Let's simplify the given rational expressions step by step.

3. $\dfrac{2x - 4y}{x^2 - 4y^2}$

• The numerator is $2x - 4y$. We can factor out the common factor of 2:
  $2(x - 2y)$
• The denominator is $x^2 - 4y^2$, which is a difference of squares:
  $x^2 - (2y)^2 = (x - 2y)(x + 2y)$

Thus, the expression becomes: $\dfrac{2(x - 2y)}{(x - 2y)(x + 2y)}$ Cancel out the common factor $(x - 2y)$: $\dfrac{2}{x + 2y}$

4. $\dfrac{12x - 12y}{18y - 18x}$

• The numerator is $12x - 12y$, which can be factored as: $12(x - y)$
• The denominator is $18y - 18x$, which can be factored as: $18(y - x)$ Notice that $y - x$ is the negative of $x - y$: $18(y - x) = -18(x - y)$

So the expression becomes: $\dfrac{12(x - y)}{-18(x - y)}$ Cancel out $(x - y)$, and you're left with: $\dfrac{12}{-18} = -\dfrac{2}{3}$

5. $\dfrac{3x + 6}{5x + 10}$

• The numerator is $3x + 6$, which can be factored as: $3(x + 2)$
• The denominator is $5x + 10$, which can be factored as: $5(x + 2)$

Thus, the expression becomes: $\dfrac{3(x + 2)}{5(x + 2)}$ Cancel out $(x + 2)$, and you're left with: $\dfrac{3}{5}$

Do you want further details or have any questions?

Here are 5 related questions:

1. How do you identify common factors in a rational expression?
2. What are the key steps for simplifying a difference of squares?
3. When can we cancel out factors in rational expressions?
4. Why does changing the order of subtraction introduce a negative sign?
5. How do you simplify a rational expression when both terms are linear?

Tip: Always check for common factors in both the numerator and denominator before canceling to avoid errors in simplification.