Solve 2/v+9+2/v-5/frac{6{v-5}+1/v+9}

$$\frac { \frac { 2 } { v + 9 } + \frac { 2 } { v - 5 } } { \frac { 6 } { v - 5 } + \frac { 1 } { v + 9 } }$$

Evaluate

$$\frac{4\left(v+2\right)}{7\left(v+7\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of v+9 and v-5 is \left(v-5\right)\left(v+9\right). Multiply \frac{2}{v+9} times \frac{v-5}{v-5}. Multiply \frac{2}{v-5} times \frac{v+9}{v+9}.

$$\frac{\frac{2\left(v-5\right)}{\left(v-5\right)\left(v+9\right)}+\frac{2\left(v+9\right)}{\left(v-5\right)\left(v+9\right)}}{\frac{6}{v-5}+\frac{1}{v+9}}$$

Since \frac{2\left(v-5\right)}{\left(v-5\right)\left(v+9\right)} and \frac{2\left(v+9\right)}{\left(v-5\right)\left(v+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\left(v-5\right)+2\left(v+9\right)}{\left(v-5\right)\left(v+9\right)}}{\frac{6}{v-5}+\frac{1}{v+9}}$$

Do the multiplications in 2\left(v-5\right)+2\left(v+9\right).

$$\frac{\frac{2v-10+2v+18}{\left(v-5\right)\left(v+9\right)}}{\frac{6}{v-5}+\frac{1}{v+9}}$$

Combine like terms in 2v-10+2v+18.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{6}{v-5}+\frac{1}{v+9}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of v-5 and v+9 is \left(v-5\right)\left(v+9\right). Multiply \frac{6}{v-5} times \frac{v+9}{v+9}. Multiply \frac{1}{v+9} times \frac{v-5}{v-5}.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{6\left(v+9\right)}{\left(v-5\right)\left(v+9\right)}+\frac{v-5}{\left(v-5\right)\left(v+9\right)}}$$

Since \frac{6\left(v+9\right)}{\left(v-5\right)\left(v+9\right)} and \frac{v-5}{\left(v-5\right)\left(v+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{6\left(v+9\right)+v-5}{\left(v-5\right)\left(v+9\right)}}$$

Do the multiplications in 6\left(v+9\right)+v-5.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{6v+54+v-5}{\left(v-5\right)\left(v+9\right)}}$$

Combine like terms in 6v+54+v-5.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{7v+49}{\left(v-5\right)\left(v+9\right)}}$$

Divide \frac{4v+8}{\left(v-5\right)\left(v+9\right)} by \frac{7v+49}{\left(v-5\right)\left(v+9\right)} by multiplying \frac{4v+8}{\left(v-5\right)\left(v+9\right)} by the reciprocal of \frac{7v+49}{\left(v-5\right)\left(v+9\right)}.

$$\frac{\left(4v+8\right)\left(v-5\right)\left(v+9\right)}{\left(v-5\right)\left(v+9\right)\left(7v+49\right)}$$

Cancel out \left(v-5\right)\left(v+9\right) in both numerator and denominator.

$$\frac{4v+8}{7v+49}$$

Expand

$$\frac{4\left(v+2\right)}{7\left(v+7\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of v+9 and v-5 is \left(v-5\right)\left(v+9\right). Multiply \frac{2}{v+9} times \frac{v-5}{v-5}. Multiply \frac{2}{v-5} times \frac{v+9}{v+9}.

$$\frac{\frac{2\left(v-5\right)}{\left(v-5\right)\left(v+9\right)}+\frac{2\left(v+9\right)}{\left(v-5\right)\left(v+9\right)}}{\frac{6}{v-5}+\frac{1}{v+9}}$$

Since \frac{2\left(v-5\right)}{\left(v-5\right)\left(v+9\right)} and \frac{2\left(v+9\right)}{\left(v-5\right)\left(v+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2\left(v-5\right)+2\left(v+9\right)}{\left(v-5\right)\left(v+9\right)}}{\frac{6}{v-5}+\frac{1}{v+9}}$$

Do the multiplications in 2\left(v-5\right)+2\left(v+9\right).

$$\frac{\frac{2v-10+2v+18}{\left(v-5\right)\left(v+9\right)}}{\frac{6}{v-5}+\frac{1}{v+9}}$$

Combine like terms in 2v-10+2v+18.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{6}{v-5}+\frac{1}{v+9}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of v-5 and v+9 is \left(v-5\right)\left(v+9\right). Multiply \frac{6}{v-5} times \frac{v+9}{v+9}. Multiply \frac{1}{v+9} times \frac{v-5}{v-5}.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{6\left(v+9\right)}{\left(v-5\right)\left(v+9\right)}+\frac{v-5}{\left(v-5\right)\left(v+9\right)}}$$

Since \frac{6\left(v+9\right)}{\left(v-5\right)\left(v+9\right)} and \frac{v-5}{\left(v-5\right)\left(v+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{6\left(v+9\right)+v-5}{\left(v-5\right)\left(v+9\right)}}$$

Do the multiplications in 6\left(v+9\right)+v-5.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{6v+54+v-5}{\left(v-5\right)\left(v+9\right)}}$$

Combine like terms in 6v+54+v-5.

$$\frac{\frac{4v+8}{\left(v-5\right)\left(v+9\right)}}{\frac{7v+49}{\left(v-5\right)\left(v+9\right)}}$$

Divide \frac{4v+8}{\left(v-5\right)\left(v+9\right)} by \frac{7v+49}{\left(v-5\right)\left(v+9\right)} by multiplying \frac{4v+8}{\left(v-5\right)\left(v+9\right)} by the reciprocal of \frac{7v+49}{\left(v-5\right)\left(v+9\right)}.

$$\frac{\left(4v+8\right)\left(v-5\right)\left(v+9\right)}{\left(v-5\right)\left(v+9\right)\left(7v+49\right)}$$

Cancel out \left(v-5\right)\left(v+9\right) in both numerator and denominator.

$$\frac{4v+8}{7v+49}$$