Subtract $ \dfrac{i}{2} $ from $ 3 $ to get $ \dfrac{ \color{purple}{ -i+6 } }{ 2 }$.

Step 1: Write $ 3 $ 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 first fraction by $\color{blue}{ 2 }$.

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

Add $ \dfrac{-i+6}{2} $ and $ 2i $ to get $ \dfrac{ \color{purple}{ 3i+6 } }{ 2 }$.

Step 1: Write $ 2i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

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

$$ \begin{aligned} \frac{-i+6}{2} +2i & \xlongequal{\text{Step 1}} \frac{-i+6}{2} + \frac{2i}{\color{red}{1}} = \frac{ -i+6 }{ 2 } + \frac{ 2i \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -i+6 } }{ 2 } + \frac{ \color{purple}{ 4i } }{ 2 }=\frac{ \color{purple}{ 3i+6 } }{ 2 } \end{aligned} $$