To find the perimeter of the triangle, we need to add the lengths of its sides together.
The sides of the triangle are given as:
- \( \frac{3}{2}x + \frac{5}{2} \)
- \( 5x - \frac{3}{2} \)
- \( 4x + \frac{1}{4} \)
Let's add these expressions together:
\[ \text{Perimeter} = \left(\frac{3}{2}x + \frac{5}{2}\right) + \left(5x - \frac{3}{2}\right) + \left(4x + \frac{1}{4}\right) \]
Now we will combine the \( x \) terms and the constant terms separately.
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Combining the \( x \) terms: \[ \frac{3}{2}x + 5x + 4x \] Converting \( 5x \) and \( 4x \) to fractions with a common denominator: \[ 5x = \frac{10}{2}x \quad \text{and} \quad 4x = \frac{8}{2}x \] Therefore: \[ \frac{3}{2}x + \frac{10}{2}x + \frac{8}{2}x = \left(\frac{3 + 10 + 8}{2}\right)x = \frac{21}{2}x \]
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Combining the constant terms: \[ \frac{5}{2} - \frac{3}{2} + \frac{1}{4} \] First, we combine \( \frac{5}{2} - \frac{3}{2} \): \[ \frac{5}{2} - \frac{3}{2} = \frac{2}{2} = 1 \] Next, we need to add \( 1 + \frac{1}{4} \). Convert \( 1 \) into a fraction with a denominator of 4: \[ 1 = \frac{4}{4} \] So: \[ \frac{4}{4} + \frac{1}{4} = \frac{5}{4} \]
Now, putting both parts together, the perimeter is: \[ \text{Perimeter} = \frac{21}{2}x + \frac{5}{4} \]
Thus, the expression that represents the perimeter of the triangle is:
D. \( \frac{21}{2}x + \frac{5}{4} \).