To solve this equation, we can break it down into different cases based on the value of x.

Case 1: x < 1
In this case, |x - 1| = -(x - 1) = -x + 1
|x - 3| = -(x - 3) = -x + 3
|x - 6| = -(x - 6) = -x + 6

Substituting these values into the equation, we get:
(-x + 1) - 3(-x + 3) + 5(-x + 6) = -(x - 3)

Simplifying, we have:
-x + 1 + 3x - 9 - 5x + 30 = -x + 3

Combining like terms, we get:
-3x + 22 = -x + 3

Adding 3x to both sides and subtracting 3 from both sides, we get:
22 = 2x

Dividing both sides by 2, we get:
x = 11

However, since x < 1 in this case, x = 11 is not a valid solution.

Case 2: 1 ≤ x < 3
In this case, |x - 1| = x - 1
|x - 3| = -(x - 3) = -x + 3
|x - 6| = -(x - 6) = -x + 6

Substituting these values into the equation, we get:
(x - 1) - 3(-x + 3) + 5(-x + 6) = -(x - 3)

Simplifying, we have:
x - 1 + 3x - 9 - 5x + 30 = -x + 3

Combining like terms, we get:
-2x + 20 = -x + 3

Adding 2x to both sides and subtracting 3 from both sides, we get:
17 = x

Since 1 ≤ x < 3 in this case, x = 17 is not a valid solution.

Case 3: 3 ≤ x < 6
In this case, |x - 1| = x - 1
|x - 3| = x - 3
|x - 6| = -(x - 6) = -x + 6

Substituting these values into the equation, we get:
(x - 1) - 3(x - 3) + 5(-x + 6) = -(x - 3)

Simplifying, we have:
x - 1 - 3x + 9 - 5x + 30 = -x + 3

Combining like terms, we get:
-7x + 38 = -x + 3

Adding 7x to both sides and subtracting 3 from both sides, we get:
35 = 6x

Dividing both sides by 6, we get:
x = 35/6

Since 3 ≤ x < 6 in this case, x = 35/6 is a valid solution.

Case 4: x ≥ 6
In this case, |x - 1| = x - 1
|x - 3| = x - 3
|x - 6| = x - 6

Substituting these values into the equation, we get:
(x - 1) - 3(x - 3) + 5(x - 6) = -(x - 3)

Simplifying, we have:
x - 1 - 3x + 9 + 5x - 30 = -x + 3

Combining like terms, we get:
3x - 22 = -x + 3

Adding x to both sides and adding 22 to both sides, we get:
4x = 25

Dividing both sides by 4, we get:
x = 25/4

Since x ≥ 6 in this case, x = 25/4 is a valid solution.

In summary, the solutions to the equation |x - 1| - 3|x - 3| + 5|x - 6| = |x - 3| are x = 35/6 and x = 25/4.