The inequality given is:

\[
\frac{x}{x - 2} \geq \frac{2}{x - 2}
\]

To solve this, let's go through the following steps:

### Step 1: Subtract \(\frac{2}{x-2}\) from both sides
\[
\frac{x}{x - 2} - \frac{2}{x - 2} \geq 0
\]

Since both terms have the same denominator \(x - 2\), we can combine them:
\[
\frac{x - 2}{x - 2} \geq 0
\]

### Step 2: Simplify the expression
\[
\frac{x - 2}{x - 2} = 1
\]

Thus, the inequality becomes:
\[
1 \geq 0
\]

### Step 3: Interpret the result
The inequality \(1 \geq 0\) is **always true**, as 1 is indeed greater than or equal to 0. However, there is an exception: the expression is undefined when \(x = 2\) because the denominator becomes zero in both terms.

### Final Solution:
The solution to the inequality is all real numbers except \(x = 2\), which would make the expression undefined. Therefore, the solution set is:

\[
x \in (-\infty, 2) \cup (2, \infty)
\]

Let me know if you'd like further details on any step! Here are some related questions:

1. What happens when \(x = 2\) in rational expressions?
2. How do we solve rational inequalities in general?
3. Can you explain why dividing by zero is undefined?
4. How do we find the domain of a rational function?
5. What is the general approach for solving inequalities involving fractions?

**Tip:** When dealing with rational expressions, always check for values that make the denominator zero, as these points make the expression undefined.

This problem explores how to solve the rational inequality x/(x-2) >= 2/(x-2). Learn the steps for simplifying the expression, identifying undefined points, and determining the solution set. This solution is suitable for high school students learning about rational expressions and inequalities.

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