Solve 5/v+6+v+7/v^2-36+4/v-6

$$\frac { 5 } { v + 6 } + \frac { v + 7 } { v ^ { 2 } - 36 } + \frac { 4 } { v - 6 }$$

Evaluate

$$\frac{10v+1}{v^{2}-36}$$

Short Solution Steps

Factor v^{2}-36.

$$\frac{5}{v+6}+\frac{v+7}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

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

$$\frac{5\left(v-6\right)}{\left(v-6\right)\left(v+6\right)}+\frac{v+7}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

Since \frac{5\left(v-6\right)}{\left(v-6\right)\left(v+6\right)} and \frac{v+7}{\left(v-6\right)\left(v+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(v-6\right)+v+7}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

Do the multiplications in 5\left(v-6\right)+v+7.

$$\frac{5v-30+v+7}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

Combine like terms in 5v-30+v+7.

$$\frac{6v-23}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-6\right)\left(v+6\right) and v-6 is \left(v-6\right)\left(v+6\right). Multiply \frac{4}{v-6} times \frac{v+6}{v+6}.

$$\frac{6v-23}{\left(v-6\right)\left(v+6\right)}+\frac{4\left(v+6\right)}{\left(v-6\right)\left(v+6\right)}$$

Since \frac{6v-23}{\left(v-6\right)\left(v+6\right)} and \frac{4\left(v+6\right)}{\left(v-6\right)\left(v+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{6v-23+4\left(v+6\right)}{\left(v-6\right)\left(v+6\right)}$$

Do the multiplications in 6v-23+4\left(v+6\right).

$$\frac{6v-23+4v+24}{\left(v-6\right)\left(v+6\right)}$$

Combine like terms in 6v-23+4v+24.

$$\frac{10v+1}{\left(v-6\right)\left(v+6\right)}$$

Expand \left(v-6\right)\left(v+6\right).

$$\frac{10v+1}{v^{2}-36}$$

Expand

$$\frac{10v+1}{v^{2}-36}$$

Short Solution Steps

Factor v^{2}-36.

$$\frac{5}{v+6}+\frac{v+7}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

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

$$\frac{5\left(v-6\right)}{\left(v-6\right)\left(v+6\right)}+\frac{v+7}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

Since \frac{5\left(v-6\right)}{\left(v-6\right)\left(v+6\right)} and \frac{v+7}{\left(v-6\right)\left(v+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{5\left(v-6\right)+v+7}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

Do the multiplications in 5\left(v-6\right)+v+7.

$$\frac{5v-30+v+7}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

Combine like terms in 5v-30+v+7.

$$\frac{6v-23}{\left(v-6\right)\left(v+6\right)}+\frac{4}{v-6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-6\right)\left(v+6\right) and v-6 is \left(v-6\right)\left(v+6\right). Multiply \frac{4}{v-6} times \frac{v+6}{v+6}.

$$\frac{6v-23}{\left(v-6\right)\left(v+6\right)}+\frac{4\left(v+6\right)}{\left(v-6\right)\left(v+6\right)}$$

Since \frac{6v-23}{\left(v-6\right)\left(v+6\right)} and \frac{4\left(v+6\right)}{\left(v-6\right)\left(v+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{6v-23+4\left(v+6\right)}{\left(v-6\right)\left(v+6\right)}$$

Do the multiplications in 6v-23+4\left(v+6\right).

$$\frac{6v-23+4v+24}{\left(v-6\right)\left(v+6\right)}$$

Combine like terms in 6v-23+4v+24.

$$\frac{10v+1}{\left(v-6\right)\left(v+6\right)}$$

Expand \left(v-6\right)\left(v+6\right).

$$\frac{10v+1}{v^{2}-36}$$