The solution set of inequation `|x^(2)-2x|+|x-4|gt|x^(2)-3x+4|` is a subset of

Options

A. `(0, oo)`
B. `(0, 4)`
C. `(0, 2)`
D. `(4, oo)`

Explanation

To solve the inequality \( |x^2 - 2x| + |x - 4| > |x^2 - 3x + 4| \), we will break it down step by step.

### Step 1: Analyze the absolute value expressions
We need to identify the points where the expressions inside the absolute values change their signs.

1. \( x^2 - 2x = 0 \) gives \( x(x - 2) = 0 \) → roots at \( x = 0 \) and \( x = 2 \).
2. \( x - 4 = 0 \) gives \( x = 4 \).
3. \( x^2 - 3x + 4 = 0 \) does not have real roots (discriminant \( = (-3)^2 - 4 \cdot 1 \cdot 4 < 0 \)).

Thus, the critical points are \( x = 0, 2, 4 \).

### Step 2: Test intervals
We will test the intervals determined by these critical points: \( (-\infty, 0) \), \( (0, 2) \), \( (2, 4) \), and \( (4, \infty) \).

#### Interval 1: \( (-\infty, 0) \)
- Here, \( x^2 - 2x \geq 0 \), \( x - 4 < 0 \), and \( x^2 - 3x + 4 > 0 \).
- The inequality becomes:
\[ (x^2 - 2x) - (4 - x) > (x^2 - 3x + 4) \]
Simplifying gives:
\[ x^2 - 2x - 4 + x > x^2 - 3x + 4 \]
\[ 2x - 8 > 0 \implies x > 4 \]
This is not possible in this interval.

#### Interval 2: \( (0, 2) \)
- Here, \( x^2 - 2x < 0 \), \( x - 4 < 0 \), and \( x^2 - 3x + 4 > 0 \).
- The inequality becomes:
\[ -(x^2 - 2x) - (4 - x) > (x^2 - 3x + 4) \]
Simplifying gives:
\[ -x^2 + 2x - 4 + x > x^2 - 3x + 4 \]
\[ -2x^2 + 6x - 8 > 0 \implies x^2 - 3x + 4 < 0 \]
This is not possible in this interval.

#### Interval 3: \( (2, 4) \)
- Here, \( x^2 - 2x > 0 \), \( x - 4 < 0 \), and \( x^2 - 3x + 4 > 0 \).
- The inequality becomes:
\[ (x^2 - 2x) - (4 - x) > (x^2 - 3x + 4) \]
Simplifying gives:
\[ x^2 - 2x - 4 + x > x^2 - 3x + 4 \]
\[ 2x - 8 > 0 \implies x > 4 \]
This is not possible in this interval.

#### Interval 4: \( (4, \infty) \)
- Here, \( x^2 - 2x > 0 \), \( x - 4 > 0 \), and \( x^2 - 3x + 4 > 0 \).
- The inequality becomes:
\[ (x^2 - 2x) + (x - 4) > (x^2 - 3x + 4) \]
Simplifying gives:
\[ x^2 - 2x + x - 4 > x^2 - 3x + 4 \]
\[ 2x - 8 > 0 \implies x > 4 \]
This is valid in this interval.

### Conclusion
The solution set of the inequality is \( (4, \infty) \).

### Final Answer
The solution set of the inequality is a subset of \( (4, \infty) \).