To solve the inequality

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

we need to find the values of $x$ for which this inequality holds.

Step-by-step Solution

1. Rewrite the inequality:

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

2. Combine terms into a single fraction:

   We need a common denominator to combine the terms on the right-hand side. The common denominator is $x - 2$. Rewriting the expression:

   $\frac{x}{x - 2} < \frac{3(x - 2) - 2}{x - 2}.$

3. Simplify the expression on the right-hand side:

   Expand and simplify the numerator:

   $\frac{x}{x - 2} < \frac{3x - 6 - 2}{x - 2} = \frac{3x - 8}{x - 2}.$

4. Subtract $\frac{x}{x - 2}$ from both sides to bring the inequality to a single fraction:

   $\frac{x - (3x - 8)}{x - 2} < 0.$

5. Simplify the numerator:

   $\frac{x - 3x + 8}{x - 2} = \frac{-2x + 8}{x - 2}.$

   This simplifies to:

   $\frac{-2(x - 4)}{x - 2} < 0.$

6. Find the critical points:

   The critical points occur when the numerator or the denominator is zero:

   • Numerator zero: $-2(x - 4) = 0 \implies x = 4$.
   • Denominator zero: $x - 2 = 0 \implies x = 2$.

   These points, $x = 4$ and $x = 2$, divide the number line into intervals.

7. Test the intervals to determine where the inequality holds:

   The intervals to check are: $(- \infty, 2)$, $(2, 4)$, and $(4, \infty)$.

   • For $x < 2$: Choose $x = 0$: $\frac{-2(0 - 4)}{0 - 2} = \frac{-2(-4)}{-2} = \frac{8}{-2} = -4 < 0.$ (Satisfies the inequality)

   • For $2 < x < 4$: Choose $x = 3$: $\frac{-2(3 - 4)}{3 - 2} = \frac{-2(-1)}{1} = \frac{2}{1} = 2 > 0.$ (Does not satisfy the inequality)

   • For $x > 4$: Choose $x = 5$: $\frac{-2(5 - 4)}{5 - 2} = \frac{-2(1)}{3} = \frac{-2}{3} < 0.$ (Satisfies the inequality)

8. Include critical points in the solution:

   • $x = 4$ makes the numerator zero: The fraction equals 0 at $x = 4$ (not less than 0), so exclude $x = 4$.
   • $x = 2$ makes the denominator zero: The fraction is undefined, so we exclude $x = 2$.

9. Combine the intervals:

   The solution set for the inequality is:

   $x \in (-\infty, 2) \cup (4, \infty).$

Final Answer

$x \in (-\infty, 2) \cup (4, \infty).$

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

Here are 5 related questions you might find interesting:

1. How do you solve inequalities involving rational expressions?
2. What are the key steps to solve absolute value inequalities?
3. How can graphing help visualize the solutions of inequalities?
4. What is the difference between strict and non-strict inequalities, and how does it affect the solution?
5. How can you verify solutions to inequalities?

Tip: Always check the domain of the function to exclude points where the expression is undefined, such as division by zero.