Subtract $i$ from $ \dfrac{1}{2} $ to get $ \dfrac{ \color{purple}{ -2i+1 } }{ 2 }$.

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

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

Multiply $ \dfrac{1}{3} $ by $ i $ to get $ \dfrac{ i }{ 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.

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

Subtract $i$ from $ \dfrac{1}{2} $ to get $ \dfrac{ \color{purple}{ -2i+1 } }{ 2 }$.

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

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

Add $-1$ and $ \dfrac{i}{3} $ to get $ \dfrac{ \color{purple}{ i-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 first fraction by $\color{blue}{ 3 }$.

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

Divide $ \dfrac{-2i+1}{2} $ by $ \dfrac{i-3}{3} $ to get $ \dfrac{ -6i+3 }{ 2i-6 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Divide $ \, 3-6i \, $ by $ \, -6+2i \, $ to get $\,\, \dfrac{-3+3i}{4} $.
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