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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Multiply $ \dfrac{9}{4} $ by $ x $ to get $ \dfrac{ 9x }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{9}{4} \cdot x & \xlongequal{\text{Step 1}} \frac{9}{4} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 9 \cdot x }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 9x }{ 4 } \end{aligned} $$

Subtract $6$ from $ \dfrac{4x}{3} $ to get $ \dfrac{ \color{purple}{ 4x-18 } }{ 3 }$.

Step 1: Write $ 6 $ 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 $\color{blue}{3}$.

$$ \begin{aligned} \frac{4x}{3} -6 & \xlongequal{\text{Step 1}} \frac{4x}{3} - \frac{6}{\color{red}{1}} = \frac{ 4x }{ 3 } - \frac{ 6 \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4x } }{ 3 } - \frac{ \color{purple}{ 18 } }{ 3 }=\frac{ \color{purple}{ 4x-18 } }{ 3 } \end{aligned} $$

Multiply $ \dfrac{9}{4} $ by $ x $ to get $ \dfrac{ 9x }{ 4 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{9}{4} \cdot x & \xlongequal{\text{Step 1}} \frac{9}{4} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 9 \cdot x }{ 4 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 9x }{ 4 } \end{aligned} $$

Add $ \dfrac{4x-18}{3} $ and $ \dfrac{9x}{4} $ to get $ \dfrac{ \color{purple}{ 43x-72 } }{ 12 }$.

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}{ 4 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{4x-18}{3} + \frac{9x}{4} & = \frac{ \left( 4x-18 \right) \cdot \color{blue}{ 4 }}{ 3 \cdot \color{blue}{ 4 }} + \frac{ 9x \cdot \color{blue}{ 3 }}{ 4 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 16x-72 } }{ 12 } + \frac{ \color{purple}{ 27x } }{ 12 }=\frac{ \color{purple}{ 43x-72 } }{ 12 } \end{aligned} $$

Subtract $8$ from $ \dfrac{43x-72}{12} $ to get $ \dfrac{ \color{purple}{ 43x-168 } }{ 12 }$.

Step 1: Write $ 8 $ 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 $\color{blue}{12}$.

$$ \begin{aligned} \frac{43x-72}{12} -8 & \xlongequal{\text{Step 1}} \frac{43x-72}{12} - \frac{8}{\color{red}{1}} = \frac{ 43x-72 }{ 12 } - \frac{ 8 \cdot \color{blue}{ 12 }}{ 1 \cdot \color{blue}{ 12 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 43x-72 } }{ 12 } - \frac{ \color{purple}{ 96 } }{ 12 }=\frac{ \color{purple}{ 43x-168 } }{ 12 } \end{aligned} $$