The expression you provided is:

\[
\frac{x^2}{6x} - \frac{x - 1}{18x}
\]

Let's simplify this step-by-step.

### Step 1: Simplify each term
#### First term: \(\frac{x^2}{6x}\)
\[
\frac{x^2}{6x} = \frac{x \cdot x}{6x} = \frac{x}{6}
\]
So, \(\frac{x^2}{6x}\) simplifies to \(\frac{x}{6}\).

#### Second term: \(\frac{x - 1}{18x}\)
This term is already in its simplest form.

### Step 2: Find a common denominator
We need to subtract \(\frac{x - 1}{18x}\) from \(\frac{x}{6}\). The least common denominator (LCD) is \(18x\).

- Rewrite \(\frac{x}{6}\) with the common denominator \(18x\):
\[
\frac{x}{6} = \frac{x \cdot 3}{18x} = \frac{3x}{18x}
\]

### Step 3: Subtract the terms
Now, subtract the two fractions:
\[
\frac{3x}{18x} - \frac{x - 1}{18x} = \frac{3x - (x - 1)}{18x}
\]

Distribute the negative sign in the numerator:
\[
3x - (x - 1) = 3x - x + 1 = 2x + 1
\]

So, the expression becomes:
\[
\frac{2x + 1}{18x}
\]

### Final Answer:
\[
\frac{x^2}{6x} - \frac{x - 1}{18x} = \frac{2x + 1}{18x}
\]

Would you like further details on any part of this explanation? 

Here are 5 related questions for you to explore:
1. How do you find the least common denominator (LCD) of two fractions?
2. How can you simplify complex fractions with different denominators?
3. What are the rules for distributing negative signs in expressions?
4. How do you handle algebraic fractions with variables in both the numerator and denominator?
5. Can you solve similar expressions with higher-degree polynomials?

**Tip:** Always check for common factors before simplifying, as this can save steps!

Learn how to simplify the algebraic expression X² over 6x - (x-1) over 18x step-by-step. This solution covers algebraic fraction simplification, finding the least common denominator, and using the distributive property. Ideal for students in Grades 8-10.

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