To solve the rational inequality:

\[
\frac{3}{x - 2} < \frac{1}{x}
\]

### Step 1: Clear the denominators
To eliminate the fractions, multiply both sides by the common denominator, which is \( x(x - 2) \). Be cautious here, because multiplying or dividing by expressions involving \( x \) could affect the inequality sign if \( x \) is negative.

\[
\left( \frac{3}{x - 2} \right) \cdot x(x - 2) < \left( \frac{1}{x} \right) \cdot x(x - 2)
\]

This simplifies to:

\[
3x < (x - 2)
\]

### Step 2: Solve the resulting inequality
Now, solve the inequality:

\[
3x < x - 2
\]

Subtract \( x \) from both sides:

\[
2x < -2
\]

Now divide by 2:

\[
x < -1
\]

### Step 3: Consider restrictions
The original inequality has denominators \( x - 2 \) and \( x \). We need to exclude the values where these expressions equal zero:

- \( x \neq 2 \) (from \( x - 2 \neq 0 \))
- \( x \neq 0 \) (from \( x \neq 0 \))

### Step 4: Check intervals
The solution must satisfy \( x < -1 \) while excluding \( x = 0 \) and \( x = 2 \). The critical points are \( x = -1 \), \( x = 0 \), and \( x = 2 \), so we will check the intervals:

1. \( (-\infty, -1) \)
2. \( (-1, 0) \)
3. \( (0, 2) \)
4. \( (2, \infty) \)

### Step 5: Test intervals
Now test the inequality in each of the intervals to determine where it holds true:

- For \( x = -2 \) (in \( (-\infty, -1) \)):
  \[
  \frac{3}{-2 - 2} = \frac{3}{-4}, \quad \frac{1}{-2} = -\frac{1}{2}
  \]
  Since \( -\frac{3}{4} < -\frac{1}{2} \), this interval works.

- For \( x = -0.5 \) (in \( (-1, 0) \)), the inequality does not hold.

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

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

Would you like more details, or do you have any questions about the solution?

### 5 Related Questions:
1. How do we solve rational inequalities involving three terms?
2. What happens to an inequality when multiplying by negative values?
3. How do you determine critical points in rational inequalities?
4. Can you explain how to test intervals for inequalities?
5. How does domain restriction affect inequality solutions?

### Tip:
Always watch for values that make denominators zero, as these are critical points where the inequality may not be defined.

Solve the rational inequality 3/(x-2) < 1/x

Learn how to solve the rational inequality 3/(x-2) < 1/x with a detailed, step-by-step explanation. This problem focuses on key algebraic concepts such as rational inequalities, clearing denominators, and domain restrictions. Suitable for students in Grades 9-12.

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