https://file.aiquickdraw.com/mathbot/file/1725381122013hqspc6qx.jpg

The given image contains a mathematical expression that appears to be:

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

To simplify this expression, let's proceed step by step:

### Step 1: Factor the denominator in the first fraction.
The denominator \(x^2 - 1\) can be factored as a difference of squares:

\[
x^2 - 1 = (x - 1)(x + 1)
\]

So the expression becomes:

\[
\frac{2x}{(x - 1)(x + 1)} - \frac{1}{x - 1}
\]

### Step 2: Obtain a common denominator.
The common denominator between the two fractions is \((x - 1)(x + 1)\). Rewrite the second fraction with this common denominator:

\[
\frac{2x}{(x - 1)(x + 1)} - \frac{1 \cdot (x + 1)}{(x - 1)(x + 1)}
\]

### Step 3: Combine the fractions.
Now, subtract the numerators:

\[
\frac{2x - (x + 1)}{(x - 1)(x + 1)} = \frac{2x - x - 1}{(x - 1)(x + 1)}
\]

This simplifies to:

\[
\frac{x - 1}{(x - 1)(x + 1)}
\]

### Step 4: Simplify the expression.
Notice that the numerator \(x - 1\) is a factor of the denominator, so they cancel out:

\[
\frac{x - 1}{(x - 1)(x + 1)} = \frac{1}{x + 1} \quad \text{for } x \neq 1
\]

### Final Answer:
The simplified expression is:

\[
\frac{1}{x + 1}
\]

This is the simplified form of the given expression.

Would you like any further details or have any questions?

Here are five related questions you might find helpful:
1. What are the steps to factor a quadratic expression?
2. How do you find the common denominator when subtracting rational expressions?
3. What is the difference of squares, and how is it used in factoring?
4. What happens if a factor in the numerator cancels out a factor in the denominator?
5. How would the process change if the equation had a different form, such as a sum instead of a difference?

**Tip:** Always check for common factors that can be canceled when simplifying fractions; this can significantly reduce the complexity of the expression.

Simplify the expression \(\frac{2x}{x^2 - 1} - \frac{1}{x - 1}\).

Learn how to simplify the rational expression \(\frac{2x}{x^2 - 1} - \frac{1}{x - 1}\) by factoring and finding a common denominator. This step-by-step guide is perfect for students in Grades 8-10 looking to strengthen their understanding of algebra and rational expressions.

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