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

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.

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

Add $ \dfrac{-2x}{3} $ and $ \dfrac{1}{9} $ to get $ \dfrac{ \color{purple}{ -6x+1 } }{ 9 }$.

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

$$ \begin{aligned} \frac{-2x}{3} + \frac{1}{9} & = \frac{ \left( -2x \right) \cdot \color{blue}{ 3 }}{ 3 \cdot \color{blue}{ 3 }} + \frac{ 1 }{ 9 } = \\[1ex] &=\frac{ \color{purple}{ -6x } }{ 9 } + \frac{ \color{purple}{ 1 } }{ 9 }=\frac{ \color{purple}{ -6x+1 } }{ 9 } \end{aligned} $$