To solve this inequality, let's go through the process step-by-step:

Given: $\frac{6}{x-1} - \frac{6}{x} \geq 1$

1. Combine Fractions: Start by finding a common denominator for the two fractions on the left side.

   $\frac{6}{x-1} - \frac{6}{x} = \frac{6x - 6(x-1)}{x(x-1)} = \frac{6x - 6x + 6}{x(x-1)} = \frac{6}{x(x-1)}$

   So, the inequality becomes: $\frac{6}{x(x-1)} \geq 1$

2. Rearrange the Inequality: Move the 1 to the left side.

   $\frac{6}{x(x-1)} - 1 \geq 0$

3. Rewrite as a Single Fraction:

   Express the left side with a common denominator: $\frac{6 - x(x-1)}{x(x-1)} \geq 0$ Simplify the numerator: $6 - x^2 + x = -x^2 + x + 6$ The inequality now is: $\frac{-x^2 + x + 6}{x(x-1)} \geq 0$

4. Factor the Numerator: Factor $-x^2 + x + 6$.

   This can be rewritten as: $-(x^2 - x - 6) = -(x - 3)(x + 2)$

   Thus, the inequality becomes: $\frac{-(x - 3)(x + 2)}{x(x - 1)} \geq 0$

5. Identify Critical Points: The critical points are where the expression equals zero or is undefined, namely:

   • $x = 3$
   • $x = -2$
   • $x = 0$
   • $x = 1$

6. Test Intervals: Use these critical points to test the intervals:

   • $(-\infty, -2)$
   • $(-2, 0)$
   • $(0, 1)$
   • $(1, 3)$
   • $(3, \infty)$

7. Determine the Signs: Test each interval to see where the inequality holds true (i.e., where the expression is non-negative).

8. Solution: After testing, you'll find that the solution to the inequality is: $(-\infty, -2] \cup (3, \infty)$

Conclusion

The correct answer should be: $(-\infty, -2] \cup (3, \infty)$

The mistake in your original answer was including intervals that do not satisfy the inequality.

Would you like more details or have any questions about these steps?

Here are some additional questions to deepen your understanding:

1. How do you find a common denominator for rational expressions?
2. Why is it important to test intervals when solving inequalities?
3. What does each interval represent in the context of this inequality?
4. How do you handle inequalities that involve undefined points in their domain?
5. How does factoring help simplify and solve inequalities?

Tip: Always check each interval around the critical points to ensure they satisfy the inequality before finalizing your answer.