To solve this equation, we need to consider the different cases when the absolute values are positive or negative.

Case 1:
When both | x - 3 | and | x - 5 | are positive:
x - 3 + x - 5 = 2
2x - 8 = 2
2x = 10
x = 5

Case 2:
When | x - 3 | is positive and | x - 5 | is negative:
x - 3 - (x - 5) = 2
x - 3 - x + 5 = 2
2 = 2 (This case is not valid)

Case 3:
When | x - 3 | is negative and | x - 5 | is positive:
-(x - 3) + x - 5 = 2

x + 3 + x - 5 = 2
-2 = 2 (This case is not valid)

Case 4:
When both | x - 3 | and | x - 5 | are negative:
-(x - 3) - (x - 5) = 2

x + 3 - x + 5 = 2
8 = 2 (This case is not valid)

Therefore, the only solution to the equation | x - 3 | + | x - 5 | = 2 is x = 5.