Subtract $ \dfrac{1}{2} $ from $ x $ to get $ \dfrac{ \color{purple}{ 2x-1 } }{ 2 }$.

Step 1: Write $ x $ 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 first fraction by $\color{blue}{ 2 }$.

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

Add $x$ and $ \dfrac{1}{3} $ to get $ \dfrac{ \color{purple}{ 3x+1 } }{ 3 }$.

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 $\color{blue}{ 3 }$.

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

Multiply $ \dfrac{2x-1}{2} $ by $ \dfrac{3x+1}{3} $ to get $ \dfrac{6x^2-x-1}{6} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2x-1}{2} \cdot \frac{3x+1}{3} & \xlongequal{\text{Step 1}} \frac{ \left( 2x-1 \right) \cdot \left( 3x+1 \right) }{ 2 \cdot 3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 6x^2+2x-3x-1 }{ 6 } = \frac{6x^2-x-1}{6} \end{aligned} $$