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?

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.