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

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $2$ by $ \dfrac{i}{9} $ to get $ \dfrac{ 2i }{ 9 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{i}{9} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{i}{9} \xlongequal{\text{Step 2}} \frac{ 2 \cdot i }{ 1 \cdot 9 } \xlongequal{\text{Step 3}} \frac{ 2i }{ 9 } \end{aligned} $$

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

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

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

Multiply $2$ by $ \dfrac{i}{9} $ to get $ \dfrac{ 2i }{ 9 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 2 \cdot \frac{i}{9} & \xlongequal{\text{Step 1}} \frac{2}{\color{red}{1}} \cdot \frac{i}{9} \xlongequal{\text{Step 2}} \frac{ 2 \cdot i }{ 1 \cdot 9 } \xlongequal{\text{Step 3}} \frac{ 2i }{ 9 } \end{aligned} $$

Add $ \dfrac{-4i+3}{3} $ and $ \dfrac{2i}{9} $ to get $ \dfrac{ \color{purple}{ -10i+9 } }{ 9 }$.

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

$$ \begin{aligned} \frac{-4i+3}{3} + \frac{2i}{9} & = \frac{ \left( -4i+3 \right) \cdot \color{blue}{ 3 }}{ 3 \cdot \color{blue}{ 3 }} + \frac{ 2i }{ 9 } = \\[1ex] &=\frac{ \color{purple}{ -12i+9 } }{ 9 } + \frac{ \color{purple}{ 2i } }{ 9 }=\frac{ \color{purple}{ -10i+9 } }{ 9 } \end{aligned} $$

Subtract $ \dfrac{10}{3} $ from $ \dfrac{-10i+9}{9} $ to get $ \dfrac{ \color{purple}{ -10i-21 } }{ 9 }$.

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

$$ \begin{aligned} \frac{-10i+9}{9} - \frac{10}{3} & = \frac{ -10i+9 }{ 9 } - \frac{ 10 \cdot \color{blue}{ 3 }}{ 3 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ -10i+9 } }{ 9 } - \frac{ \color{purple}{ 30 } }{ 9 }=\frac{ \color{purple}{ -10i-21 } }{ 9 } \end{aligned} $$