Add $ \dfrac{1}{x} $ and $ 4 $ to get $ \dfrac{ \color{purple}{ 4x+1 } }{ x }$.

Step 1: Write $ 4 $ 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 $\color{blue}{x}$.

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

Subtract $1$ from $ \dfrac{4x+1}{x} $ to get $ \dfrac{ \color{purple}{ 3x+1 } }{ x }$.

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

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x}$.

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