Given equation:
|x^2-2x| |x-4| > |x^2-3x-4|

Overview:
We are required to solve the inequality |x^2-2x| |x-4| > |x^2-3x-4|.

Breakdown of the Solution:

1. Solving for x^2-2x > 0:
To determine the values of x for which the expression |x^2-2x| is positive, we need to solve the inequality x^2-2x > 0.

1.1. Finding the critical points:
To find the critical points of the inequality, we set x^2-2x equal to zero and solve for x:

x^2-2x = 0
x(x-2) = 0

The critical points are x = 0 and x = 2.

1.2. Analyzing the intervals:
We need to analyze the intervals between the critical points to determine the sign of x^2-2x:

Interval 1: x < />
Choose a test point within this interval, such as x = -1. Substitute it into x^2-2x:

(-1)^2 - 2(-1) = 1 + 2 = 3

Since the result is positive, x^2-2x > 0 for x < />

Interval 2: 0 < x="" />< />
Choose a test point within this interval, such as x = 1. Substitute it into x^2-2x:

(1)^2 - 2(1) = 1 - 2 = -1

Since the result is negative, x^2-2x < 0="" for="" 0="" />< x="" />< />

Interval 3: x > 2
Choose a test point within this interval, such as x = 3. Substitute it into x^2-2x:

(3)^2 - 2(3) = 9 - 6 = 3

Since the result is positive, x^2-2x > 0 for x > 2.

1.3. Conclusion:
From the analysis of intervals, we can conclude that x^2-2x > 0 for x < 0="" and="" x="" /> 2.

2. Solving for x-4 > 0:
To determine the values of x for which the expression |x-4| is positive, we need to solve the inequality x-4 > 0.

2.1. Finding the critical points:
To find the critical points of the inequality, we set x-4 equal to zero and solve for x:

x-4 = 0
x = 4

The critical point is x = 4.

2.2. Analyzing the intervals:
We need to analyze the intervals between the critical points to determine the sign of x-4:

Interval 1: x < />
Choose a test point within this interval, such as x = 3. Substitute it into x-4:

3 - 4 = -1