https://file.aiquickdraw.com/mathbot/file/17253999354532sxcl98r.jpg

To solve the expression provided in the image:

\[
\frac{2x-1}{x+3} - \frac{x+1}{x-3}
\]

we will follow these steps:

### Step 1: Identify the common denominator
The two fractions have different denominators: \( x+3 \) and \( x-3 \). The common denominator for these fractions would be the product of these two denominators:

\[
(x + 3)(x - 3)
\]

### Step 2: Rewrite each fraction with the common denominator
We'll rewrite each fraction so that they have the same denominator:

\[
\frac{2x-1}{x+3} = \frac{(2x-1)(x-3)}{(x+3)(x-3)}
\]

\[
\frac{x+1}{x-3} = \frac{(x+1)(x+3)}{(x-3)(x+3)}
\]

### Step 3: Subtract the fractions
Now that the fractions have the same denominator, we can subtract them:

\[
\frac{(2x-1)(x-3)}{(x+3)(x-3)} - \frac{(x+1)(x+3)}{(x-3)(x+3)}
\]

This becomes:

\[
\frac{(2x-1)(x-3) - (x+1)(x+3)}{(x+3)(x-3)}
\]

### Step 4: Expand the numerators
We need to expand both numerators:

1. Expand \((2x-1)(x-3)\):

\[
(2x-1)(x-3) = 2x(x) - 2x(3) - 1(x) + 1(3) = 2x^2 - 6x - x + 3 = 2x^2 - 7x + 3
\]

2. Expand \((x+1)(x+3)\):

\[
(x+1)(x+3) = x(x) + x(3) + 1(x) + 1(3) = x^2 + 3x + x + 3 = x^2 + 4x + 3
\]

### Step 5: Combine and simplify the numerator
Now we subtract the expanded forms:

\[
(2x^2 - 7x + 3) - (x^2 + 4x + 3)
\]

This results in:

\[
2x^2 - 7x + 3 - x^2 - 4x - 3 = x^2 - 11x
\]

### Step 6: Write the final simplified expression
So, the expression simplifies to:

\[
\frac{x^2 - 11x}{(x+3)(x-3)}
\]

This is the simplified result for the given expression.

Do you have any questions or need further details?

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### 5 Related Questions:
1. How do you factor a quadratic expression like \( x^2 - 11x \)?
2. What are the steps to find the least common denominator for rational expressions?
3. How do you simplify complex fractions with multiple terms in the numerator and denominator?
4. How can you identify when a rational expression can be simplified further?
5. What are the potential restrictions on the variable \( x \) in the expression \( \frac{2x-1}{x+3} - \frac{x+1}{x-3} \)?

### Tip:
Always check for common factors in both the numerator and the denominator to see if the expression can be simplified further after performing operations.

How to solve the expression \(\frac{2x-1}{x+3} - \frac{x+1}{x-3}\)?

This detailed guide explains how to simplify the rational expression \(\frac{2x-1}{x+3} - \frac{x+1}{x-3}\) step by step. Learn about finding a common denominator, expanding and simplifying numerators, and combining the results. Suitable for high school students in Grades 9-12.

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