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The given equation is:

\[
\frac{1 - 3x}{x^2} + \frac{3}{2x} = \frac{4}{x}
\]

Let's solve this step by step.

### Step 1: Simplify the equation
We aim to eliminate the denominators by multiplying the entire equation by the least common denominator (LCD), which is \( x^2 \). Multiplying both sides by \( x^2 \), we get:

\[
x^2 \left( \frac{1 - 3x}{x^2} + \frac{3}{2x} \right) = x^2 \left( \frac{4}{x} \right)
\]

This simplifies to:

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

### Step 2: Simplify terms
Next, simplify the second term on the left-hand side:

\[
1 - 3x + \frac{3x}{2} = 4x
\]

### Step 3: Eliminate fractions
To eliminate the fraction \(\frac{3x}{2}\), multiply the entire equation by 2:

\[
2(1 - 3x) + 3x = 8x
\]

Simplifying:

\[
2 - 6x + 3x = 8x
\]

### Step 4: Solve for \(x\)
Combine like terms:

\[
2 - 3x = 8x
\]

Add \(3x\) to both sides:

\[
2 = 11x
\]

Finally, divide by 11:

\[
x = \frac{2}{11}
\]

### Final Answer:
\[
x = \frac{2}{11}
\]

Would you like further clarification or additional details on any step?

Here are 5 related questions to explore further:
1. What would happen if the equation had an extra term?
2. How can you verify the solution by substituting \(x = \frac{2}{11}\)?
3. How would the solution change if the constant on the right was different?
4. Can you solve this using another algebraic method like factoring?
5. How would you graph the functions involved to visualize the solution?

**Tip:** Always check the final solution by plugging it back into the original equation to verify its correctness.

Solve the equation: (1 - 3x)/x^2 + 3/(2x) = 4/x

Learn how to solve the rational equation (1 - 3x)/x^2 + 3/(2x) = 4/x step by step. This problem involves simplifying rational expressions, multiplying by the least common denominator (LCD), and solving linear equations. Suitable for high school algebra students (Grades 9-12).

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