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

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 first fraction by $\color{blue}{ 2 }$.

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

Add $ \dfrac{2x+3}{2} $ and $ \dfrac{2x-3}{2} $ to get $ \dfrac{4x}{2} $.

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

$$ \begin{aligned} \frac{2x+3}{2} + \frac{2x-3}{2} & = \frac{2x+3}{\color{blue}{2}} + \frac{2x-3}{\color{blue}{2}} =\frac{ 2x+3 + \left( 2x-3 \right) }{ \color{blue}{ 2 }} = \\[1ex] &= \frac{4x}{2} \end{aligned} $$