Multiply $3$ by $ \dfrac{x}{x} $ to get $ \dfrac{ 3x }{ x } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{6}{x} $ by $ 2 $ to get $ \dfrac{ 12 }{ x } $.

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.

$$ \begin{aligned} \frac{6}{x} \cdot 2 & \xlongequal{\text{Step 1}} \frac{6}{x} \cdot \frac{2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 6 \cdot 2 }{ x \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 12 }{ x } \end{aligned} $$

Multiply $ \dfrac{3x}{x} $ by $ 2 $ to get $ \dfrac{ 6x }{ x } $.

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.

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

Multiply $ \dfrac{6}{x} $ by $ 2 $ to get $ \dfrac{ 12 }{ x } $.

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.

$$ \begin{aligned} \frac{6}{x} \cdot 2 & \xlongequal{\text{Step 1}} \frac{6}{x} \cdot \frac{2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 6 \cdot 2 }{ x \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 12 }{ x } \end{aligned} $$

Subtract $9$ from $ \dfrac{6x}{x} $ to get $ \dfrac{ \color{purple}{ -3x } }{ x }$.

Step 1: Write $ 9 $ 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 $\color{blue}{x}$.

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

Multiply $ \dfrac{6}{x} $ by $ 2 $ to get $ \dfrac{ 12 }{ x } $.

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.

$$ \begin{aligned} \frac{6}{x} \cdot 2 & \xlongequal{\text{Step 1}} \frac{6}{x} \cdot \frac{2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 6 \cdot 2 }{ x \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 12 }{ x } \end{aligned} $$

Subtract $ \dfrac{12}{x} $ from $ \dfrac{-3x}{x} $ to get $ \dfrac{ -3x - 12 }{ \color{blue}{ x }}$.

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

$$ \begin{aligned} \frac{-3x}{x} - \frac{12}{x} & = \frac{-3x}{\color{blue}{x}} - \frac{12}{\color{blue}{x}} =\frac{ -3x - 12 }{ \color{blue}{ x }} \end{aligned} $$

Add $ \dfrac{-3x-12}{x} $ and $ x $ to get $ \dfrac{ \color{purple}{ x^2-3x-12 } }{ x }$.

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

$$ \begin{aligned} \frac{-3x-12}{x} +x & \xlongequal{\text{Step 1}} \frac{-3x-12}{x} + \frac{x}{\color{red}{1}} = \frac{ -3x-12 }{ x } + \frac{ x \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -3x-12 } }{ x } + \frac{ \color{purple}{ x^2 } }{ x }=\frac{ \color{purple}{ x^2-3x-12 } }{ x } \end{aligned} $$

Subtract $12$ from $ \dfrac{x^2-3x-12}{x} $ to get $ \dfrac{ \color{purple}{ x^2-15x-12 } }{ x }$.

Step 1: Write $ 12 $ 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 $\color{blue}{x}$.

$$ \begin{aligned} \frac{x^2-3x-12}{x} -12 & \xlongequal{\text{Step 1}} \frac{x^2-3x-12}{x} - \frac{12}{\color{red}{1}} = \frac{ x^2-3x-12 }{ x } - \frac{ 12 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2-3x-12 } }{ x } - \frac{ \color{purple}{ 12x } }{ x }=\frac{ \color{purple}{ x^2-15x-12 } }{ x } \end{aligned} $$