Thank you for clarifying that it's a rational inequality. Let's approach it with the appropriate method for solving rational inequalities. The inequality given is:

$\frac{3}{x-2} < \frac{1}{2}$

Steps to solve the rational inequality:

1. Move all terms to one side to have zero on the other side: $\frac{3}{x-2} - \frac{1}{2} < 0$

2. Find a common denominator to combine the fractions: The common denominator for the fractions $\frac{3}{x-2}$ and $\frac{1}{2}$ is $2(x-2)$. $\frac{6 - (x-2)}{2(x-2)} < 0$

3. Simplify the numerator: $\frac{6 - x + 2}{2(x-2)} < 0$ $\frac{8 - x}{2(x-2)} < 0$

4. Identify the critical points by setting the numerator and denominator equal to zero**:

   • $8 - x = 0$ gives $x = 8$.
   • $x - 2 = 0$ gives $x = 2$.

   These are the points where the expression can change sign.

5. Test the intervals determined by the critical points $x = 2$ and $x = 8$:

   • For $x < 2$, say $x = 0$: $\frac{8 - 0}{2(0-2)} = \frac{8}{-4} = -2$ (negative).
   • For $2 < x < 8$, say $x = 4$: $\frac{8 - 4}{2(4-2)} = \frac{4}{4} = 1$ (positive).
   • For $x > 8$, say $x = 10$: $\frac{8 - 10}{2(10-2)} = \frac{-2}{16} = -\frac{1}{8}$ (negative).

6. Determine the solution intervals:

   • The inequality $\frac{8-x}{2(x-2)} < 0$ is negative in the intervals $(-\infty, 2)$ and $(8, \infty)$.

   However, since $x = 2$ makes the denominator zero (undefined), and $x = 8$ makes the numerator zero (where the expression equals zero, not negative), we exclude these points from the solution.

Final Solution:

The solution to the inequality is: $x \in (-\infty, 2) \cup (8, \infty)$

Let me know if you'd like any more details or if you have questions!

Here are five related questions to consider:

1. How do you solve rational inequalities that have more than two factors in the denominator?
2. What is the difference between solving a rational inequality and a polynomial inequality?
3. Can you explain how to graph the solution set of a rational inequality?
4. How do you determine the intervals of test points for more complex rational inequalities?
5. What happens when the inequality involves a non-strict inequality, like $\geq$ or $\leq$?

Tip: Always be cautious about points where the denominator is zero, as these points are not included in the solution set.