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.

Learn how to add fractions 1/x and 1/(x^2 - x) step-by-step with a detailed solution. Understand the concept of finding a common denominator, rewriting fractions, and simplifying the result effectively. Suitable for students in Grades 7-9.

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