algebra
posted by beth .
5/4x+1/8x=5/8+x

Please use parentheses
do you mean
(5/4)x
or
5/(4x) etc 
5/4x+1/8x=5/8+x
10/8x + 1/8x = 5/8 + x
11/8x = 5/8 + x
11/8x + x = 5/8
19/8x = 5/8
x = (5/8) / (19/8)
x = (5/8)* (8/19)
x = 40/152 = 5/19 
Asuming the problem to be
5/4x + 1/8x = 5/8 + x
(10 + 1) / 8x = (5 + 8x) / 8
ie. 11/ 8x = (8x + 5) / 8
11 = 8x (8x +5) / 8
ie. 11 = 8x ^2 + 5x
We will now have to handle it as a quadratic equation:
8x ^2 + 5x  11 = 0
Apply the formula to get the roots:
(b + Sqrt ( b^2  4ac) ) / 2a

Hope this clarifies enough! 
If you meant
5/(4x) + 1/(8x) = 5/(8+x) then follow Kay's solution
if you meant it the way you typed it, the follow Ms. Sue's way, except I noted an error in her 3rd line, should have been ....
11/8x  x = 5/8
3/8x = 5/8
x = (5/8) / (3/8)
x = (5/8)* (8/3)
x = 40/152 = 5/3 
Oops! Thanks for the correction, Reiny.