If ` log_(1//2)(4-x)gelog_(1//2)2-log_(1//2)(x-1)`,then x belongs to

Options

A. `(1, 2]`
B. `[3, 4)`
C. `(1, 3]`
D. `[1, 4)`

Explanation

To solve the inequality \( \log_{1/2}(4-x) \geq \log_{1/2}(2) - \log_{1/2}(x-1) \), we will follow these steps:

### Step 1: Rewrite the logarithmic expression
We start with the given inequality:
\[ \log_{1/2}(4-x) \geq \log_{1/2}(2) - \log_{1/2}(x-1) \]
Using the property of logarithms that states \( \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \), we can rewrite the right side:
\[ \log_{1/2}(4-x) \geq \log_{1/2}\left(\frac{2}{x-1}\right) \]

### Step 2: Eliminate the logarithm
Since the base of the logarithm \( \frac{1}{2} \) is less than 1, we need to reverse the inequality when we remove the logarithm:
\[ 4-x \leq \frac{2}{x-1} \]

### Step 3: Cross-multiply to eliminate the fraction
To eliminate the fraction, we cross-multiply:
\[ (4-x)(x-1) \leq 2 \]

### Step 4: Expand the left side
Expanding the left side gives us:
\[ 4x - 4 - x^2 + x \leq 2 \]
This simplifies to:
\[ -x^2 + 5x - 4 \leq 2 \]

### Step 5: Rearrange the inequality
Rearranging the inequality results in:
\[ -x^2 + 5x - 6 \leq 0 \]
Multiplying through by -1 (which reverses the inequality) gives:
\[ x^2 - 5x + 6 \geq 0 \]

### Step 6: Factor the quadratic
Factoring the quadratic expression:
\[ (x-2)(x-3) \geq 0 \]

### Step 7: Determine the intervals
To find the intervals where this inequality holds, we analyze the sign of the product:
1. The roots are \( x = 2 \) and \( x = 3 \).
2. The test intervals are \( (-\infty, 2) \), \( (2, 3) \), and \( (3, \infty) \).

Testing these intervals:
- For \( x < 2 \) (e.g., \( x = 1 \)): \( (1-2)(1-3) = (-)(-) > 0 \) (True)
- For \( 2 < x < 3 \) (e.g., \( x = 2.5 \)): \( (2.5-2)(2.5-3) = (+)(-) < 0 \) (False)
- For \( x > 3 \) (e.g., \( x = 4 \)): \( (4-2)(4-3) = (+)(+) > 0 \) (True)

### Step 8: Conclusion
The solution to the inequality is:
\[ x \leq 2 \quad \text{or} \quad x \geq 3 \]
Thus, the intervals where \( x \) belongs are:
\[ (-\infty, 2] \cup [3, \infty) \]

### Final Answer
The values of \( x \) belong to the intervals \( (1, 2] \cup [3, 4) \).