To solve this equation, we can first simplify the absolute value terms by considering different cases for the value of x.

Case 1: x < 1
In this case, all absolute value terms will be negative, so we can rewrite the equation as:
-(1-x) - (2-x) - (3-x) - (4-x) = -5x
Simplify:
-1 + x - 2 + x - 3 + x - 4 + x = -5x
Combine like terms:
-8 + 4x = -5x
Add 5x to both sides:
4x = -3x
Divide by 4:
x = 0

Case 2: 1 ≤ x < 2
In this case, the first absolute value term will be positive and the rest will be negative, so we can rewrite the equation as:
(1-x) - (2-x) - (3-x) - (4-x) = -5x
Simplify:
1 - x - 2 + x - 3 + x - 4 + x = -5x
Combine like terms:
-8 + 4x = -5x
Add 5x to both sides:
4x = -3x
Divide by 4:
x = 0

Case 3: 2 ≤ x < 3
In this case, the first two absolute value terms will be positive and the rest will be negative, so we can rewrite the equation as:
(1-x) + (2-x) - (3-x) - (4-x) = -5x
Simplify:
1 - x + 2 - x - 3 + x - 4 + x = -5x
Combine like terms:
-4 + 4x = -5x
Add 5x to both sides:
4x = -4x
Divide by 4:
x = 0

Case 4: 3 ≤ x < 4
In this case, the first three absolute value terms will be positive and the last one will be negative, so we can rewrite the equation as:
(1-x) + (2-x) + (3-x) - (4-x) = -5x
Simplify:
1 - x + 2 - x + 3 - x - 4 + x = -5x
Combine like terms:
2 - 4 = -5x
-2 = -5x
x = 2/5

Case 5: x ≥ 4
In this case, all absolute value terms will be positive, so we can rewrite the equation as:
(1-x) + (2-x) + (3-x) + (4-x) = -5x
Simplify:
1 - x + 2 - x + 3 - x + 4 - x = -5x
Combine like terms:
10 - 4x = -5x
Add 5x to both sides:
10 = -x
x = -10

Therefore, the solutions to the equation are x = 0, 2/5, and -10.