Question 117089
I'm not exactly sure what the equations are
 from your description, but I'll guess.
{{{x / ((6 - x) / 8) = 1}}}
Think of this as:
{{{(1 / ((6 - x) / 8)) * x = 1}}}
This is like {{{(1 / (a/b)) * x = 1}}}
{{{1 / (a/b) = b / a}}} so,
{{{1 / ((6 - x) / 8) = 8 / (6 - x)}}}
Now put it back together
{{{ (8 / (6 - x)) * x = 1}}}
Now multiply both sides by {{{6 - x}}}
{{{8x = 6 - x}}}
{{{9x = 6}}}
{{{x = 2/3}}} answer
Now check the answer by plugging it back into the original problem
{{{x / ((6 - x) / 8) = 1}}}
{{{(2/3) / ((6 - (2/3)) / 8) = 1}}}
{{{(2/3) / (((18/3) - (2/3)) / 8) = 1}}}
{{{(2/3) / ((16 / 3)/8) = 1}}}
OK, just as before , this is the same as
{{{(1 / ((16/3) / 8)) *  (2/3) = 1}}}
{{{(8 / (16/3)) * (2/3) = 1}}}
multiply both sides by {{{16/3}}}
{{{8 * (2/3) = 16/3}}}
Now multiply both sides by {{{3}}}
{{{8 * 2 = 16}}}
{{{16 = 16}}}
OK
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My guess for the other equation is
{{{(x / (x - 2)) - ((x - 1) / x) = 8 / (x^2 - 2x)}}}
The first step is to factor the right side
{{{(x / (x - 2)) - ((x - 1) / x) = 8 / (x(x - 2))}}} 
Now multiply both sides by {{{x(x - 2)}}}
{{{x^2 - (x - 1)(x - 2) = 8}}}
{{{x^2 - x^2 + 3x - 2 = 8}}}
{{{3x - 2 = 8}}}
{{{3x = 10}}}
{{{x = 10/3}}} answer
check answer
{{{(x / (x - 2)) - ((x - 1) / x) = 8 / (x^2 - 2x)}}}
{{{((10/3)/((10/3) - 2)) - (((10/3) - 1)/(10/3)) = 8 / ((10/3)^2 - 2*(10/3))}}}
{{{((10/3)/(4/3)) - ((7/3)/(10/3)) = 8 / ((100/9) - (20/3))}}}
{{{(10/4) - (7/10) = 8 / (40/9)}}}
{{{10/4 - 7/10 = 72 / 40}}}
multiply both sides by {{{40}}}
{{{100 - 28 = 72}}}
{{{72 = 72}}}
OK