Let's work through determining the LCD (Least Common Denominator) for each pair of rational expressions in the image:

1. $\frac{1}{3}$ and $\frac{2}{x+1}$:

   • The denominators are $3$ and $x + 1$.
   • The LCD is simply the product of these distinct terms: $3(x + 1)$.

2. $\frac{1}{x}$ and $\frac{3}{2x}$:

   • The denominators are $x$ and $2x$. Since they share the factor $x$, the LCD is the highest degree of the shared factor $2x$: $2x$.

3. $\frac{2}{x}$ and $\frac{1}{x - 2}$:

   • The denominators are $x$ and $x - 2$.
   • Since these terms are distinct, the LCD is their product: $x(x - 2)$.

4. $\frac{1}{4}$ and $\frac{1}{4x + 8}$:

   • The denominators are $4$ and $4x + 8$. The second denominator can be factored as $4(x + 2)$.
   • The LCD is the highest power of the factor $4(x + 2)$, so the LCD is: $4(x + 2)$.

5. $\frac{1}{x + 1}$ and $\frac{2}{x - 1}$:

   • The denominators are $x + 1$ and $x - 1$.
   • Since these terms are distinct, the LCD is their product: $(x + 1)(x - 1)$.

Would you like a deeper explanation on any of these steps or additional details?

Here are 5 related questions to help solidify your understanding:

1. How do you find the LCD when the denominators contain polynomial expressions?
2. Why is it important to factor the denominators before determining the LCD?
3. What would the LCD be for $\frac{1}{2x}$ and $\frac{3}{3x + 6}$?
4. How does the LCD help in adding or subtracting rational expressions?
5. How would the LCD change if both denominators shared more than one factor?

Tip: Always factor expressions in the denominator to simplify finding the LCD, especially when dealing with polynomials.