Subtract $1$ from $ \dfrac{1+3iqrst}{2} $ to get $ \dfrac{ \color{purple}{ 3iqrst-1 } }{ 2 }$.

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 $\color{blue}{2}$.

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

Divide $ \dfrac{1+3iqrst}{2} $ by $ \dfrac{3iqrst-1}{2} $ to get $ \dfrac{3iqrst+1}{3iqrst-1} $.

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

Step 2: Cancel $ \color{red}{ 2 } $ in first and second fraction.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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