To solve the equation \(-\frac{1}{2}\left(\frac{3}{4}w - 2 + \frac{3}{2}w\right) = -3w\), let's follow these steps:

### Step 1: Distribute \(-\frac{1}{2}\) across the terms inside the parentheses.
\[
-\frac{1}{2} \times \frac{3}{4}w - \frac{1}{2} \times (-2) - \frac{1}{2} \times \frac{3}{2}w
\]
This simplifies to:
\[
-\frac{3}{8}w + 1 - \frac{3}{4}w
\]
So the equation becomes:
\[
-\frac{3}{8}w + 1 - \frac{3}{4}w = -3w
\]

### Step 2: Combine like terms on the left side.
\[
-\frac{3}{8}w - \frac{6}{8}w + 1 = -3w
\]
\[
-\frac{9}{8}w + 1 = -3w
\]

### Step 3: Move the \(w\) terms to one side of the equation.
\[
1 = -3w + \frac{9}{8}w
\]

### Step 4: Combine the \(w\) terms on the right side.
\[
1 = -\frac{24}{8}w + \frac{9}{8}w
\]
\[
1 = -\frac{15}{8}w
\]

### Step 5: Solve for \(w\).
\[
w = \frac{1 \times 8}{-15}
\]
\[
w = -\frac{8}{15}
\]

So, the solution is \(w = -\frac{8}{15}\).

Would you like more details on any of these steps?

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Here are 5 related questions to further explore this topic:

1. How would the solution change if the equation was \(-\frac{1}{3}(3/4w - 2 + 3/2w) = -3w\)?
2. What is the general approach to solving linear equations with fractions?
3. How can you verify the solution to a linear equation?
4. What happens if the equation had been an inequality instead of an equality?
5. How do you solve for \(w\) if the equation had an additional constant term on the right side?

**Tip:** When dealing with fractions, it's often easier to multiply the entire equation by the least common denominator (LCD) to eliminate the fractions early in the process.

Explore how to solve the equation -1/2(3/4w - 2 + 3/2w) = -3w step-by-step. This problem involves distributing, combining like terms, and solving equations with fractions. Suitable for high school students and above.

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