Multiply $6i$ by $ \dfrac{9}{2} $ to get $ \dfrac{ 54i }{ 2 } $.

Step 1: Write $ 6i $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 6i \cdot \frac{9}{2} & \xlongequal{\text{Step 1}} \frac{6i}{\color{red}{1}} \cdot \frac{9}{2} \xlongequal{\text{Step 2}} \frac{ 6i \cdot 9 }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 54i }{ 2 } \end{aligned} $$

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

Step 1: Write $ 5i $ 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 $\color{blue}{4}$.

$$ \begin{aligned} \frac{3}{4} +5i & \xlongequal{\text{Step 1}} \frac{3}{4} + \frac{5i}{\color{red}{1}} = \frac{ 3 }{ 4 } + \frac{ 5i \cdot \color{blue}{ 4 }}{ 1 \cdot \color{blue}{ 4 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 3 } }{ 4 } + \frac{ \color{purple}{ 20i } }{ 4 }=\frac{ \color{purple}{ 20i+3 } }{ 4 } \end{aligned} $$

Multiply $6i$ by $ \dfrac{9}{2} $ to get $ \dfrac{ 54i }{ 2 } $.

Step 1: Write $ 6i $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 6i \cdot \frac{9}{2} & \xlongequal{\text{Step 1}} \frac{6i}{\color{red}{1}} \cdot \frac{9}{2} \xlongequal{\text{Step 2}} \frac{ 6i \cdot 9 }{ 1 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 54i }{ 2 } \end{aligned} $$

Multiply $4$ by $ \dfrac{20i+3}{4} $ to get $ 20i+3$.

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

Step 2: Cancel $ \color{blue}{ 4 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{20i+3}{4} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{20i+3}{4} \xlongequal{\text{Step 2}} \frac{\color{blue}{1}}{1} \cdot \frac{20i+3}{\color{blue}{1}} = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 1 \cdot \left( 20i+3 \right) }{ 1 \cdot 1 } \xlongequal{\text{Step 4}} \frac{ 20i+3 }{ 1 } =20i+3 \end{aligned} $$

Subtract $20i+3$ from $ \dfrac{54i}{2} $ to get $ \dfrac{ \color{purple}{ 14i-6 } }{ 2 }$.

Step 1: Write $ 20i+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 second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{54i}{2} -20i+3 & \xlongequal{\text{Step 1}} \frac{54i}{2} - \frac{20i+3}{\color{red}{1}} = \frac{ 54i }{ 2 } - \frac{ \left( 20i+3 \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 54i } }{ 2 } - \frac{ \color{purple}{ 40i+6 } }{ 2 }=\frac{ \color{purple}{ 54i - \left( 40i+6 \right) } }{ 2 } = \\[1ex] &=\frac{ \color{purple}{ 14i-6 } }{ 2 } \end{aligned} $$