Multiply $5$ by $ \dfrac{x}{6} $ to get $ \dfrac{ 5x }{ 6 } $.

Step 1: Write $ 5 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 5 \cdot \frac{x}{6} & \xlongequal{\text{Step 1}} \frac{5}{\color{red}{1}} \cdot \frac{x}{6} \xlongequal{\text{Step 2}} \frac{ 5 \cdot x }{ 1 \cdot 6 } \xlongequal{\text{Step 3}} \frac{ 5x }{ 6 } \end{aligned} $$

Multiply $5$ by $ \dfrac{x}{3} $ to get $ \dfrac{ 5x }{ 3 } $.

Step 1: Write $ 5 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 5 \cdot \frac{x}{3} & \xlongequal{\text{Step 1}} \frac{5}{\color{red}{1}} \cdot \frac{x}{3} \xlongequal{\text{Step 2}} \frac{ 5 \cdot x }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ 5x }{ 3 } \end{aligned} $$

Subtract $ \dfrac{5x}{6} $ from $ \dfrac{x}{9} $ to get $ \dfrac{ \color{purple}{ -13x } }{ 18 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 2 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{x}{9} - \frac{5x}{6} & = \frac{ x \cdot \color{blue}{ 2 }}{ 9 \cdot \color{blue}{ 2 }} - \frac{ 5x \cdot \color{blue}{ 3 }}{ 6 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 2x } }{ 18 } - \frac{ \color{purple}{ 15x } }{ 18 }=\frac{ \color{purple}{ -13x } }{ 18 } \end{aligned} $$

Multiply $5$ by $ \dfrac{x}{3} $ to get $ \dfrac{ 5x }{ 3 } $.

Step 1: Write $ 5 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 5 \cdot \frac{x}{3} & \xlongequal{\text{Step 1}} \frac{5}{\color{red}{1}} \cdot \frac{x}{3} \xlongequal{\text{Step 2}} \frac{ 5 \cdot x }{ 1 \cdot 3 } \xlongequal{\text{Step 3}} \frac{ 5x }{ 3 } \end{aligned} $$

Add $ \dfrac{-13x}{18} $ and $ \dfrac{5x}{3} $ to get $ \dfrac{ \color{purple}{ 17x } }{ 18 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{6}$.

$$ \begin{aligned} \frac{-13x}{18} + \frac{5x}{3} & = \frac{ -13x }{ 18 } + \frac{ 5x \cdot \color{blue}{ 6 }}{ 3 \cdot \color{blue}{ 6 }} = \\[1ex] &=\frac{ \color{purple}{ -13x } }{ 18 } + \frac{ \color{purple}{ 30x } }{ 18 }=\frac{ \color{purple}{ 17x } }{ 18 } \end{aligned} $$