$$-6\cdot\dfrac{\dfrac{x}{x-2}}{-3\dfrac{x}{x-2}+3}=\dfrac{6x}{-6}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-6 \cdot \frac{\frac{x}{x-2}}{-3\frac{x}{x-2}+3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}6 \cdot \frac{\frac{x}{x-2}}{(-\frac{x}{x-2}+1)\cdot3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}6 \cdot \frac{\frac{x}{x-2}}{(-\frac{2}{x-2})\cdot3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}6 \cdot \frac{\frac{x}{x-2}}{-\frac{6}{x-2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}6 \cdot \frac{x}{-6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{6x}{-6}\end{aligned} $$ | |
| ① | Use the distributive property. |
| ② | Add $ \dfrac{-x}{x-2} $ and $ 1 $ to get $ \dfrac{ \color{purple}{ -2 } }{ x-2 }$. Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |
| ③ | Multiply $ \dfrac{-2}{x-2} $ by $ 3 $ to get $ \dfrac{ -6 }{ x-2 } $. Step 1: Write $ 3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{-2}{x-2} \cdot 3 & \xlongequal{\text{Step 1}} \frac{-2}{x-2} \cdot \frac{3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( -2 \right) \cdot 3 }{ \left( x-2 \right) \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -6 }{ x-2 } \end{aligned} $$ |
| ④ | Divide $ \dfrac{x}{x-2} $ by $ \dfrac{-6}{x-2} $ to get $ \dfrac{ x }{ -6 } $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Cancel $ \color{red}{ x-2 } $ in first and second fraction. Step 3: Multiply numerators and denominators. Step 4: Simplify numerator and denominator. $$ \begin{aligned} \frac{ \frac{x}{x-2} }{ \frac{\color{blue}{-6}}{\color{blue}{x-2}} } & \xlongequal{\text{Step 1}} \frac{x}{x-2} \cdot \frac{\color{blue}{x-2}}{\color{blue}{-6}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{x}{\color{red}{1}} \cdot \frac{\color{red}{1}}{-6} \xlongequal{\text{Step 3}} \frac{ x \cdot 1 }{ 1 \cdot \left( -6 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ x }{ -6 } \end{aligned} $$ |
| ⑤ | Multiply $6$ by $ \dfrac{x}{-6} $ to get $ \dfrac{ 6x }{ -6 } $. Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 6 \cdot \frac{x}{-6} & \xlongequal{\text{Step 1}} \frac{6}{\color{red}{1}} \cdot \frac{x}{-6} \xlongequal{\text{Step 2}} \frac{ 6 \cdot x }{ 1 \cdot \left( -6 \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 6x }{ -6 } \end{aligned} $$ |