To find the perimeter of the triangle given the lengths of its sides as polynomials, we need to sum up the lengths of all three sides:
- \( \frac{3}{2}x + \frac{5}{2} \)
- \( 5x - \frac{3}{2} \)
- \( 4x + \frac{1}{4} \)
Let's add these expressions together step by step.
First, we can write down the sum of the sides:
\[ \text{Perimeter} = \left(\frac{3}{2}x + \frac{5}{2}\right) + \left(5x - \frac{3}{2}\right) + \left(4x + \frac{1}{4}\right) \]
Now, we will combine the coefficients of \(x\) and the constant terms:
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Coefficients of \(x\):
- From the first side: \( \frac{3}{2}x \)
- From the second side: \( 5x \)
- From the third side: \( 4x \)
Summing these:
\[ \frac{3}{2} + 5 + 4 = \frac{3}{2} + \frac{10}{2} + \frac{8}{2} = \frac{21}{2}x \]
-
Constant terms:
- From the first side: \( \frac{5}{2} \)
- From the second side: \( -\frac{3}{2} \)
- From the third side: \( \frac{1}{4} \)
Summing these:
\[ \frac{5}{2} - \frac{3}{2} + \frac{1}{4} = \frac{2}{2} + \frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \]
Now, combining both parts gives us the total perimeter:
\[ \text{Perimeter} = \frac{21}{2}x + \frac{5}{4} \]
Therefore, the expression that represents the perimeter of the triangle in units is:
\[ \boxed{\frac{21}{2}x + \frac{5}{4}} \]