Which of the following can be added without altering any of the fractions? (4x)/(x-2)+(7)/(x-2), (2x)/(x+2)+(2x)/(x-2), (3x+4)/(2x-5)+(1)/(3x+4),(2x)/(x-8)+(x-8)/(2x)

To determine which of the given expressions can be added without altering any of the fractions, we need to see if the fractions have a common denominator. Let's analyze each pair:

1. **Expression 1:** \(\frac{4x}{x-2} + \frac{7}{x-2}\)

   Both fractions have the same denominator \(x-2\), so they can be directly added:
   \[
   \frac{4x}{x-2} + \frac{7}{x-2} = \frac{4x + 7}{x-2}
   \]

2. **Expression 2:** \(\frac{2x}{x+2} + \frac{2x}{x-2}\)

   The fractions have different denominators, \(x+2\) and \(x-2\), so they cannot be directly added without finding a common denominator.

3. **Expression 3:** \(\frac{3x+4}{2x-5} + \frac{1}{3x+4}\)

   The fractions have different denominators, \(2x-5\) and \(3x+4\), so they cannot be directly added without finding a common denominator.

4. **Expression 4:** \(\frac{2x}{x-8} + \frac{x-8}{2x}\)

   The fractions have different denominators, \(x-8\) and \(2x\), so they cannot be directly added without finding a common denominator.

**Conclusion:** 

The only expression that can be added without altering any of the fractions is:

- **Expression 1:** \(\frac{4x}{x-2} + \frac{7}{x-2}\)

Would you like further details on any of the other expressions? Here are some related questions:

1. How do you find a common denominator for fractions with different denominators?
2. What happens if you add fractions with unlike denominators?
3. Can the expression \(\frac{2x}{x+2} + \frac{2x}{x-2}\) be simplified in another way?
4. How can you check if two fractions are equivalent?
5. What are some common mistakes to avoid when adding algebraic fractions?
6. How do you simplify complex fractions?
7. What is the role of the Least Common Denominator (LCD) in adding fractions?
8. How do you solve equations involving fractional expressions?

**Tip:** When adding fractions, always check the denominators first; they must be the same for direct addition.

Explore how to add fractions without altering any of the fractions using common denominators. This detailed explanation covers algebraic fractions and essential concepts in algebra. Suitable for students in Grades 7-9.

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