Solve 6/v+4+v+5/v^2-16+5/v-4

$$\frac { 6 } { v + 4 } + \frac { v + 5 } { v ^ { 2 } - 16 } + \frac { 5 } { v - 4 }$$

Evaluate

$$\frac{12v+1}{v^{2}-16}$$

Short Solution Steps

Factor v^{2}-16.

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

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

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

Since \frac{6\left(v-4\right)}{\left(v-4\right)\left(v+4\right)} and \frac{v+5}{\left(v-4\right)\left(v+4\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 6\left(v-4\right)+v+5.

$$\frac{6v-24+v+5}{\left(v-4\right)\left(v+4\right)}+\frac{5}{v-4}$$

Combine like terms in 6v-24+v+5.

$$\frac{7v-19}{\left(v-4\right)\left(v+4\right)}+\frac{5}{v-4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-4\right)\left(v+4\right) and v-4 is \left(v-4\right)\left(v+4\right). Multiply \frac{5}{v-4} times \frac{v+4}{v+4}.

$$\frac{7v-19}{\left(v-4\right)\left(v+4\right)}+\frac{5\left(v+4\right)}{\left(v-4\right)\left(v+4\right)}$$

Since \frac{7v-19}{\left(v-4\right)\left(v+4\right)} and \frac{5\left(v+4\right)}{\left(v-4\right)\left(v+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{7v-19+5\left(v+4\right)}{\left(v-4\right)\left(v+4\right)}$$

Do the multiplications in 7v-19+5\left(v+4\right).

$$\frac{7v-19+5v+20}{\left(v-4\right)\left(v+4\right)}$$

Combine like terms in 7v-19+5v+20.

$$\frac{12v+1}{\left(v-4\right)\left(v+4\right)}$$

Expand \left(v-4\right)\left(v+4\right).

$$\frac{12v+1}{v^{2}-16}$$

Expand

$$\frac{12v+1}{v^{2}-16}$$

Short Solution Steps

Factor v^{2}-16.

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

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

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

Since \frac{6\left(v-4\right)}{\left(v-4\right)\left(v+4\right)} and \frac{v+5}{\left(v-4\right)\left(v+4\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 6\left(v-4\right)+v+5.

$$\frac{6v-24+v+5}{\left(v-4\right)\left(v+4\right)}+\frac{5}{v-4}$$

Combine like terms in 6v-24+v+5.

$$\frac{7v-19}{\left(v-4\right)\left(v+4\right)}+\frac{5}{v-4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(v-4\right)\left(v+4\right) and v-4 is \left(v-4\right)\left(v+4\right). Multiply \frac{5}{v-4} times \frac{v+4}{v+4}.

$$\frac{7v-19}{\left(v-4\right)\left(v+4\right)}+\frac{5\left(v+4\right)}{\left(v-4\right)\left(v+4\right)}$$

Since \frac{7v-19}{\left(v-4\right)\left(v+4\right)} and \frac{5\left(v+4\right)}{\left(v-4\right)\left(v+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{7v-19+5\left(v+4\right)}{\left(v-4\right)\left(v+4\right)}$$

Do the multiplications in 7v-19+5\left(v+4\right).

$$\frac{7v-19+5v+20}{\left(v-4\right)\left(v+4\right)}$$

Combine like terms in 7v-19+5v+20.

$$\frac{12v+1}{\left(v-4\right)\left(v+4\right)}$$

Expand \left(v-4\right)\left(v+4\right).

$$\frac{12v+1}{v^{2}-16}$$