To solve the equation \(\frac{1}{2}h - 3 = \frac{3}{2} - 3h\), follow these steps:
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Eliminate fractions by multiplying everything by 2 (the least common multiple of the denominators): \[ 2 \left(\frac{1}{2}h\right) - 2(3) = 2\left(\frac{3}{2}\right) - 2(3h) \]
This simplifies to: \[ h - 6 = 3 - 6h \]
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Add \(6h\) to both sides: \[ h + 6h - 6 = 3 \] which simplifies to: \[ 7h - 6 = 3 \]
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Add 6 to both sides: \[ 7h = 3 + 6 \] which simplifies to: \[ 7h = 9 \]
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Divide both sides by 7: \[ h = \frac{9}{7} \]
Thus, the solution is: \[ \boxed{\frac{9}{7}} \]
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