https://file.aiquickdraw.com/mathbot/file/17240731440930pchr9pz.jpg

It's a general mathematics of grade 11 humss 

The inequality shown in the image is:

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

Let's solve this inequality step by step.

### Step 1: Subtract 1 from both sides
To eliminate the 1 on the right-hand side, subtract 1 from both sides:

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

### Step 2: Combine into a single fraction
To combine the left-hand side into a single fraction, express 1 as a fraction with the same denominator:

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

Simplifying the numerator:

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

which simplifies further to:

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

### Step 3: Analyze the sign of the fraction
Now, let's analyze the sign of the fraction \(\frac{2-x}{x-1}\). The fraction will be non-negative (greater than or equal to 0) when:

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

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

So, \(1 < x < 2\).

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

This case is not possible because \(x\) cannot be both greater than 2 and less than 1 simultaneously.

#### Case 3: Fraction equals zero
- \(2-x = 0 \Rightarrow x = 2\).

### Step 4: Combine the intervals
From the analysis:
- The solution includes \(x\) in the interval \(1 < x \leq 2\).

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

\[
x \in (1, 2].
\]

Would you like more details on any part of the solution? 

Here are some related questions for further exploration:
1. How would the solution change if the inequality was \(\frac{1}{x-1} > 1\)?
2. What happens to the solution if the inequality is \(\frac{1}{x-1} \leq 1\)?
3. Can you solve \(\frac{1}{x-1} \geq k\) for some 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 linear inequalities?

**Tip:** Always consider the domain of the expression when dealing with inequalities, especially when the variable is in the denominator.

Learn how to solve the rational inequality 1/(x-1) >= 1 step-by-step with detailed explanations. This problem covers key concepts in inequalities, rational expressions, and interval notation, suitable for Grade 11 (HUMSS) students.

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