Thank you for the clarification. The inequality in the image is:

\[
\frac{2}{x-1} \geq 1
\]

Let's solve this inequality step by step.

### Step 1: Subtract 1 from both sides
We start by subtracting 1 from both sides of the inequality:

\[
\frac{2}{x-1} - 1 \geq 0.
\]

### Step 2: Combine into a single fraction
Next, we combine the left-hand side into a single fraction by expressing 1 as a fraction with the same denominator:

\[
\frac{2 - (x-1)}{x-1} \geq 0.
\]

Simplify the numerator:

\[
\frac{2 - x + 1}{x-1} \geq 0,
\]

which simplifies further to:

\[
\frac{3-x}{x-1} \geq 0.
\]

### Step 3: Analyze the sign of the fraction
Now, let's analyze when the fraction \(\frac{3-x}{x-1}\) is non-negative (greater than or equal to 0). This will occur when:

1. **Both the numerator and denominator are positive:** \(3-x > 0\) and \(x-1 > 0\).
2. **Both the numerator and denominator are negative:** \(3-x < 0\) and \(x-1 < 0\).
3. **The fraction equals zero:** \(3-x = 0\).

#### Case 1: Both positive
- \(3-x > 0 \Rightarrow x < 3\).
- \(x-1 > 0 \Rightarrow x > 1\).

So, \(1 < x < 3\).

#### Case 2: Both negative
- \(3-x < 0 \Rightarrow x > 3\).
- \(x-1 < 0 \Rightarrow x < 1\).

This case is not possible because \(x\) cannot be both greater than 3 and less than 1 simultaneously.

#### Case 3: Fraction equals zero
- \(3-x = 0 \Rightarrow x = 3\).

### Step 4: Combine the intervals
From the analysis:
- The solution includes \(x\) in the interval \(1 < x \leq 3\).

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

\[
x \in (1, 3].
\]

If you have any further questions or need additional explanations, feel free to ask!

Here are some related questions to consider:
1. How would the solution change if the inequality was \(\frac{2}{x-1} > 1\)?
2. What happens if the inequality is \(\frac{2}{x-1} \leq 1\)?
3. Can you solve \(\frac{2}{x-1} \geq k\) for a constant \(k\)?
4. What is the solution to \(\frac{a}{x-b} \geq c\) where \(a\), \(b\), and \(c\) are constants?
5. How does solving rational inequalities differ from solving polynomial inequalities?

**Tip:** When dealing with rational inequalities, it's crucial to consider the domain where the denominator is not zero.

Learn how to solve the rational inequality 2/(x-1) >= 1 with a detailed step-by-step solution. This problem covers key concepts in inequalities and rational functions. Suitable for advanced high school students in Grades 11-12.

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