Add $ \dfrac{1}{2} $ and $ 3i $ to get $ \dfrac{ \color{purple}{ 6i+1 } }{ 2 }$.

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 $\color{blue}{2}$.

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

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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Add $ \dfrac{1}{2} $ and $ 3i $ to get $ \dfrac{ \color{purple}{ 6i+1 } }{ 2 }$.

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 $\color{blue}{2}$.

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

Add $ \dfrac{1}{5} $ and $ \dfrac{4i}{3} $ to get $ \dfrac{ \color{purple}{ 20i+3 } }{ 15 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 3 }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{1}{5} + \frac{4i}{3} & = \frac{ 1 \cdot \color{blue}{ 3 }}{ 5 \cdot \color{blue}{ 3 }} + \frac{ 4i \cdot \color{blue}{ 5 }}{ 3 \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ 3 } }{ 15 } + \frac{ \color{purple}{ 20i } }{ 15 }=\frac{ \color{purple}{ 20i+3 } }{ 15 } \end{aligned} $$

Subtract $ \dfrac{20i+3}{15} $ from $ \dfrac{6i+1}{2} $ to get $ \dfrac{ \color{purple}{ 50i+9 } }{ 30 }$.

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}{ 15 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{6i+1}{2} - \frac{20i+3}{15} & = \frac{ \left( 6i+1 \right) \cdot \color{blue}{ 15 }}{ 2 \cdot \color{blue}{ 15 }} - \frac{ \left( 20i+3 \right) \cdot \color{blue}{ 2 }}{ 15 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 90i+15 } }{ 30 } - \frac{ \color{purple}{ 40i+6 } }{ 30 }=\frac{ \color{purple}{ 90i+15 - \left( 40i+6 \right) } }{ 30 } = \\[1ex] &=\frac{ \color{purple}{ 50i+9 } }{ 30 } \end{aligned} $$