Multiply $ \dfrac{1}{2} $ by $ x $ to get $ \dfrac{ x }{ 2 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

Step 1: Write $ x^2 $ 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 $\color{blue}{ 2 }$.

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

Subtract $3$ from $ \dfrac{2x^2-x}{2} $ to get $ \dfrac{ \color{purple}{ 2x^2-x-6 } }{ 2 }$.

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 $\color{blue}{2}$.

$$ \begin{aligned} \frac{2x^2-x}{2} -3 & \xlongequal{\text{Step 1}} \frac{2x^2-x}{2} - \frac{3}{\color{red}{1}} = \frac{ 2x^2-x }{ 2 } - \frac{ 3 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2x^2-x } }{ 2 } - \frac{ \color{purple}{ 6 } }{ 2 }=\frac{ \color{purple}{ 2x^2-x-6 } }{ 2 } \end{aligned} $$

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

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

Step 2: Write $ x-2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Factor numerators and denominators.

Step 4: Cancel common factors.

Step 5: Multiply numerators and denominators.

Step 6: Simplify numerator and denominator.

$$ \begin{aligned} \frac{x-2}{ \frac{\color{blue}{2x^2-x-6}}{\color{blue}{2}} } & \xlongequal{\text{Step 1}} x-2 \cdot \frac{\color{blue}{2}}{\color{blue}{2x^2-x-6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{x-2}{\color{red}{1}} \cdot \frac{2}{2x^2-x-6} \xlongequal{\text{Step 3}} \frac{ 1 \cdot \color{blue}{ \left( x-2 \right) } }{ 1 } \cdot \frac{ 2 }{ \left( 2x+3 \right) \cdot \color{blue}{ \left( x-2 \right) } } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 1 }{ 1 } \cdot \frac{ 2 }{ 2x+3 } \xlongequal{\text{Step 5}} \frac{ 1 \cdot 2 }{ 1 \cdot \left( 2x+3 \right) } \xlongequal{\text{Step 6}} \frac{ 2 }{ 2x+3 } \end{aligned} $$