x+5)}}} so you now have:
{{{(9(x+5)-8(x-5))/(x-5)(x+5) = 1}}} Simplify the numerator.
{{{(9x+45-8x+40)/(x-5)(x+5) = 1}}} Continue to simplify.
{{{(x+85)/(x-5)(x+5) = 1}}} Now multiply both sides by {{{(x-5)(x+5)}}} to get:
{{{x+85 = (x-5)(x+5)}}} Expand the right side.
{{{x+85 = x^2-25}}} Now subtract {{{x+85}}} from both sides.
{{{0 = x^2-x-110}}} or {{{x^2-x-110 = 0}}} This can be factored:
{{{(x+10)(x-11) = 0}}} Apply the zero products principle to get:
{{{x = -10}}} or {{{x = 11}}} These are the solutions!
Let's check the solutions by substituting the solutions, one-by-one into the original equation:
{{{(9/(x-5))-1 = 8/(x+5)}}} Substitute x = -10.
{{{(9/(-10-5))-1 = 8/(-10+5)}}}
{{{(9/(-15)) - 1 = 8/(-5)}}}
{{{(9/(-15)) - ((-15)/(-15)) = 8/(-5)}}}
{{{-24/15 = -(8/5)}}} Reduce the fraction on the left side.
{{{-(8/5) = -(8/5)}}} It checks!
Try the other solution yourself and see how you do.