Let's solve the equation:

\[
\frac{1}{2}h - 3 = \frac{3}{2} - 3h
\]

### Step 1: Eliminate fractions by multiplying by 2

Multiply every term by 2 to clear the fractions:

\[
2 \times \left(\frac{1}{2}h\right) - 2 \times 3 = 2 \times \left(\frac{3}{2}\right) - 2 \times 3h
\]

Simplify:

\[
h - 6 = 3 - 6h
\]

### Step 2: Combine like terms

Add \(6h\) to both sides to get all \(h\) terms on one side:

\[
h + 6h - 6 = 3
\]

Simplify:

\[
7h - 6 = 3
\]

### Step 3: Isolate \(h\)

Add 6 to both sides:

\[
7h = 9
\]

### Step 4: Solve for \(h\)

Divide by 7:

\[
h = \frac{9}{7}
\]

### Final Answer:

\[
h = \frac{9}{7}
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How would the solution change if the equation were \(\frac{1}{3}h - 3 = \frac{4}{2} - 3h\)?
2. What steps are involved in solving equations with variables on both sides?
3. How can you verify the solution \(h = \frac{9}{7}\) by substitution?
4. What happens if both sides of the equation are multiplied by a different number?
5. How does distributing and combining like terms simplify an equation?

### Tip:
Always clear fractions first in an equation by multiplying through by the least common multiple (LCM) of the denominators.

Learn how to solve the linear equation 1/2h - 3 = 3/2 - 3h with a detailed, step-by-step solution. This problem focuses on key algebraic concepts like eliminating fractions, combining like terms, and solving for a variable on both sides. Suitable for students in Grades 6-8.

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