Add $ \dfrac{4}{x} $ and $ 3 $ to get $ \dfrac{ \color{purple}{ 3x+4 } }{ x }$.

Step 1: Write $ 3 $ 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 $\color{blue}{x}$.

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

Multiply $3$ by $ \dfrac{x}{x} $ to get $ \dfrac{ 3x }{ x } $.

Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 3 \cdot \frac{x}{x} & \xlongequal{\text{Step 1}} \frac{3}{\color{red}{1}} \cdot \frac{x}{x} \xlongequal{\text{Step 2}} \frac{ 3 \cdot x }{ 1 \cdot x } \xlongequal{\text{Step 3}} \frac{ 3x }{ x } \end{aligned} $$

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

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

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

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

Step 1: Write $ 3 $ 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 $\color{blue}{x}$.

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