Solve 3/v+2(x+1)-1/x-2

$$\frac { 3 } { v + 2 ( x + 1 ) } - \frac { 1 } { x - 2 }$$

Evaluate

$$\frac{x-v-8}{\left(x-2\right)\left(2x+v+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of v+2\left(x+1\right) and x-2 is \left(x-2\right)\left(2\left(x+1\right)+v\right). Multiply \frac{3}{v+2\left(x+1\right)} times \frac{x-2}{x-2}. Multiply \frac{1}{x-2} times \frac{2\left(x+1\right)+v}{2\left(x+1\right)+v}.

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

Since \frac{3\left(x-2\right)}{\left(x-2\right)\left(2\left(x+1\right)+v\right)} and \frac{2\left(x+1\right)+v}{\left(x-2\right)\left(2\left(x+1\right)+v\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(x-2\right)-\left(2\left(x+1\right)+v\right).

$$\frac{3x-6-2-2x-v}{\left(x-2\right)\left(2\left(x+1\right)+v\right)}$$

Combine like terms in 3x-6-2-2x-v.

$$\frac{x-8-v}{\left(x-2\right)\left(2\left(x+1\right)+v\right)}$$

Expand \left(x-2\right)\left(2\left(x+1\right)+v\right).

$$\frac{x-8-v}{2x^{2}+vx-2x-2v-4}$$

Expand

$$\frac{x-v-8}{\left(x-2\right)\left(2x+v+2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of v+2\left(x+1\right) and x-2 is \left(x-2\right)\left(2\left(x+1\right)+v\right). Multiply \frac{3}{v+2\left(x+1\right)} times \frac{x-2}{x-2}. Multiply \frac{1}{x-2} times \frac{2\left(x+1\right)+v}{2\left(x+1\right)+v}.

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

Since \frac{3\left(x-2\right)}{\left(x-2\right)\left(2\left(x+1\right)+v\right)} and \frac{2\left(x+1\right)+v}{\left(x-2\right)\left(2\left(x+1\right)+v\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(x-2\right)-\left(2\left(x+1\right)+v\right).

$$\frac{3x-6-2-2x-v}{\left(x-2\right)\left(2\left(x+1\right)+v\right)}$$

Combine like terms in 3x-6-2-2x-v.

$$\frac{x-8-v}{\left(x-2\right)\left(2\left(x+1\right)+v\right)}$$

Expand \left(x-2\right)\left(2\left(x+1\right)+v\right).

$$\frac{x-8-v}{2x^{2}+vx-2x-2v-4}$$