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Question

Answer

$$2x-\dfrac{7}{3}+3x+\dfrac{5}{2}=\dfrac{30x+1}{6}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}2x-\frac{7}{3}+3x+\frac{5}{2}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{6x-7}{3}+3x+\frac{5}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{15x-7}{3}+\frac{5}{2} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30x+1}{6}\end{aligned} $$
①

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

Step 1: Write $ 2x $ 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 $\color{blue}{ 3 }$.

$$ \begin{aligned} 2x- \frac{7}{3} & \xlongequal{\text{Step 1}} \frac{2x}{\color{red}{1}} - \frac{7}{3} = \frac{ 2x \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} - \frac{ 7 }{ 3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6x } }{ 3 } - \frac{ \color{purple}{ 7 } }{ 3 }=\frac{ \color{purple}{ 6x-7 } }{ 3 } \end{aligned} $$
②

Add $ \dfrac{6x-7}{3} $ and $ 3x $ to get $ \dfrac{ \color{purple}{ 15x-7 } }{ 3 }$.

Step 1: Write $ 3x $ 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 second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{6x-7}{3} +3x & \xlongequal{\text{Step 1}} \frac{6x-7}{3} + \frac{3x}{\color{red}{1}} = \frac{ 6x-7 }{ 3 } + \frac{ 3x \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 6x-7 } }{ 3 } + \frac{ \color{purple}{ 9x } }{ 3 }=\frac{ \color{purple}{ 15x-7 } }{ 3 } \end{aligned} $$
③

Add $ \dfrac{15x-7}{3} $ and $ \dfrac{5}{2} $ to get $ \dfrac{ \color{purple}{ 30x+1 } }{ 6 }$.

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}{ 2 }$ and the second by $\color{blue}{ 3 }$.

$$ \begin{aligned} \frac{15x-7}{3} + \frac{5}{2} & = \frac{ \left( 15x-7 \right) \cdot \color{blue}{ 2 }}{ 3 \cdot \color{blue}{ 2 }} + \frac{ 5 \cdot \color{blue}{ 3 }}{ 2 \cdot \color{blue}{ 3 }} = \\[1ex] &=\frac{ \color{purple}{ 30x-14 } }{ 6 } + \frac{ \color{purple}{ 15 } }{ 6 }=\frac{ \color{purple}{ 30x+1 } }{ 6 } \end{aligned} $$
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