Solution for washing all dirty people's backs (3)
I believe the answer is:
lye
'solution for washing' is the definition.
(I've seen this before)
'all dirty people's backs' is the wordplay.
'backs' indicates one should take the final letters (I've seen 'back' mean this).
The final letters of 'all dirty people' are 'LYE'.
(Other definitions for lye that I've seen before include "solution that comes out in the wash?" , "Solution obtained by leaching" , "caustic liquid" , "Strong alkaline solution" , "Alkaline liquid" .)