Solve (1+1/x-1+1/x+2):x^2+3x-1/x^2-4+9-x^2/x^2-4x+3

$$( 1 + \frac { 1 } { x - 1 } + \frac { 1 } { x + 2 } ) : \frac { x ^ { 2 } + 3 x - 1 } { x ^ { 2 } - 4 } + \frac { 9 - x ^ { 2 } } { x ^ { 2 } - 4 x + 3 }$$

Evaluate

$$-\frac{5}{x-1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-1}{x-1}.

$$\frac{\frac{x-1}{x-1}+\frac{1}{x-1}+\frac{1}{x+2}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Since \frac{x-1}{x-1} and \frac{1}{x-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x-1+1}{x-1}+\frac{1}{x+2}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Combine like terms in x-1+1.

$$\frac{\frac{x}{x-1}+\frac{1}{x+2}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

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

$$\frac{\frac{x\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}+\frac{x-1}{\left(x-1\right)\left(x+2\right)}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

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

$$\frac{\frac{x\left(x+2\right)+x-1}{\left(x-1\right)\left(x+2\right)}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

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

$$\frac{\frac{x^{2}+2x+x-1}{\left(x-1\right)\left(x+2\right)}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

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

$$\frac{\frac{x^{2}+3x-1}{\left(x-1\right)\left(x+2\right)}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Divide \frac{x^{2}+3x-1}{\left(x-1\right)\left(x+2\right)} by \frac{x^{2}+3x-1}{x^{2}-4} by multiplying \frac{x^{2}+3x-1}{\left(x-1\right)\left(x+2\right)} by the reciprocal of \frac{x^{2}+3x-1}{x^{2}-4}.

$$\frac{\left(x^{2}+3x-1\right)\left(x^{2}-4\right)}{\left(x-1\right)\left(x+2\right)\left(x^{2}+3x-1\right)}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Cancel out x^{2}+3x-1 in both numerator and denominator.

$$\frac{x^{2}-4}{\left(x-1\right)\left(x+2\right)}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Factor the expressions that are not already factored in \frac{x^{2}-4}{\left(x-1\right)\left(x+2\right)}.

$$\frac{\left(x-2\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{x-2}{x-1}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Factor the expressions that are not already factored in \frac{9-x^{2}}{x^{2}-4x+3}.

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

Cancel out x-3 in both numerator and denominator.

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

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

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

Combine like terms in x-2-x-3.

$$\frac{-5}{x-1}$$

Expand

$$-\frac{5}{x-1}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-1}{x-1}.

$$\frac{\frac{x-1}{x-1}+\frac{1}{x-1}+\frac{1}{x+2}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Since \frac{x-1}{x-1} and \frac{1}{x-1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x-1+1}{x-1}+\frac{1}{x+2}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Combine like terms in x-1+1.

$$\frac{\frac{x}{x-1}+\frac{1}{x+2}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

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

$$\frac{\frac{x\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}+\frac{x-1}{\left(x-1\right)\left(x+2\right)}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

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

$$\frac{\frac{x\left(x+2\right)+x-1}{\left(x-1\right)\left(x+2\right)}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

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

$$\frac{\frac{x^{2}+2x+x-1}{\left(x-1\right)\left(x+2\right)}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

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

$$\frac{\frac{x^{2}+3x-1}{\left(x-1\right)\left(x+2\right)}}{\frac{x^{2}+3x-1}{x^{2}-4}}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Divide \frac{x^{2}+3x-1}{\left(x-1\right)\left(x+2\right)} by \frac{x^{2}+3x-1}{x^{2}-4} by multiplying \frac{x^{2}+3x-1}{\left(x-1\right)\left(x+2\right)} by the reciprocal of \frac{x^{2}+3x-1}{x^{2}-4}.

$$\frac{\left(x^{2}+3x-1\right)\left(x^{2}-4\right)}{\left(x-1\right)\left(x+2\right)\left(x^{2}+3x-1\right)}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Cancel out x^{2}+3x-1 in both numerator and denominator.

$$\frac{x^{2}-4}{\left(x-1\right)\left(x+2\right)}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Factor the expressions that are not already factored in \frac{x^{2}-4}{\left(x-1\right)\left(x+2\right)}.

$$\frac{\left(x-2\right)\left(x+2\right)}{\left(x-1\right)\left(x+2\right)}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{x-2}{x-1}+\frac{9-x^{2}}{x^{2}-4x+3}$$

Factor the expressions that are not already factored in \frac{9-x^{2}}{x^{2}-4x+3}.

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

Cancel out x-3 in both numerator and denominator.

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

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

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

Combine like terms in x-2-x-3.

$$\frac{-5}{x-1}$$