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.

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

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

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.

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

Add $ \dfrac{2x}{3} $ and $ \dfrac{x}{2} $ to get $ \dfrac{ \color{purple}{ 7x } }{ 6 }$.

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

$$ \begin{aligned} \frac{2x}{3} + \frac{x}{2} & = \frac{ 2x \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} + \frac{ x \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 4x } }{ 6 } + \frac{ \color{purple}{ 3x } }{ 6 }=\frac{ \color{purple}{ 7x } }{ 6 } \end{aligned} $$

Subtract $ \dfrac{5}{2} $ from $ \dfrac{7x}{6} $ to get $ \dfrac{ \color{purple}{ 7x-15 } }{ 6 }$.

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

$$ \begin{aligned} \frac{7x}{6} - \frac{5}{2} & = \frac{ 7x }{ 6 } - \frac{ 5 \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 7x } }{ 6 } - \frac{ \color{purple}{ 15 } }{ 6 }=\frac{ \color{purple}{ 7x-15 } }{ 6 } \end{aligned} $$