Let's simplify the given expression step by step. The problem is:

$\frac{x^2 - 1}{x - 7} - \frac{x^2 - 9}{x - 3}.$

Step 1: Factorize the numerators

• $x^2 - 1$ is a difference of squares: $x^2 - 1 = (x - 1)(x + 1).$
• $x^2 - 9$ is also a difference of squares: $x^2 - 9 = (x - 3)(x + 3).$

So the expression becomes: $\frac{(x - 1)(x + 1)}{x - 7} - \frac{(x - 3)(x + 3)}{x - 3}.$

Step 2: Simplify the second term

In the second term, $x - 3$ cancels out: $\frac{(x - 3)(x + 3)}{x - 3} = x + 3 \quad (\text{for } x \neq 3).$

Thus, the expression becomes: $\frac{(x - 1)(x + 1)}{x - 7} - (x + 3).$

Step 3: Combine the terms

The common denominator is $x - 7$. Rewrite $x + 3$ as a fraction: $(x + 3) = \frac{(x + 3)(x - 7)}{x - 7}.$

Now the expression is: $\frac{(x - 1)(x + 1)}{x - 7} - \frac{(x + 3)(x - 7)}{x - 7}.$

Step 4: Simplify the numerator

Combine the fractions: $\frac{(x - 1)(x + 1) - (x + 3)(x - 7)}{x - 7}.$

Expand both terms in the numerator:

1. Expand $(x - 1)(x + 1)$: $(x - 1)(x + 1) = x^2 - 1.$
2. Expand $(x + 3)(x - 7)$: $(x + 3)(x - 7) = x^2 - 7x + 3x - 21 = x^2 - 4x - 21.$

Subtract the two: $(x^2 - 1) - (x^2 - 4x - 21).$

Simplify: $x^2 - 1 - x^2 + 4x + 21 = 4x + 20.$

So the numerator becomes: $4x + 20.$

Step 5: Factorize the numerator

Factor $4x + 20$: $4x + 20 = 4(x + 5).$

Thus, the expression is: $\frac{4(x + 5)}{x - 7}.$

Final Answer:

$\frac{4(x + 5)}{x - 7}.$

Let me know if you need further clarification or additional steps explained! Here are five related questions for practice:

1. How do you factorize a difference of squares?
2. What are the restrictions on the values of $x$ in the original expression?
3. How do you find a common denominator for fractions with different denominators?
4. What is the importance of simplifying numerators and denominators before combining fractions?
5. How would the solution change if the second term had a denominator of $x - 7$ instead of $x - 3$?

Tip: Always check for restrictions in the domain when simplifying rational expressions, especially when variables are in the denominator!