Add $x$ and $ \dfrac{5}{2} $ to get $ \dfrac{ \color{purple}{ 2x+5 } }{ 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 $\color{blue}{ 2 }$.

$$ \begin{aligned} x+ \frac{5}{2} & \xlongequal{\text{Step 1}} \frac{x}{\color{red}{1}} + \frac{5}{2} = \frac{ x \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 5 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x } }{ 2 } + \frac{ \color{purple}{ 5 } }{ 2 }=\frac{ \color{purple}{ 2x+5 } }{ 2 } \end{aligned} $$

Add $ \dfrac{2x+5}{2} $ and $ x $ to get $ \dfrac{ \color{purple}{ 4x+5 } }{ 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 second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{2x+5}{2} +x & \xlongequal{\text{Step 1}} \frac{2x+5}{2} + \frac{x}{\color{red}{1}} = \frac{ 2x+5 }{ 2 } + \frac{ x \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x+5 } }{ 2 } + \frac{ \color{purple}{ 2x } }{ 2 }=\frac{ \color{purple}{ 4x+5 } }{ 2 } \end{aligned} $$

Add $ \dfrac{4x+5}{2} $ and $ \dfrac{2}{7} $ to get $ \dfrac{ \color{purple}{ 28x+39 } }{ 14 }$.

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}{ 7 }$ and the second by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{4x+5}{2} + \frac{2}{7} & = \frac{ \left( 4x+5 \right) \cdot \color{blue}{ 7 }}{ 2 \cdot \color{blue}{ 7 }} + \frac{ 2 \cdot \color{blue}{ 2 }}{ 7 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ 28x+35 } }{ 14 } + \frac{ \color{purple}{ 4 } }{ 14 }=\frac{ \color{purple}{ 28x+39 } }{ 14 } \end{aligned} $$