Solve 6x/(x+2)(x-2)-2/x+2+3/2-x

$$\frac { 6 x } { ( x + 2 ) ( x - 2 ) } - \frac { 2 } { x + 2 } + \frac { 3 } { 2 - x }$$

Evaluate

$$\frac{1}{x+2}$$

Short Solution Steps

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

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

Since \frac{6x}{\left(x-2\right)\left(x+2\right)} and \frac{2\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 6x-2\left(x-2\right).

$$\frac{6x-2x+4}{\left(x-2\right)\left(x+2\right)}+\frac{3}{2-x}$$

Combine like terms in 6x-2x+4.

$$\frac{4x+4}{\left(x-2\right)\left(x+2\right)}+\frac{3}{2-x}$$

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

$$\frac{4x+4}{\left(x-2\right)\left(x+2\right)}+\frac{3\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}$$

Since \frac{4x+4}{\left(x-2\right)\left(x+2\right)} and \frac{3\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{4x+4+3\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}$$

Do the multiplications in 4x+4+3\left(-1\right)\left(x+2\right).

$$\frac{4x+4-3x-6}{\left(x-2\right)\left(x+2\right)}$$

Combine like terms in 4x+4-3x-6.

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

Cancel out x-2 in both numerator and denominator.

$$\frac{1}{x+2}$$

Expand

$$\frac{1}{x+2}$$

Short Solution Steps

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

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

Since \frac{6x}{\left(x-2\right)\left(x+2\right)} and \frac{2\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 6x-2\left(x-2\right).

$$\frac{6x-2x+4}{\left(x-2\right)\left(x+2\right)}+\frac{3}{2-x}$$

Combine like terms in 6x-2x+4.

$$\frac{4x+4}{\left(x-2\right)\left(x+2\right)}+\frac{3}{2-x}$$

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

$$\frac{4x+4}{\left(x-2\right)\left(x+2\right)}+\frac{3\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}$$

Since \frac{4x+4}{\left(x-2\right)\left(x+2\right)} and \frac{3\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{4x+4+3\left(-1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}$$

Do the multiplications in 4x+4+3\left(-1\right)\left(x+2\right).

$$\frac{4x+4-3x-6}{\left(x-2\right)\left(x+2\right)}$$

Combine like terms in 4x+4-3x-6.

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

Cancel out x-2 in both numerator and denominator.

$$\frac{1}{x+2}$$