Multiply $7$ by $ \dfrac{x}{10} $ to get $ \dfrac{ 7x }{ 10 } $.

Step 1: Write $ 7 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 7 \cdot \frac{x}{10} & \xlongequal{\text{Step 1}} \frac{7}{\color{red}{1}} \cdot \frac{x}{10} \xlongequal{\text{Step 2}} \frac{ 7 \cdot x }{ 1 \cdot 10 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 7x }{ 10 } \end{aligned} $$

Add $ \dfrac{x}{5} $ and $ \dfrac{7x}{10} $ to get $ \dfrac{ \color{purple}{ 9x } }{ 10 }$.

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

$$ \begin{aligned} \frac{x}{5} + \frac{7x}{10} & = \frac{ x \cdot \color{blue}{ 2 }}{ 5 \cdot \color{blue}{ 2 }} + \frac{ 7x }{ 10 } = \\[1ex] &=\frac{ \color{purple}{ 2x } }{ 10 } + \frac{ \color{purple}{ 7x } }{ 10 }=\frac{ \color{purple}{ 9x } }{ 10 } \end{aligned} $$