To find the perimeter of a rectangle, we use the formula:
\[ \text{Perimeter} = 2 \times (\text{Width} + \text{Height}) \]
Given that the width is \( \frac{3}{2}x + 3 \) and the height is \( 3x - \frac{1}{2} \), let's first calculate \( \text{Width} + \text{Height} \):
\[ \text{Width} + \text{Height} = \left(\frac{3}{2}x + 3\right) + \left(3x - \frac{1}{2}\right) \]
Now, combine the terms:
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Combine the \( x \)-terms: \[ \frac{3}{2}x + 3x = \frac{3}{2}x + \frac{6}{2}x = \frac{9}{2}x \]
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Combine the constant terms: \[ 3 - \frac{1}{2} = \frac{6}{2} - \frac{1}{2} = \frac{5}{2} \]
So, we have:
\[ \text{Width} + \text{Height} = \frac{9}{2}x + \frac{5}{2} \]
Now substitute this back into the perimeter formula:
\[ \text{Perimeter} = 2 \times \left(\frac{9}{2}x + \frac{5}{2}\right) \]
Distribute the \( 2 \):
\[ \text{Perimeter} = 2 \times \frac{9}{2}x + 2 \times \frac{5}{2} \]
This simplifies to:
\[ \text{Perimeter} = 9x + 5 \]
Thus, the expression that represents the perimeter of the rectangle in units is:
\[ \boxed{9x + 5} \]