Subtract $3i$ from $ \dfrac{5+i}{2} $ to get $ \dfrac{ \color{purple}{ -5i+5 } }{ 2 }$.

Step 1: Write $ 3i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{5+i}{2} -3i & \xlongequal{\text{Step 1}} \frac{5+i}{2} - \frac{3i}{\color{red}{1}} = \frac{ 5+i }{ 2 } - \frac{ 3i \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 5+i } }{ 2 } - \frac{ \color{purple}{ 6i } }{ 2 }=\frac{ \color{purple}{ -5i+5 } }{ 2 } \end{aligned} $$