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 $\color{blue}{ 2 }$.

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

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

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 $\color{blue}{ 4 }$.

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

Add $ \dfrac{2x-1}{2} $ and $ \dfrac{y}{2} $ to get $ \dfrac{2x+y-1}{2} $.

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

$$ \begin{aligned} \frac{2x-1}{2} + \frac{y}{2} & = \frac{2x-1}{\color{blue}{2}} + \frac{y}{\color{blue}{2}} =\frac{ 2x-1 + y }{ \color{blue}{ 2 }} = \\[1ex] &= \frac{2x+y-1}{2} \end{aligned} $$

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

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

$$ \begin{aligned} \frac{2x-1}{2} - \frac{y}{2} & = \frac{2x-1}{\color{blue}{2}} - \frac{y}{\color{blue}{2}} =\frac{ 2x-1 - y }{ \color{blue}{ 2 }} = \\[1ex] &= \frac{2x-y-1}{2} \end{aligned} $$

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

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 $\color{blue}{ 4 }$.

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

Multiply $ \dfrac{2x+y-1}{2} $ by $ \dfrac{2x-y-1}{2} $ to get $ \dfrac{4x^2-y^2-4x+1}{4} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2x+y-1}{2} \cdot \frac{2x-y-1}{2} & \xlongequal{\text{Step 1}} \frac{ \left( 2x+y-1 \right) \cdot \left( 2x-y-1 \right) }{ 2 \cdot 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 4x^2 -\cancel{2xy}-2x+ \cancel{2xy}-y^2 -\cancel{y}-2x+ \cancel{y}+1 }{ 4 } = \frac{4x^2-y^2-4x+1}{4} \end{aligned} $$

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

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 $\color{blue}{ 4 }$.

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

Multiply $ \dfrac{4x^2-y^2-4x+1}{4} $ by $ \dfrac{4x+3}{4} $ to get $ \dfrac{16x^3-4xy^2-4x^2-3y^2-8x+3}{16} $.

Step 1: Multiply numerators and denominators.

Step 2: Simplify numerator and denominator.

$$ \begin{aligned} \frac{4x^2-y^2-4x+1}{4} \cdot \frac{4x+3}{4} & \xlongequal{\text{Step 1}} \frac{ \left( 4x^2-y^2-4x+1 \right) \cdot \left( 4x+3 \right) }{ 4 \cdot 4 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 16x^3+12x^2-4xy^2-3y^2-16x^2-12x+4x+3 }{ 16 } = \frac{16x^3-4xy^2-4x^2-3y^2-8x+3}{16} \end{aligned} $$