Multiply $3$ by $ \dfrac{i}{4} $ to get $ \dfrac{ 3i }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{i}{4} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{i}{4} \xlongequal{\text{Step 2}} \frac{ 3 \cdot i }{ 1 \cdot 4 } \xlongequal{\text{Step 3}} \frac{ 3i }{ 4 } \end{aligned} $$

Subtract $ \dfrac{3i}{4} $ from $ 2 $ to get $ \dfrac{ \color{purple}{ -3i+8 } }{ 4 }$.

Step 1: Write $ 2 $ 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 $\color{blue}{ 4 }$.

$$ \begin{aligned} 2- \frac{3i}{4} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} - \frac{3i}{4} = \frac{ 2 \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} - \frac{ 3i }{ 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 8 } }{ 4 } - \frac{ \color{purple}{ 3i } }{ 4 }=\frac{ \color{purple}{ -3i+8 } }{ 4 } \end{aligned} $$

Add $ \dfrac{-3i+8}{4} $ and $ 3i $ to get $ \dfrac{ \color{purple}{ 9i+8 } }{ 4 }$.

Step 1: Write $ 3i $ 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 $\color{blue}{4}$.

$$ \begin{aligned} \frac{-3i+8}{4} +3i & \xlongequal{\text{Step 1}} \frac{-3i+8}{4} + \frac{3i}{\color{red}{1}} = \frac{ -3i+8 }{ 4 } + \frac{ 3i \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -3i+8 } }{ 4 } + \frac{ \color{purple}{ 12i } }{ 4 }=\frac{ \color{purple}{ 9i+8 } }{ 4 } \end{aligned} $$