Which of the following is not the solution `(logx)(5/2-1/x) gt (5/2-1/x)` ?

Options

A. `(2/5,1/2)`
B. `(1,2)`
C. ` (2/5, 1)`
D. none of these

Explanation

To solve the inequality \( (\log x)\left(\frac{5}{2} - \frac{1}{x}\right) > \left(\frac{5}{2} - \frac{1}{x}\right) \), we will follow these steps:

### Step 1: Rearranging the Inequality
We start by moving all terms to one side of the inequality:
\[ (\log x)\left(\frac{5}{2} - \frac{1}{x}\right) - \left(\frac{5}{2} - \frac{1}{x}\right) > 0 \]
This can be factored as:
\[ \left(\frac{5}{2} - \frac{1}{x}\right)(\log x - 1) > 0 \]

### Step 2: Identifying Critical Points
Next, we identify the critical points where each factor is zero:
1. \( \frac{5}{2} - \frac{1}{x} = 0 \) gives \( x = \frac{2}{5} \).
2. \( \log x - 1 = 0 \) gives \( x = 10 \).

### Step 3: Testing Intervals
We will test the intervals determined by the critical points \( x = \frac{2}{5} \) and \( x = 10 \):
- **Interval 1**: \( (-\infty, \frac{2}{5}) \)
- **Interval 2**: \( \left(\frac{2}{5}, 10\right) \)
- **Interval 3**: \( (10, \infty) \)

### Step 4: Analyzing Each Interval
1. **For \( x < \frac{2}{5} \)**:
- \( \frac{5}{2} - \frac{1}{x} > 0 \) (since \( x \) is negative)
- \( \log x - 1 < 0 \) (since \( x < 1 \))
- Product: Negative

2. **For \( \frac{2}{5} < x < 10 \)**:
- \( \frac{5}{2} - \frac{1}{x} > 0 \)
- \( \log x - 1 < 0 \) (since \( x < 10 \))
- Product: Negative

3. **For \( x > 10 \)**:
- \( \frac{5}{2} - \frac{1}{x} > 0 \)
- \( \log x - 1 > 0 \)
- Product: Positive

### Step 5: Conclusion
The inequality \( \left(\frac{5}{2} - \frac{1}{x}\right)(\log x - 1) > 0 \) holds true when \( x > 10 \).

### Step 6: Finding the Non-Solution
Now we check the options provided to find which is not a solution:
- **Option 1**: \( x > 10 \) (solution)
- **Option 2**: \( 0 < x < \frac{2}{5} \) (not a solution)
- **Option 3**: \( 1 < x < 2 \) (not a solution)
- **Option 4**: \( \frac{2}{5} < x < 10 \) (not a solution)

The option that is not a solution is any of those that fall within the intervals \( (0, \frac{2}{5}) \) and \( (2/5, 10) \).

### Final Answer
The answer is any option that suggests \( x \) is in the intervals \( (0, \frac{2}{5}) \) or \( (2/5, 10) \).