Solve x-1/x-2+x+3/x+1-x+3/x+2-x+1/x-1

$$\frac { x - 1 } { x - 2 } + \frac { x + 3 } { x + 1 } - \frac { x + 3 } { x + 2 } - \frac { x + 1 } { x - 1 }$$

Evaluate

$$\frac{12}{\left(x^{2}-4\right)\left(x^{2}-1\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}+x-x-1+x^{2}-2x+3x-6.

$$\frac{2x^{2}+x-7}{\left(x-2\right)\left(x+1\right)}-\frac{x+3}{x+2}-\frac{x+1}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+1\right) and x+2 is \left(x-2\right)\left(x+1\right)\left(x+2\right). Multiply \frac{2x^{2}+x-7}{\left(x-2\right)\left(x+1\right)} times \frac{x+2}{x+2}. Multiply \frac{x+3}{x+2} times \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}.

$$\frac{\left(2x^{2}+x-7\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{\left(x+3\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x+1}{x-1}$$

Since \frac{\left(2x^{2}+x-7\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(x+3\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x^{2}+x-7\right)\left(x+2\right)-\left(x+3\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x+1}{x-1}$$

Do the multiplications in \left(2x^{2}+x-7\right)\left(x+2\right)-\left(x+3\right)\left(x-2\right)\left(x+1\right).

$$\frac{2x^{3}+4x^{2}+x^{2}+2x-7x-14-x^{3}+x^{2}+2x-3x^{2}+3x+6}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x+1}{x-1}$$

Combine like terms in 2x^{3}+4x^{2}+x^{2}+2x-7x-14-x^{3}+x^{2}+2x-3x^{2}+3x+6.

$$\frac{x^{3}+3x^{2}-8}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x+1}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+1\right)\left(x+2\right) and x-1 is \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right). Multiply \frac{x^{3}+3x^{2}-8}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} times \frac{x-1}{x-1}. Multiply \frac{x+1}{x-1} times \frac{\left(x-2\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(x^{3}+3x^{2}-8\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}-\frac{\left(x+1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Since \frac{\left(x^{3}+3x^{2}-8\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(x+1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{3}+3x^{2}-8\right)\left(x-1\right)-\left(x+1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Do the multiplications in \left(x^{3}+3x^{2}-8\right)\left(x-1\right)-\left(x+1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right).

$$\frac{x^{4}-x^{3}+3x^{3}-3x^{2}-8x+8-x^{4}-x^{3}+4x^{2}+4x-x^{3}-x^{2}+4x+4}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in x^{4}-x^{3}+3x^{3}-3x^{2}-8x+8-x^{4}-x^{3}+4x^{2}+4x-x^{3}-x^{2}+4x+4.

$$\frac{12}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{12}{x^{4}-5x^{2}+4}$$

Expand

$$\frac{12}{\left(x^{2}-4\right)\left(x^{2}-1\right)}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x^{2}+x-x-1+x^{2}-2x+3x-6.

$$\frac{2x^{2}+x-7}{\left(x-2\right)\left(x+1\right)}-\frac{x+3}{x+2}-\frac{x+1}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+1\right) and x+2 is \left(x-2\right)\left(x+1\right)\left(x+2\right). Multiply \frac{2x^{2}+x-7}{\left(x-2\right)\left(x+1\right)} times \frac{x+2}{x+2}. Multiply \frac{x+3}{x+2} times \frac{\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}.

$$\frac{\left(2x^{2}+x-7\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{\left(x+3\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x+1}{x-1}$$

Since \frac{\left(2x^{2}+x-7\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(x+3\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x^{2}+x-7\right)\left(x+2\right)-\left(x+3\right)\left(x-2\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x+1}{x-1}$$

Do the multiplications in \left(2x^{2}+x-7\right)\left(x+2\right)-\left(x+3\right)\left(x-2\right)\left(x+1\right).

$$\frac{2x^{3}+4x^{2}+x^{2}+2x-7x-14-x^{3}+x^{2}+2x-3x^{2}+3x+6}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x+1}{x-1}$$

Combine like terms in 2x^{3}+4x^{2}+x^{2}+2x-7x-14-x^{3}+x^{2}+2x-3x^{2}+3x+6.

$$\frac{x^{3}+3x^{2}-8}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x+1}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+1\right)\left(x+2\right) and x-1 is \left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right). Multiply \frac{x^{3}+3x^{2}-8}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} times \frac{x-1}{x-1}. Multiply \frac{x+1}{x-1} times \frac{\left(x-2\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(x^{3}+3x^{2}-8\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}-\frac{\left(x+1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Since \frac{\left(x^{3}+3x^{2}-8\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)} and \frac{\left(x+1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x^{3}+3x^{2}-8\right)\left(x-1\right)-\left(x+1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Do the multiplications in \left(x^{3}+3x^{2}-8\right)\left(x-1\right)-\left(x+1\right)\left(x-2\right)\left(x+1\right)\left(x+2\right).

$$\frac{x^{4}-x^{3}+3x^{3}-3x^{2}-8x+8-x^{4}-x^{3}+4x^{2}+4x-x^{3}-x^{2}+4x+4}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in x^{4}-x^{3}+3x^{3}-3x^{2}-8x+8-x^{4}-x^{3}+4x^{2}+4x-x^{3}-x^{2}+4x+4.

$$\frac{12}{\left(x-2\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{12}{x^{4}-5x^{2}+4}$$