The solution set of the inequality `log_(x)((x+3)/(1-2x))gt1` is

Options

A. `0ltxlt(1)/(2)`
B. `xgt3`
C. null set
D. `(1)/(2)ltxlt1`

Answer

To solve the inequality \( \log_{x}\left(\frac{x+3}{1-2x}\right) > 1 \), we will follow these steps:

### Step 1: Understand the properties of logarithms
For the logarithm \( \log_{x}(A) \) to be defined, the base \( x \) must be greater than 0 and not equal to 1. Additionally, the argument \( A \) must be greater than 0.

### Step 2: Set up the conditions for the logarithm
1. **Base condition**: \( x > 0 \) and \( x \neq 1 \)
2. **Argument condition**:
\[ \frac{x+3}{1-2x} > 0 \]

### Step 3: Solve the argument condition
To solve \( \frac{x+3}{1-2x} > 0 \), we need to determine when the numerator and denominator are both positive or both negative.

- **Numerator**: \( x + 3 > 0 \) implies \( x > -3 \)
- **Denominator**: \( 1 - 2x > 0 \) implies \( x < \frac{1}{2} \)

### Step 4: Analyze the sign of the fraction
Now we can analyze the sign of the fraction:
- The fraction \( \frac{x+3}{1-2x} \) is positive when both conditions are satisfied:
- \( x > -3 \)
- \( x < \frac{1}{2} \)

Combining these gives us:
\[ -3 < x < \frac{1}{2} \]

### Step 5: Consider the base conditions
From the base conditions, we have:
1. \( x > 0 \)
2. \( x \neq 1 \)

### Step 6: Combine the conditions
Combining \( -3 < x < \frac{1}{2} \) with \( x > 0 \) gives:
\[ 0 < x < \frac{1}{2} \]

### Step 7: Solve the inequality
Now we will solve the inequality \( \log_{x}\left(\frac{x+3}{1-2x}\right) > 1 \). This can be rewritten as:
\[ \frac{x+3}{1-2x} > x^1 \]

### Step 8: Rearranging the inequality
Rearranging gives:
\[ \frac{x+3}{1-2x} - x > 0 \]
Finding a common denominator, we have:
\[ \frac{x+3 - x(1-2x)}{1-2x} > 0 \]
This simplifies to:
\[ \frac{x + 3 - x + 2x^2}{1-2x} > 0 \implies \frac{2x^2 + 3}{1-2x} > 0 \]

### Step 9: Analyze the new inequality
1. **Numerator**: \( 2x^2 + 3 > 0 \) is always true since \( 2x^2 \) is non-negative and 3 is positive.
2. **Denominator**: \( 1 - 2x > 0 \) implies \( x < \frac{1}{2} \).

### Step 10: Conclusion
Thus, the solution set for the inequality \( \log_{x}\left(\frac{x+3}{1-2x}\right) > 1 \) is:
\[ 0 < x < \frac{1}{2} \]

### Final Answer
The solution set is \( (0, \frac{1}{2}) \).