Multiply $ \dfrac{1}{2} $ by $ a $ to get $ \dfrac{ a }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{3}{2} $ by $ a $ to get $ \dfrac{ 3a }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{a}{2} $ by $ b $ to get $ \dfrac{ ab }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{3a}{2} $ by $ b $ to get $ \dfrac{ 3ab }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{3a}{2} \cdot b & \xlongequal{\text{Step 1}} \frac{3a}{2} \cdot \frac{b}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 3a \cdot b }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 3ab }{ 2 } \end{aligned} $$

Subtract $4bc$ from $ \dfrac{ab}{2} $ to get $ \dfrac{ \color{purple}{ ab-8bc } }{ 2 }$.

Step 1: Write $ 4bc $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{ab}{2} -4bc & \xlongequal{\text{Step 1}} \frac{ab}{2} - \frac{4bc}{\color{red}{1}} = \frac{ ab }{ 2 } - \frac{ 4bc \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ ab } }{ 2 } - \frac{ \color{purple}{ 8bc } }{ 2 }=\frac{ \color{purple}{ ab-8bc } }{ 2 } \end{aligned} $$

Add $ac-2bc$ and $ \dfrac{3ab}{2} $ to get $ \dfrac{ \color{purple}{ 3ab+2ac-4bc } }{ 2 }$.

Step 1: Write $ ac-2bc $ 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 $\color{blue}{ 2 }$.

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

Add $ \dfrac{ab-8bc}{2} $ and $ 0ac $ to get $ \dfrac{ \color{purple}{ ab-8bc } }{ 2 }$.

Step 1: Write $ 0ac $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{ab-8bc}{2} +0ac & \xlongequal{\text{Step 1}} \frac{ab-8bc}{2} + \frac{0ac}{\color{red}{1}} = \frac{ ab-8bc }{ 2 } + \frac{ 0ac \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ ab-8bc } }{ 2 } + \frac{ \color{purple}{ 0ac } }{ 2 }=\frac{ \color{purple}{ ab-8bc } }{ 2 } \end{aligned} $$

Add $ac-2bc$ and $ \dfrac{3ab}{2} $ to get $ \dfrac{ \color{purple}{ 3ab+2ac-4bc } }{ 2 }$.

Step 1: Write $ ac-2bc $ 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 $\color{blue}{ 2 }$.

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

Subtract $ \dfrac{3ab+2ac-4bc}{2} $ from $ \dfrac{ab-8bc}{2} $ to get $ \dfrac{-2ab-2ac-4bc}{2} $.

To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{ab-8bc}{2} - \frac{3ab+2ac-4bc}{2} & = \frac{ab-8bc}{\color{blue}{2}} - \frac{3ab+2ac-4bc}{\color{blue}{2}} = \\[1ex] &=\frac{ ab-8bc - \left( 3ab+2ac-4bc \right) }{ \color{blue}{ 2 }}= \frac{-2ab-2ac-4bc}{2} \end{aligned} $$

Add $ \dfrac{-2ab-2ac-4bc}{2} $ and $ 3bc+2ac $ to get $ \dfrac{ \color{purple}{ -2ab+2ac+2bc } }{ 2 }$.

Step 1: Write $ 3bc+2ac $ 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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{-2ab-2ac-4bc}{2} +3bc+2ac & \xlongequal{\text{Step 1}} \frac{-2ab-2ac-4bc}{2} + \frac{3bc+2ac}{\color{red}{1}} = \frac{ -2ab-2ac-4bc }{ 2 } + \frac{ \left( 3bc+2ac \right) \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -2ab-2ac-4bc } }{ 2 } + \frac{ \color{purple}{ 6bc+4ac } }{ 2 }=\frac{ \color{purple}{ -2ab+2ac+2bc } }{ 2 } \end{aligned} $$