To add the fractions $\frac{1}{x}$ and $\frac{1}{x^2 - x}$, you'll need to find a common denominator. Here's how you can do it step by step:

Step 1: Factor the second denominator

First, factor the denominator $x^2 - x$: $x^2 - x = x(x - 1)$ So, the expression becomes: $\frac{1}{x} + \frac{1}{x(x-1)}$

Step 2: Find the common denominator

The common denominator between $\frac{1}{x}$ and $\frac{1}{x(x-1)}$ is $x(x - 1)$.

Step 3: Rewrite each fraction with the common denominator

For the first fraction $\frac{1}{x}$, multiply both the numerator and denominator by $x - 1$ to make the denominator $x(x - 1)$: $\frac{1}{x} = \frac{1 \cdot (x-1)}{x \cdot (x-1)} = \frac{x-1}{x(x-1)}$ The second fraction already has the common denominator $x(x-1)$.

Step 4: Add the fractions

Now that both fractions have the same denominator, you can add them: $\frac{x-1}{x(x-1)} + \frac{1}{x(x-1)} = \frac{(x-1) + 1}{x(x-1)}$

Step 5: Simplify the numerator

Simplify the numerator by combining like terms: $\frac{x-1 + 1}{x(x-1)} = \frac{x}{x(x-1)}$

Step 6: Simplify the overall expression

Finally, simplify the fraction: $\frac{x}{x(x-1)} = \frac{1}{x-1}$

So, the result of adding $\frac{1}{x}$ and $\frac{1}{x^2 - x}$ is: $\frac{1}{x - 1}$

Would you like more details or have any questions?

Here are five related questions to expand on this topic:

1. How would you add $\frac{1}{x+1}$ and $\frac{1}{x^2-1}$?
2. What is the process for adding rational expressions with different denominators?
3. How do you simplify complex rational expressions?
4. What happens if the denominator cannot be factored easily?
5. How do you solve equations that involve adding fractions like $\frac{1}{x}$ and $\frac{1}{x^2 - x}$?

Tip: When adding fractions, always start by finding the least common denominator (LCD) to combine them effectively.