Solve ((3x)/(x ^ 2 - 49) + 7/(x + 7))/(1/(x - 7) - 7/(2x + 14)) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{\frac{3x}{x^{2}-49}+\frac{7}{x+7}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Evaluate

$\frac{2\left(10x-49\right)}{63-5x}$

Short Solution Steps

Factor $x^{2}-49$.

$$\frac{\frac{3x}{\left(x-7\right)\left(x+7\right)}+\frac{7}{x+7}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-7\right)\left(x+7\right)$ and $x+7$ is $\left(x-7\right)\left(x+7\right)$. Multiply $\frac{7}{x+7}$ times $\frac{x-7}{x-7}$.

$$\frac{\frac{3x}{\left(x-7\right)\left(x+7\right)}+\frac{7\left(x-7\right)}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Since $\frac{3x}{\left(x-7\right)\left(x+7\right)}$ and $\frac{7\left(x-7\right)}{\left(x-7\right)\left(x+7\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{3x+7\left(x-7\right)}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Do the multiplications in $3x+7\left(x-7\right)$.

$$\frac{\frac{3x+7x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Combine like terms in $3x+7x-49$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Factor $2x+14$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2\left(x+7\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-7$ and $2\left(x+7\right)$ is $2\left(x-7\right)\left(x+7\right)$. Multiply $\frac{1}{x-7}$ times $\frac{2\left(x+7\right)}{2\left(x+7\right)}$. Multiply $\frac{7}{2\left(x+7\right)}$ times $\frac{x-7}{x-7}$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{2\left(x+7\right)}{2\left(x-7\right)\left(x+7\right)}-\frac{7\left(x-7\right)}{2\left(x-7\right)\left(x+7\right)}}$$

Since $\frac{2\left(x+7\right)}{2\left(x-7\right)\left(x+7\right)}$ and $\frac{7\left(x-7\right)}{2\left(x-7\right)\left(x+7\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{2\left(x+7\right)-7\left(x-7\right)}{2\left(x-7\right)\left(x+7\right)}}$$

Do the multiplications in $2\left(x+7\right)-7\left(x-7\right)$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{2x+14-7x+49}{2\left(x-7\right)\left(x+7\right)}}$$

Combine like terms in $2x+14-7x+49$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{-5x+63}{2\left(x-7\right)\left(x+7\right)}}$$

Divide $\frac{10x-49}{\left(x-7\right)\left(x+7\right)}$ by $\frac{-5x+63}{2\left(x-7\right)\left(x+7\right)}$ by multiplying $\frac{10x-49}{\left(x-7\right)\left(x+7\right)}$ by the reciprocal of $\frac{-5x+63}{2\left(x-7\right)\left(x+7\right)}$.

$$\frac{\left(10x-49\right)\times 2\left(x-7\right)\left(x+7\right)}{\left(x-7\right)\left(x+7\right)\left(-5x+63\right)}$$

Cancel out $\left(x-7\right)\left(x+7\right)$ in both numerator and denominator.

$$\frac{2\left(10x-49\right)}{-5x+63}$$

Use the distributive property to multiply $2$ by $10x-49$.

$$\frac{20x-98}{-5x+63}$$

Expand

$\frac{2\left(10x-49\right)}{63-5x}$

Short Solution Steps

Factor $x^{2}-49$.

$$\frac{\frac{3x}{\left(x-7\right)\left(x+7\right)}+\frac{7}{x+7}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-7\right)\left(x+7\right)$ and $x+7$ is $\left(x-7\right)\left(x+7\right)$. Multiply $\frac{7}{x+7}$ times $\frac{x-7}{x-7}$.

$$\frac{\frac{3x}{\left(x-7\right)\left(x+7\right)}+\frac{7\left(x-7\right)}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Since $\frac{3x}{\left(x-7\right)\left(x+7\right)}$ and $\frac{7\left(x-7\right)}{\left(x-7\right)\left(x+7\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\frac{3x+7\left(x-7\right)}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Do the multiplications in $3x+7\left(x-7\right)$.

$$\frac{\frac{3x+7x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Combine like terms in $3x+7x-49$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2x+14}}$$

Factor $2x+14$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{1}{x-7}-\frac{7}{2\left(x+7\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-7$ and $2\left(x+7\right)$ is $2\left(x-7\right)\left(x+7\right)$. Multiply $\frac{1}{x-7}$ times $\frac{2\left(x+7\right)}{2\left(x+7\right)}$. Multiply $\frac{7}{2\left(x+7\right)}$ times $\frac{x-7}{x-7}$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{2\left(x+7\right)}{2\left(x-7\right)\left(x+7\right)}-\frac{7\left(x-7\right)}{2\left(x-7\right)\left(x+7\right)}}$$

Since $\frac{2\left(x+7\right)}{2\left(x-7\right)\left(x+7\right)}$ and $\frac{7\left(x-7\right)}{2\left(x-7\right)\left(x+7\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{2\left(x+7\right)-7\left(x-7\right)}{2\left(x-7\right)\left(x+7\right)}}$$

Do the multiplications in $2\left(x+7\right)-7\left(x-7\right)$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{2x+14-7x+49}{2\left(x-7\right)\left(x+7\right)}}$$

Combine like terms in $2x+14-7x+49$.

$$\frac{\frac{10x-49}{\left(x-7\right)\left(x+7\right)}}{\frac{-5x+63}{2\left(x-7\right)\left(x+7\right)}}$$

Divide $\frac{10x-49}{\left(x-7\right)\left(x+7\right)}$ by $\frac{-5x+63}{2\left(x-7\right)\left(x+7\right)}$ by multiplying $\frac{10x-49}{\left(x-7\right)\left(x+7\right)}$ by the reciprocal of $\frac{-5x+63}{2\left(x-7\right)\left(x+7\right)}$.

$$\frac{\left(10x-49\right)\times 2\left(x-7\right)\left(x+7\right)}{\left(x-7\right)\left(x+7\right)\left(-5x+63\right)}$$

Cancel out $\left(x-7\right)\left(x+7\right)$ in both numerator and denominator.

$$\frac{2\left(10x-49\right)}{-5x+63}$$

Use the distributive property to multiply $2$ by $10x-49$.

$$\frac{20x-98}{-5x+63}$$