To solve this equation, we need to consider two cases: when the expression inside the absolute value is positive and when it is negative.

Case 1: When 4x - 2 is positive (4x - 2 > 0)
In this case, we can remove the absolute value signs without changing the equation:
4x - 2 = x + 10

Simplifying this equation, we get:
3x = 12
x = 4

Case 2: When 4x - 2 is negative (4x - 2 < 0)
In this case, we need to change the sign of the expression inside the absolute value:
-(4x - 2) = x + 10

Simplifying this equation, we get:
-4x + 2 = x + 10

Combining like terms, we get:
-5x = 8
x = -8/5

Therefore, the solutions to the equation |4x-2| = |x+10| are x = 4 and x = -8/5.