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.