(2 + x+1/x-1) / (1 – x – 1/ x+1) = 3x/2

Let's solve the equation step by step:

Given equation:

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

### Step 1: Simplify the numerator and the denominator.

For the numerator:
\[
2 + x + \frac{1}{x} - 1 = (x + \frac{1}{x}) + 1
\]

For the denominator:
\[
1 - x - \frac{1}{x} + 1 = -(x + \frac{1}{x}) + 2
\]

So the equation becomes:

\[
\frac{x + \frac{1}{x} + 1}{2 - \left(x + \frac{1}{x}\right)} = \frac{3x}{2}
\]

### Step 2: Cross multiply to eliminate the fraction.

\[
2 \left(x + \frac{1}{x} + 1\right) = 3x \left(2 - \left(x + \frac{1}{x}\right)\right)
\]

Expanding both sides:

\[
2x + \frac{2}{x} + 2 = 6x - 3x^2 - \frac{3x}{x}
\]

Simplifying further:

\[
2x + \frac{2}{x} + 2 = 6x - 3x^2 - 3
\]

### Step 3: Move all terms to one side of the equation.

\[
3x^2 + 2x - 6x + \frac{2}{x} + 2 + 3 = 0
\]

Simplify:

\[
3x^2 - 4x + \frac{2}{x} + 5 = 0
\]

### Step 4: Multiply the entire equation by \(x\) to eliminate the fraction.

\[
3x^3 - 4x^2 + 2 + 5x = 0
\]

Rearrange to form a standard cubic equation:

\[
3x^3 - 4x^2 + 5x + 2 = 0
\]

### Step 5: Solve the cubic equation.

This equation can be solved using numerical methods or factoring, if possible.

Would you like me to solve the cubic equation further using numerical methods or discuss the methods available to solve cubic equations?

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Here are five related questions:
1. How do you solve cubic equations using numerical methods like Newton's method?
2. What is the significance of each term in a cubic equation for its graph?
3. How do you approach factoring a cubic polynomial?
4. What is the Rational Root Theorem, and how can it help solve cubic equations?
5. How can complex roots be identified and interpreted in cubic equations?

**Tip:** When dealing with complex rational expressions, cross-multiplication is a powerful tool to simplify the equation and eliminate fractions.

Learn how to solve the rational expression equation (2 + x + 1/x - 1) / (1 - x - 1/x + 1) = 3x/2 with a detailed step-by-step solution. This problem covers key algebraic concepts, including rational expressions and cross-multiplication. Suitable for students in Grades 11-12.

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