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 follow these steps:

### Step 1: Analyze the expressions inside the absolute values
We need to identify the points where the expressions inside the absolute values change their signs. The expressions are:
1. \( x^2 - 2x = x(x - 2) \)
2. \( x - 4 \)
3. \( x^2 - 3x + 4 \)

**Finding the roots:**
- For \( x^2 - 2x = 0 \):
- \( x(x - 2) = 0 \) gives \( x = 0 \) and \( x = 2 \).
- For \( x - 4 = 0 \):
- \( x = 4 \).
- For \( x^2 - 3x + 4 = 0 \):
- The discriminant is \( (-3)^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7 \) (no real roots).

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