Let's simplify the expression step by step:

Given: $\left( \dfrac{ 1+x }{ 1-x } - \dfrac{ 1-x }{ 1+x } \right) \left( \dfrac{ 3 }{ 4x } + \dfrac{ x }{ 4 } - x \right)$

Step 1: Simplify the first term

First, we'll simplify the expression inside the first set of parentheses: $\left( \dfrac{ 1+x }{ 1-x } - \dfrac{ 1-x }{ 1+x } \right)$ We need to find a common denominator. The common denominator between $\frac{1+x}{1-x}$ and $\frac{1-x}{1+x}$ is $(1-x)(1+x)$.

So: $\dfrac{ 1+x }{ 1-x } = \dfrac{ (1+x)^2 }{ (1-x)(1+x) }$ and $\dfrac{ 1-x }{ 1+x } = \dfrac{ (1-x)^2 }{ (1-x)(1+x) }$ Now, subtract the fractions: $\left( \dfrac{ (1+x)^2 }{ (1-x)(1+x) } - \dfrac{ (1-x)^2 }{ (1-x)(1+x) } \right)$ $= \dfrac{ (1+x)^2 - (1-x)^2 }{ (1-x)(1+x) }$ Using the difference of squares: $(1+x)^2 - (1-x)^2 = \left( (1+x) - (1-x) \right) \left( (1+x) + (1-x) \right)$ Simplify each term: $(1+x) - (1-x) = 2x$ $(1+x) + (1-x) = 2$ So: $(1+x)^2 - (1-x)^2 = 2x \times 2 = 4x$ Thus, the first term becomes: $\dfrac{ 4x }{ (1-x)(1+x) }$

Step 2: Simplify the second term

Now simplify the expression inside the second set of parentheses: $\left( \dfrac{ 3 }{ 4x } + \dfrac{ x }{ 4 } - x \right)$ We can combine these terms by getting a common denominator. The common denominator between $\frac{3}{4x}$, $\frac{x}{4}$, and $x$ is $4x$.

So: $\dfrac{ 3 }{ 4x } + \dfrac{ x }{ 4 } - x = \dfrac{ 3 }{ 4x } + \dfrac{ x^2 }{ 4x } - \dfrac{ 4x^2 }{ 4x }$ Combine the terms: $= \dfrac{ 3 + x^2 - 4x^2 }{ 4x }$ $= \dfrac{ 3 - 3x^2 }{ 4x }$ Factor out the common factor of 3: $= \dfrac{ 3(1 - x^2) }{ 4x }$

Step 3: Multiply the two simplified expressions

Now, multiply the two simplified expressions: $\left( \dfrac{ 4x }{ (1-x)(1+x) } \right) \times \left( \dfrac{ 3(1 - x^2) }{ 4x } \right)$ Simplify the multiplication: $= \dfrac{ 4x \times 3(1 - x^2) }{ (1-x)(1+x) \times 4x }$ The $4x$ terms cancel out, leaving: $= \dfrac{ 3(1 - x^2) }{ (1-x)(1+x) }$ Since $1 - x^2 = (1-x)(1+x)$, we can cancel out $(1-x)(1+x)$ from the numerator and denominator: $= 3$

Final Answer:

The simplified expression is $\boxed{3}$.

---

Would you like any further details or have any questions about this solution?

Here are 5 related questions to explore further:

1. How do you simplify expressions involving the difference of squares?
2. Can you explain how to handle expressions with fractions and common denominators?
3. What are other ways to simplify expressions like $\dfrac{(1+x)^2}{(1-x)(1+x)}$?
4. How do you perform operations with rational expressions in algebra?
5. Can you show an example where such simplifications appear in real-world problems?

Tip: When simplifying complex expressions, always try to look for patterns like the difference of squares or common denominators to reduce complexity!