Solve x-2/x-3+x/x+2+3x-4/x^2-x-6=

$$\frac { x - 2 } { x - 3 } + \frac { x } { x + 2 } + \frac { 3 x - 4 } { x ^ { 2 } - x - 6 } =$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}+2x-2x-4+x^{2}-3x.

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

Factor x^{2}-x-6.

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

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

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

Combine like terms in 2x^{2}-3x-4+3x-4.

$$\frac{2x^{2}-8}{\left(x-3\right)\left(x+2\right)}$$

Factor the expressions that are not already factored in \frac{2x^{2}-8}{\left(x-3\right)\left(x+2\right)}.

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

Cancel out x+2 in both numerator and denominator.

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

Use the distributive property to multiply 2 by x-2.

$$\frac{2x-4}{x-3}$$

Expand

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}+2x-2x-4+x^{2}-3x.

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

Factor x^{2}-x-6.

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

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

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

Combine like terms in 2x^{2}-3x-4+3x-4.

$$\frac{2x^{2}-8}{\left(x-3\right)\left(x+2\right)}$$

Factor the expressions that are not already factored in \frac{2x^{2}-8}{\left(x-3\right)\left(x+2\right)}.

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

Cancel out x+2 in both numerator and denominator.

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

Use the distributive property to multiply 2 by x-2.

$$\frac{2x-4}{x-3}$$