The inequality given in the image is:

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

To solve this inequality, follow these steps:

Step 1: Consider the domain

The expressions $\frac{1}{x}$ and $\frac{1}{x - 3}$ are undefined when the denominators are zero. Therefore, $x \neq 0$ and $x \neq 3$.

Step 2: Subtract $\frac{1}{x-3}$ from both sides

Subtract $\frac{1}{x-3}$ from both sides to bring all terms to one side of the inequality:

$\frac{1}{x} - \frac{1}{x-3} < 0$

Step 3: Combine the fractions

To combine the fractions, find a common denominator:

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

Simplify the numerator:

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

Step 4: Analyze the sign of the expression

The inequality $\frac{-3}{x(x-3)} < 0$ is negative when the product $x(x-3)$ is positive.

• $x(x-3)$ is positive when both $x > 3$ or $x < 0$.

Step 5: Determine the solution

Given that $-3$ is negative, for the entire fraction to be negative, the denominator must be positive.

• $x > 3$ or $x < 0$

Thus, the solution to the inequality is:

$x < 0 \quad \text{or} \quad x > 3$

Conclusion

The solution to the inequality $\frac{1}{x} < \frac{1}{x - 3}$ is:

$x \in (-\infty, 0) \cup (3, \infty)$

Would you like any further details or have any questions?

Here are some related questions:

1. What happens if the inequality sign is reversed?
2. How would the solution change if the inequality was $\frac{1}{x} \leq \frac{1}{x - 3}$?
3. What is the significance of $x = 0$ and $x = 3$ in this inequality?
4. How can inequalities like these be solved graphically?
5. What other types of inequalities can involve rational functions?

Tip: When solving inequalities involving fractions, always consider the critical points where the denominators become zero, as these points often split the number line into regions that need to be tested separately.