https://file.aiquickdraw.com/mathbot/file/17255699618310bb2t7do.jpg

The inequality in the image is:

\[
\frac{x - 10}{x - 1} \geq 4
\]

To solve this, let's follow these steps:

### Step 1: Move all terms to one side
Rewrite the inequality by subtracting 4 from both sides:

\[
\frac{x - 10}{x - 1} - 4 \geq 0
\]

### Step 2: Combine the expression into one fraction
Convert the 4 into a fraction with a denominator of \(x - 1\):

\[
\frac{x - 10}{x - 1} - \frac{4(x - 1)}{x - 1} \geq 0
\]

Simplify the numerator:

\[
\frac{(x - 10) - 4(x - 1)}{x - 1} \geq 0
\]

Expand the terms in the numerator:

\[
\frac{x - 10 - 4x + 4}{x - 1} \geq 0
\]

Simplify further:

\[
\frac{-3x - 6}{x - 1} \geq 0
\]

### Step 3: Analyze the inequality
We now need to solve:

\[
\frac{-3(x + 2)}{x - 1} \geq 0
\]

This fraction will be non-negative when the numerator and denominator have the same sign. Let's consider the critical points: \(x = -2\) and \(x = 1\), where either the numerator or the denominator is zero.

### Step 4: Critical points and sign analysis
The critical points are \(x = -2\) and \(x = 1\). Now, we perform a sign analysis for the intervals determined by these critical points: \( (-\infty, -2), (-2, 1), (1, \infty) \).

- For \(x < -2\), both the numerator and denominator are negative, so the fraction is positive.
- For \(-2 < x < 1\), the numerator is positive, and the denominator is negative, so the fraction is negative.
- For \(x > 1\), both the numerator and denominator are positive, so the fraction is positive.

### Step 5: Conclusion
The inequality is satisfied when the expression is non-negative. So, the solution includes:

\[
x \in (-\infty, -2] \cup (1, \infty)
\]

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

\[
x \in (-\infty, -2] \cup (1, \infty)
\]

Do you need more details or have any questions?

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Here are five related questions:
1. How can we solve rational inequalities that involve more complicated expressions?
2. What is the significance of critical points in solving inequalities?
3. How can we graph the solution of a rational inequality on a number line?
4. How do we check for extraneous solutions when solving rational equations or inequalities?
5. How can we use test points to determine the sign of a rational function in an interval?

**Tip:** Always remember to exclude points where the denominator is zero from your solution set, as the function is undefined at those points.

Solve the inequality: (x - 10) / (x - 1) ≥ 4

Learn how to solve the rational inequality (x - 10) / (x - 1) ≥ 4 with a detailed step-by-step solution. This problem involves key algebraic concepts, including rational inequalities, sign analysis, and solving by critical points. Suitable for students in Grades 10-12.

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