The expression given is:

$\frac{x + 7}{x - 7} - \frac{x}{7 - x}$

Step 1: Simplify the second fraction

Notice that $7 - x$ can be rewritten as $-(x - 7)$. So:

$\frac{x}{7 - x} = -\frac{x}{x - 7}$

Step 2: Rewrite the expression

Substitute this back into the expression:

$\frac{x + 7}{x - 7} - \left( -\frac{x}{x - 7} \right) = \frac{x + 7}{x - 7} + \frac{x}{x - 7}$

Step 3: Combine the fractions

Since both fractions have the same denominator, we can combine them:

$\frac{(x + 7) + x}{x - 7} = \frac{2x + 7}{x - 7}$

The simplified expression is:

$\frac{2x + 7}{x - 7}$

Do you want further explanation on any step, or would you like to explore more related problems?

Here are 5 related questions:

1. How would the expression change if the signs in the denominators were reversed?
2. What happens to the expression as $x$ approaches 7?
3. How can you simplify a similar expression with different constants?
4. Can you solve this expression for a specific value of $x$?
5. How do you handle complex fractions with different denominators?

Tip: Always look out for opportunities to factor or recognize patterns in denominators and numerators—they can simplify your work!