Solve x-2/x+1-x+3/x-2/frac{x-1{x+2}+x-5/x+1}

$$\frac { \frac { x - 2 } { x + 1 } - \frac { x + 3 } { x - 2 } } { \frac { x - 1 } { x + 2 } + \frac { x - 5 } { x + 1 } }$$

Evaluate

$$\frac{\left(1-8x\right)\left(x+2\right)}{\left(x-2\right)\left(2x^{2}-3x-11\right)}$$

Short Solution Steps

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

$$\frac{\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}-\frac{\left(x+3\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\frac{x-1}{x+2}+\frac{x-5}{x+1}}$$

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

$$\frac{\frac{\left(x-2\right)\left(x-2\right)-\left(x+3\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\frac{x-1}{x+2}+\frac{x-5}{x+1}}$$

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

$$\frac{\frac{x^{2}-2x-2x+4-x^{2}-x-3x-3}{\left(x-2\right)\left(x+1\right)}}{\frac{x-1}{x+2}+\frac{x-5}{x+1}}$$

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

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{x-1}{x+2}+\frac{x-5}{x+1}}$$

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+1\right)\left(x+2\right). Multiply \frac{x-1}{x+2} times \frac{x+1}{x+1}. Multiply \frac{x-5}{x+1} times \frac{x+2}{x+2}.

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{\left(x-1\right)\left(x+1\right)}{\left(x+1\right)\left(x+2\right)}+\frac{\left(x-5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}}$$

Since \frac{\left(x-1\right)\left(x+1\right)}{\left(x+1\right)\left(x+2\right)} and \frac{\left(x-5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{\left(x-1\right)\left(x+1\right)+\left(x-5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}}$$

Do the multiplications in \left(x-1\right)\left(x+1\right)+\left(x-5\right)\left(x+2\right).

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+x-x-1+x^{2}+2x-5x-10}{\left(x+1\right)\left(x+2\right)}}$$

Combine like terms in x^{2}+x-x-1+x^{2}+2x-5x-10.

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{2x^{2}-3x-11}{\left(x+1\right)\left(x+2\right)}}$$

Divide \frac{-8x+1}{\left(x-2\right)\left(x+1\right)} by \frac{2x^{2}-3x-11}{\left(x+1\right)\left(x+2\right)} by multiplying \frac{-8x+1}{\left(x-2\right)\left(x+1\right)} by the reciprocal of \frac{2x^{2}-3x-11}{\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(-8x+1\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(2x^{2}-3x-11\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{\left(x+2\right)\left(-8x+1\right)}{\left(x-2\right)\left(2x^{2}-3x-11\right)}$$

Apply the distributive property by multiplying each term of x+2 by each term of -8x+1.

$$\frac{-8x^{2}+x-16x+2}{\left(x-2\right)\left(2x^{2}-3x-11\right)}$$

Combine x and -16x to get -15x.

$$\frac{-8x^{2}-15x+2}{\left(x-2\right)\left(2x^{2}-3x-11\right)}$$

Apply the distributive property by multiplying each term of x-2 by each term of 2x^{2}-3x-11.

$$\frac{-8x^{2}-15x+2}{2x^{3}-3x^{2}-11x-4x^{2}+6x+22}$$

Combine -3x^{2} and -4x^{2} to get -7x^{2}.

$$\frac{-8x^{2}-15x+2}{2x^{3}-7x^{2}-11x+6x+22}$$

Combine -11x and 6x to get -5x.

$$\frac{-8x^{2}-15x+2}{2x^{3}-7x^{2}-5x+22}$$

Expand

$$\frac{2-15x-8x^{2}}{\left(x-2\right)\left(2x^{2}-3x-11\right)}$$

Short Solution Steps

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

$$\frac{\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}-\frac{\left(x+3\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\frac{x-1}{x+2}+\frac{x-5}{x+1}}$$

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

$$\frac{\frac{\left(x-2\right)\left(x-2\right)-\left(x+3\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\frac{x-1}{x+2}+\frac{x-5}{x+1}}$$

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

$$\frac{\frac{x^{2}-2x-2x+4-x^{2}-x-3x-3}{\left(x-2\right)\left(x+1\right)}}{\frac{x-1}{x+2}+\frac{x-5}{x+1}}$$

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

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{x-1}{x+2}+\frac{x-5}{x+1}}$$

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+1\right)\left(x+2\right). Multiply \frac{x-1}{x+2} times \frac{x+1}{x+1}. Multiply \frac{x-5}{x+1} times \frac{x+2}{x+2}.

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{\left(x-1\right)\left(x+1\right)}{\left(x+1\right)\left(x+2\right)}+\frac{\left(x-5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}}$$

Since \frac{\left(x-1\right)\left(x+1\right)}{\left(x+1\right)\left(x+2\right)} and \frac{\left(x-5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{\left(x-1\right)\left(x+1\right)+\left(x-5\right)\left(x+2\right)}{\left(x+1\right)\left(x+2\right)}}$$

Do the multiplications in \left(x-1\right)\left(x+1\right)+\left(x-5\right)\left(x+2\right).

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{x^{2}+x-x-1+x^{2}+2x-5x-10}{\left(x+1\right)\left(x+2\right)}}$$

Combine like terms in x^{2}+x-x-1+x^{2}+2x-5x-10.

$$\frac{\frac{-8x+1}{\left(x-2\right)\left(x+1\right)}}{\frac{2x^{2}-3x-11}{\left(x+1\right)\left(x+2\right)}}$$

Divide \frac{-8x+1}{\left(x-2\right)\left(x+1\right)} by \frac{2x^{2}-3x-11}{\left(x+1\right)\left(x+2\right)} by multiplying \frac{-8x+1}{\left(x-2\right)\left(x+1\right)} by the reciprocal of \frac{2x^{2}-3x-11}{\left(x+1\right)\left(x+2\right)}.

$$\frac{\left(-8x+1\right)\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+1\right)\left(2x^{2}-3x-11\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{\left(x+2\right)\left(-8x+1\right)}{\left(x-2\right)\left(2x^{2}-3x-11\right)}$$

Apply the distributive property by multiplying each term of x+2 by each term of -8x+1.

$$\frac{-8x^{2}+x-16x+2}{\left(x-2\right)\left(2x^{2}-3x-11\right)}$$

Combine x and -16x to get -15x.

$$\frac{-8x^{2}-15x+2}{\left(x-2\right)\left(2x^{2}-3x-11\right)}$$

Apply the distributive property by multiplying each term of x-2 by each term of 2x^{2}-3x-11.

$$\frac{-8x^{2}-15x+2}{2x^{3}-3x^{2}-11x-4x^{2}+6x+22}$$

Combine -3x^{2} and -4x^{2} to get -7x^{2}.

$$\frac{-8x^{2}-15x+2}{2x^{3}-7x^{2}-11x+6x+22}$$

Combine -11x and 6x to get -5x.

$$\frac{-8x^{2}-15x+2}{2x^{3}-7x^{2}-5x+22}$$