Solve 4/x^2-25+2/x+5/frac{1{x+5}+1/x-5}

$$\frac { \frac { 4 } { x ^ { 2 } - 25 } + \frac { 2 } { x + 5 } } { \frac { 1 } { x + 5 } + \frac { 1 } { x - 5 } }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-25.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+5\right) and x+5 is \left(x-5\right)\left(x+5\right). Multiply \frac{2}{x+5} times \frac{x-5}{x-5}.

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

Since \frac{4}{\left(x-5\right)\left(x+5\right)} and \frac{2\left(x-5\right)}{\left(x-5\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4+2\left(x-5\right).

$$\frac{\frac{4+2x-10}{\left(x-5\right)\left(x+5\right)}}{\frac{1}{x+5}+\frac{1}{x-5}}$$

Combine like terms in 4+2x-10.

$$\frac{\frac{-6+2x}{\left(x-5\right)\left(x+5\right)}}{\frac{1}{x+5}+\frac{1}{x-5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+5 and x-5 is \left(x-5\right)\left(x+5\right). Multiply \frac{1}{x+5} times \frac{x-5}{x-5}. Multiply \frac{1}{x-5} times \frac{x+5}{x+5}.

$$\frac{\frac{-6+2x}{\left(x-5\right)\left(x+5\right)}}{\frac{x-5}{\left(x-5\right)\left(x+5\right)}+\frac{x+5}{\left(x-5\right)\left(x+5\right)}}$$

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

$$\frac{\frac{-6+2x}{\left(x-5\right)\left(x+5\right)}}{\frac{x-5+x+5}{\left(x-5\right)\left(x+5\right)}}$$

Combine like terms in x-5+x+5.

$$\frac{\frac{-6+2x}{\left(x-5\right)\left(x+5\right)}}{\frac{2x}{\left(x-5\right)\left(x+5\right)}}$$

Divide \frac{-6+2x}{\left(x-5\right)\left(x+5\right)} by \frac{2x}{\left(x-5\right)\left(x+5\right)} by multiplying \frac{-6+2x}{\left(x-5\right)\left(x+5\right)} by the reciprocal of \frac{2x}{\left(x-5\right)\left(x+5\right)}.

$$\frac{\left(-6+2x\right)\left(x-5\right)\left(x+5\right)}{\left(x-5\right)\left(x+5\right)\times 2x}$$

Cancel out \left(x-5\right)\left(x+5\right) in both numerator and denominator.

$$\frac{2x-6}{2x}$$

Factor the expressions that are not already factored.

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

Cancel out 2 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor x^{2}-25.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-5\right)\left(x+5\right) and x+5 is \left(x-5\right)\left(x+5\right). Multiply \frac{2}{x+5} times \frac{x-5}{x-5}.

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

Since \frac{4}{\left(x-5\right)\left(x+5\right)} and \frac{2\left(x-5\right)}{\left(x-5\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 4+2\left(x-5\right).

$$\frac{\frac{4+2x-10}{\left(x-5\right)\left(x+5\right)}}{\frac{1}{x+5}+\frac{1}{x-5}}$$

Combine like terms in 4+2x-10.

$$\frac{\frac{-6+2x}{\left(x-5\right)\left(x+5\right)}}{\frac{1}{x+5}+\frac{1}{x-5}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+5 and x-5 is \left(x-5\right)\left(x+5\right). Multiply \frac{1}{x+5} times \frac{x-5}{x-5}. Multiply \frac{1}{x-5} times \frac{x+5}{x+5}.

$$\frac{\frac{-6+2x}{\left(x-5\right)\left(x+5\right)}}{\frac{x-5}{\left(x-5\right)\left(x+5\right)}+\frac{x+5}{\left(x-5\right)\left(x+5\right)}}$$

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

$$\frac{\frac{-6+2x}{\left(x-5\right)\left(x+5\right)}}{\frac{x-5+x+5}{\left(x-5\right)\left(x+5\right)}}$$

Combine like terms in x-5+x+5.

$$\frac{\frac{-6+2x}{\left(x-5\right)\left(x+5\right)}}{\frac{2x}{\left(x-5\right)\left(x+5\right)}}$$

Divide \frac{-6+2x}{\left(x-5\right)\left(x+5\right)} by \frac{2x}{\left(x-5\right)\left(x+5\right)} by multiplying \frac{-6+2x}{\left(x-5\right)\left(x+5\right)} by the reciprocal of \frac{2x}{\left(x-5\right)\left(x+5\right)}.

$$\frac{\left(-6+2x\right)\left(x-5\right)\left(x+5\right)}{\left(x-5\right)\left(x+5\right)\times 2x}$$

Cancel out \left(x-5\right)\left(x+5\right) in both numerator and denominator.

$$\frac{2x-6}{2x}$$

Factor the expressions that are not already factored.

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

Cancel out 2 in both numerator and denominator.

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