https://file.aiquickdraw.com/mathbot/file/172616713660934d33m6n.jpg

To find which expression is equivalent to the given expression for all values of \( x \) and \( h \), we will analyze the provided image. The expression simplifies as follows:

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

Now, let's compare this to the provided options:
- **Option A:** \(\frac{-h}{(x-1+h)(x-1)}\) 
- **Option B:** \(\frac{-1}{(x-1+h)(x-1)}\) 
- **Option C:** \(\frac{-1}{(x-1)(x-1)}\) 
- **Option D:** \( h(x-1+h)(x-1) \)

Given the original expression simplifies to \(\frac{1}{(x-1)(x-1+h)}\), and we have negative signs in the fraction terms, **Option B** \(\frac{-1}{(x-1+h)(x-1)}\) is the correct choice as it matches the simplified form.

Would you like further clarification on any specific part of this solution?

Here are some related questions you might find useful:
1. How would the expression change if \( h \) approaches zero?
2. What happens to the expression if \( x = 1 \) or \( x = 1-h \)?
3. Can you explain how the common denominator was determined in the solution?
4. How does the sign of \( h \) (positive or negative) affect the simplified expression?
5. What are some applications of this type of algebraic manipulation in calculus?

**Tip:** Always check for potential undefined points or discontinuities when simplifying expressions, especially those involving variables in denominators.

Which of the following expressions is equivalent to the expression above for all values of x and h?

Learn how to simplify and find equivalent expressions for rational functions involving x and h. This problem involves common denominator and simplification techniques. Suitable for high school students studying algebra.

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