Solve 1-1/x-2-1/x^2-4/frac{x+1{x^2-4x+4}}

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

Evaluate

$$\frac{4}{x+2}$$

Short Solution Steps

Factor x^{2}-4.

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

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

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

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

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

Combine like terms in x+2-1.

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

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

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

Cancel out x+1 in both numerator and denominator.

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

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

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

Cancel out x-2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+2}{x+2}.

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

Since \frac{x+2}{x+2} and \frac{x-2}{x+2} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

$$\frac{4}{x+2}$$

Expand

$$\frac{4}{x+2}$$

Short Solution Steps

Factor x^{2}-4.

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

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

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

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

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

Combine like terms in x+2-1.

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

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

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

Cancel out x+1 in both numerator and denominator.

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

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

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

Cancel out x-2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x+2}{x+2}.

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

Since \frac{x+2}{x+2} and \frac{x-2}{x+2} have the same denominator, subtract them by subtracting their numerators.

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

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

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

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

$$\frac{4}{x+2}$$