To solve the inequality \( 2x \ln\left(\frac{2}{x}\right) - x > 0 \), let's go through the steps:

### Step 1: Rewrite the expression
Start by simplifying the logarithmic term:

\[
2x \ln\left(\frac{2}{x}\right) = 2x \left(\ln(2) - \ln(x)\right)
\]

So the inequality becomes:

\[
2x (\ln(2) - \ln(x)) - x > 0
\]

### Step 2: Distribute the \( 2x \)
Expanding the terms:

\[
2x \ln(2) - 2x \ln(x) - x > 0
\]

Group similar terms:

\[
x(2 \ln(2) - 2 \ln(x) - 1) > 0
\]

### Step 3: Analyze the inequality
Now we need to determine when \( x(2 \ln(2) - 2 \ln(x) - 1) > 0 \).

1. \( x > 0 \) (since the logarithmic function \( \ln(x) \) is only defined for \( x > 0 \))
2. Solve the inequality \( 2 \ln(2) - 2 \ln(x) - 1 > 0 \).

Let’s now isolate the logarithmic term. Divide by 2:

\[
\ln(2) - \ln(x) - \frac{1}{2} > 0
\]

### Step 4: Simplify further
Combine the logarithms:

\[
\ln\left(\frac{2}{x}\right) > \frac{1}{2}
\]

Exponentiate both sides:

\[
\frac{2}{x} > e^{1/2}
\]

Since \( e^{1/2} = \sqrt{e} \), we get:

\[
\frac{2}{x} > \sqrt{e}
\]

### Step 5: Solve for \( x \)
Multiply both sides by \( x \) and divide by \( \sqrt{e} \):

\[
x < \frac{2}{\sqrt{e}}
\]

Thus, the inequality holds when \( 0 < x < \frac{2}{\sqrt{e}} \).

### Final Answer:
The solution to the inequality is:

\[
0 < x < \frac{2}{\sqrt{e}}
\]

Would you like to go into more detail about any of these steps?

### Related Questions:
1. How do we solve inequalities involving logarithms in general?
2. What are the domain restrictions for logarithmic functions?
3. How do we simplify expressions involving multiple logarithmic terms?
4. How does exponentiating both sides of an inequality affect the solution?
5. What properties of \( e \) are useful in solving inequalities?

### Tip:
Always check the domain of a logarithmic function before solving the inequality or equation.

This guide explains how to solve the inequality 2x ln(2/x) - x > 0 step by step, involving logarithmic functions, inequalities, and exponential functions. Suitable for advanced high school students or undergraduate learners.

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