The inequality given is:

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

Let's solve this step by step:

### Step 1: Analyze the domain
- For \(\frac{1}{x}\), \(x \neq 0\), because division by zero is undefined.
- For \(\frac{1}{x - 3}\), \(x \neq 3\), as this would also result in division by zero.

Thus, the domain of the inequality is \(x \neq 0\) and \(x \neq 3\).

### Step 2: Cross-multiply
Since both sides are fractions, we can cross-multiply. However, we must be cautious about the signs of \(x\) and \(x-3\), because multiplying by a negative number flips the inequality. Let's break this into cases:

#### Case 1: \(x > 3\)
When \(x > 3\), both \(x\) and \(x - 3\) are positive, so the inequality remains the same:
\[
1 \cdot (x - 3) < 1 \cdot x
\]
Simplifying:
\[
x - 3 < x
\]
This simplifies to:
\[
-3 < 0
\]
This is **always true**, so for \(x > 3\), the inequality holds.

#### Case 2: \(0 < x < 3\)
In this case, \(x\) is positive, but \(x - 3\) is negative. This means we need to flip the inequality when multiplying:
\[
1 \cdot (x - 3) > 1 \cdot x
\]
Simplifying:
\[
x - 3 > x
\]
This simplifies to:
\[
-3 > 0
\]
This is **never true**, so there are no solutions in the interval \(0 < x < 3\).

#### Case 3: \(x < 0\)
For \(x < 0\), both \(x\) and \(x - 3\) are negative, so the inequality remains the same:
\[
1 \cdot (x - 3) < 1 \cdot x
\]
Simplifying:
\[
x - 3 < x
\]
This simplifies to:
\[
-3 < 0
\]
This is **always true**, so for \(x < 0\), the inequality holds.

### Step 3: Conclusion
The inequality holds when \(x < 0\) or \(x > 3\). So the solution is:

\[
x \in (-\infty, 0) \cup (3, \infty)
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you handle inequalities when cross-multiplying?
2. What happens when multiplying or dividing by a negative number in inequalities?
3. How do you determine the domain of a rational function?
4. Can you graph the solution of a rational inequality?
5. How can we solve inequalities involving quadratic expressions?

**Tip:** Always be cautious when multiplying or dividing inequalities by variables, as their signs affect the direction of the inequality.

Understand how to solve the rational inequality 1/x < 1/(x - 3) by analyzing its domain and applying inequality rules. Learn the step-by-step solution and the key concepts of rational functions and inequalities. Suitable for students in Grades 9-12.

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