Divide $ \dfrac{7}{x} $ by $ 7 $ to get $ \dfrac{ 1 }{ x } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Cancel $ \color{blue}{ 7 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{7}{x} }{7} & \xlongequal{\text{Step 1}} \frac{7}{x} \cdot \frac{\color{blue}{1}}{\color{blue}{7}} \xlongequal{\text{Step 2}} \frac{\color{blue}{1}}{x} \cdot \frac{1}{\color{blue}{1}} = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 1 \cdot 1 }{ x \cdot 1 } \xlongequal{\text{Step 4}} \frac{ 1 }{ x } \end{aligned} $$

Add $x$ and $ \dfrac{1}{x} $ to get $ \dfrac{ \color{purple}{ x^2+1 } }{ x }$.

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 $\color{blue}{ x }$.

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

Add $ \dfrac{x^2+1}{x} $ and $ 7 $ to get $ \dfrac{ \color{purple}{ x^2+7x+1 } }{ x }$.

Step 1: Write $ 7 $ 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 $\color{blue}{x}$.

$$ \begin{aligned} \frac{x^2+1}{x} +7 & \xlongequal{\text{Step 1}} \frac{x^2+1}{x} + \frac{7}{\color{red}{1}} = \frac{ x^2+1 }{ x } + \frac{ 7 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ x^2+1 } }{ x } + \frac{ \color{purple}{ 7x } }{ x }=\frac{ \color{purple}{ x^2+7x+1 } }{ x } \end{aligned} $$