The set of all the solution of the inequality `log_(2-x) (x-3) ge 1` is :

Options

A. `x lt 2`
B. `x gt 3`
C. `(x lt 2) cup (x gt 3)`
D. none of these

Answer

To solve the inequality \( \log_{(2-x)}(x-3) \geq 1 \), we will follow these steps:

### Step 1: Rewrite the Inequality
The inequality \( \log_{(2-x)}(x-3) \geq 1 \) can be rewritten in exponential form:
\[ x - 3 \geq (2 - x)^1 \]
This means:
\[ x - 3 \geq 2 - x \]

### Step 2: Solve the Exponential Inequality
Now, we will solve the inequality \( x - 3 \geq 2 - x \):
\[ x - 3 + x \geq 2 \]
\[ 2x - 3 \geq 2 \]
\[ 2x \geq 5 \]
\[ x \geq \frac{5}{2} \]

### Step 3: Determine the Domain of the Logarithm
Next, we need to ensure that the logarithm is defined. For \( \log_{(2-x)}(x-3) \) to be defined:
1. \( x - 3 > 0 \) (the argument of the logarithm must be positive)
\[ x > 3 \]
2. \( 2 - x > 0 \) (the base of the logarithm must be positive)
\[ x < 2 \]

### Step 4: Combine the Conditions
From the previous steps, we have:
- From the inequality, \( x \geq \frac{5}{2} \)
- From the domain of the logarithm, \( x > 3 \) and \( x < 2 \)

However, \( x > 3 \) and \( x < 2 \) cannot both be true simultaneously. Thus, there is no value of \( x \) that satisfies all conditions.

### Conclusion
The set of all solutions of the inequality \( \log_{(2-x)}(x-3) \geq 1 \) is:
\[ \text{No solution} \]

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