yes that is the correct expression, yes please simplify it 

To simplify the expression:

\[
\frac{3 - \frac{2x}{5x - 2}}{2x - \frac{x}{1 - \frac{x}{1 - \frac{x}{2 + 2x}}}}
\]

We'll start by simplifying the innermost fractions and work our way outwards.

### Step 1: Simplify the innermost fraction \(\frac{x}{2 + 2x}\)
\[
\frac{x}{2 + 2x} = \frac{x}{2(1 + x)} = \frac{1}{2(1 + x)}
\]

### Step 2: Substitute this into the next fraction and simplify
\[
\frac{x}{1 - \frac{1}{2(1 + x)}} = \frac{x}{\frac{2(1 + x) - 1}{2(1 + x)}} = \frac{2x(1 + x)}{2x + 1}
\]

### Step 3: Substitute into the next fraction and simplify
\[
\frac{x}{1 - \frac{2x(1 + x)}{2x + 1}} = \frac{x(2x + 1)}{(2x + 1) - 2x(1 + x)} = \frac{x(2x + 1)}{1 - 2x^2}
\]

### Step 4: Substitute into the outer expression
Now that we have the simplified form of the denominator, we can rewrite the whole expression:
\[
\frac{3 - \frac{2x}{5x - 2}}{2x - \frac{x(2x + 1)}{1 - 2x^2}}
\]

### Step 5: Simplify the numerator and denominator
\[
3 - \frac{2x}{5x - 2}
\]

We'll get the common denominator in the numerator:
\[
\frac{3(5x - 2) - 2x}{5x - 2} = \frac{15x - 6 - 2x}{5x - 2} = \frac{13x - 6}{5x - 2}
\]

Now, the denominator is:
\[
2x - \frac{x(2x + 1)}{1 - 2x^2} = \frac{2x(1 - 2x^2) - x(2x + 1)}{1 - 2x^2} = \frac{2x - 4x^3 - 2x^2 - x(2x + 1)}{1 - 2x^2} = \frac{2x - 4x^3 - 2x^2 - 2x^2 - x}{1 - 2x^2} = \frac{2x - 4x^3 - 2x^2}{1 - 2x^2}
\]

### Step 6: Combine all
We finally arrive at the simplified expression:

\[
\frac{\frac{13x - 6}{5x - 2}}{\frac{2x - 4x^3 - 2x^2}{1 - 2x^2}} = \frac{(13x - 6)(1 - 2x^2)}{(5x - 2)(2x - 4x^3 - 2x^2)}
\]

Further simplification could involve factorization or cancellation, depending on the goal.

Would you like to simplify or solve this further, or do you have any questions? 

---

Here are some follow-up questions you might consider:

1. How would the expression change if there were no fractions in the denominator?
2. What if \( x = 0 \)? What happens to the expression?
3. How can we simplify the expression further for specific values of \( x \)?
4. How do these algebraic manipulations translate into graphical representations?
5. What would the impact be on the expression if \( x \) were negative?

**Tip:** Always start simplifying from the innermost fractions or parentheses and work your way outwards.

Learn how to simplify the complex fractional expression \( \frac{3 - \frac{2x}{5x - 2}}{2x - \frac{x(2x + 1)}{1 - 2x^2}} \) with a detailed step-by-step solution. This problem covers advanced algebraic concepts and techniques for simplifying complex fractions. Suitable for advanced high school and college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation