Add $ \dfrac{2i-4}{5} $ and $ 2i $ to get $ \dfrac{ \color{purple}{ 12i-4 } }{ 5 }$.

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 $\color{blue}{5}$.

$$ \begin{aligned} \frac{2i-4}{5} +2i & \xlongequal{\text{Step 1}} \frac{2i-4}{5} + \frac{2i}{\color{red}{1}} = \frac{ 2i-4 }{ 5 } + \frac{ 2i \cdot \color{blue}{ 5 }}{ 1 \cdot \color{blue}{ 5 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2i-4 } }{ 5 } + \frac{ \color{purple}{ 10i } }{ 5 }=\frac{ \color{purple}{ 12i-4 } }{ 5 } \end{aligned} $$

Subtract $2$ from $ \dfrac{12i-4}{5} $ to get $ \dfrac{ \color{purple}{ 12i-14 } }{ 5 }$.

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 second fraction by $\color{blue}{5}$.

$$ \begin{aligned} \frac{12i-4}{5} -2 & \xlongequal{\text{Step 1}} \frac{12i-4}{5} - \frac{2}{\color{red}{1}} = \frac{ 12i-4 }{ 5 } - \frac{ 2 \cdot \color{blue}{ 5 }}{ 1 \cdot \color{blue}{ 5 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 12i-4 } }{ 5 } - \frac{ \color{purple}{ 10 } }{ 5 }=\frac{ \color{purple}{ 12i-14 } }{ 5 } \end{aligned} $$