Solving for x in the first equation:
|5-x| + |x-1| = 10

Case 1: For 5-x ≥ 0 and x-1 ≥ 0 (x ≤ 5 and x ≥ 1)
(5-x) + (x-1) = 10
4 = 10
No solution for this case.

Case 2: For 5-x < 0 and x-1 ≥ 0 (x > 5 and x ≥ 1)
-(5-x) + (x-1) = 10
-4 = 10
No solution for this case.

Case 3: For 5-x ≥ 0 and x-1 < 0 (x ≤ 5 and x < 1)
(5-x) + -(x-1) = 10
6 = 10
No solution for this case.

Case 4: For 5-x < 0 and x-1 < 0 (x > 5 and x < 1)
-(5-x) + -(x-1) = 10
-6 = 10
No solution for this case.

Therefore, there are no solutions to the equation |5-x| + |x-1| = 10.

Solving for x in the second equation:
|4-x| + |x-2| = 2

Case 1: For 4-x ≥ 0 and x-2 ≥ 0 (x ≤ 4 and x ≥ 2)
(4-x) + (x-2) = 2
2 = 2
x = 3

Case 2: For 4-x < 0 and x-2 ≥ 0 (x > 4 and x ≥ 2)
-(4-x) + (x-2) = 2
-2 = 2
No solution for this case.

Case 3: For 4-x ≥ 0 and x-2 < 0 (x ≤ 4 and x < 2)
(4-x) + -(x-2) = 2
4 = 2
No solution for this case.

Case 4: For 4-x < 0 and x-2 < 0 (x > 4 and x < 2)
-(4-x) + -(x-2) = 2
-4 = 2
No solution for this case.

Therefore, the only solution to the equation |4-x| + |x-2| = 2 is x = 3.