Solve 2x/x+3+x/x-3-3x^2+3/x^2-9):(2x-2/x-3-1)

$$\frac { 2 x } { x + 3 } + \frac { x } { x - 3 } - \frac { 3 x ^ { 2 } + 3 } { x ^ { 2 } - 9 } ) : ( \frac { 2 x - 2 } { x - 3 } - 1 )$$

Evaluate

$$-\frac{3\left(x+1\right)}{x^{2}-9}$$

Short Solution Steps

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

$$\frac{2x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}+\frac{x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\frac{3x^{2}+3}{x^{2}-9}$$

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

$$\frac{2x\left(x-3\right)+x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\frac{3x^{2}+3}{x^{2}-9}$$

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

$$\frac{2x^{2}-6x+x^{2}+3x}{\left(x-3\right)\left(x+3\right)}-\frac{3x^{2}+3}{x^{2}-9}$$

Combine like terms in 2x^{2}-6x+x^{2}+3x.

$$\frac{3x^{2}-3x}{\left(x-3\right)\left(x+3\right)}-\frac{3x^{2}+3}{x^{2}-9}$$

Factor x^{2}-9.

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

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

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

Do the multiplications in 3x^{2}-3x-\left(3x^{2}+3\right).

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

Combine like terms in 3x^{2}-3x-3x^{2}-3.

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

Expand \left(x-3\right)\left(x+3\right).

$$\frac{-3x-3}{x^{2}-9}$$

Expand

$$-\frac{3\left(x+1\right)}{x^{2}-9}$$

Short Solution Steps

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

$$\frac{2x\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}+\frac{x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\frac{3x^{2}+3}{x^{2}-9}$$

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

$$\frac{2x\left(x-3\right)+x\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\frac{3x^{2}+3}{x^{2}-9}$$

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

$$\frac{2x^{2}-6x+x^{2}+3x}{\left(x-3\right)\left(x+3\right)}-\frac{3x^{2}+3}{x^{2}-9}$$

Combine like terms in 2x^{2}-6x+x^{2}+3x.

$$\frac{3x^{2}-3x}{\left(x-3\right)\left(x+3\right)}-\frac{3x^{2}+3}{x^{2}-9}$$

Factor x^{2}-9.

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

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

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

Do the multiplications in 3x^{2}-3x-\left(3x^{2}+3\right).

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

Combine like terms in 3x^{2}-3x-3x^{2}-3.

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

Expand \left(x-3\right)\left(x+3\right).

$$\frac{-3x-3}{x^{2}-9}$$