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

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

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

Step 1: Write $ 2x+3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

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

Subtract $ \dfrac{2x+3}{6} $ from $ \dfrac{2x-2}{3} $ to get $ \dfrac{ \color{purple}{ 2x-7 } }{ 6 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 2 }$.

$$ \begin{aligned} \frac{2x-2}{3} - \frac{2x+3}{6} & = \frac{ \left( 2x-2 \right) \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} - \frac{ 2x+3 }{ 6 } = \\[1ex] &=\frac{ \color{purple}{ 4x-4 } }{ 6 } - \frac{ \color{purple}{ 2x+3 } }{ 6 }=\frac{ \color{purple}{ 4x-4 - \left( 2x+3 \right) } }{ 6 } = \\[1ex] &=\frac{ \color{purple}{ 2x-7 } }{ 6 } \end{aligned} $$