Solve x-10/x+15+x-10/x-5/1-frac{5{x-5}}

$$\frac { \frac { x - 10 } { x + 15 } + \frac { x - 10 } { x - 5 } } { 1 - \frac { 5 } { x - 5 } }$$

Evaluate

$$\frac{2\left(x+5\right)}{x+15}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+15 and x-5 is \left(x-5\right)\left(x+15\right). Multiply \frac{x-10}{x+15} times \frac{x-5}{x-5}. Multiply \frac{x-10}{x-5} times \frac{x+15}{x+15}.

$$\frac{\frac{\left(x-10\right)\left(x-5\right)}{\left(x-5\right)\left(x+15\right)}+\frac{\left(x-10\right)\left(x+15\right)}{\left(x-5\right)\left(x+15\right)}}{1-\frac{5}{x-5}}$$

Since \frac{\left(x-10\right)\left(x-5\right)}{\left(x-5\right)\left(x+15\right)} and \frac{\left(x-10\right)\left(x+15\right)}{\left(x-5\right)\left(x+15\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x-10\right)\left(x-5\right)+\left(x-10\right)\left(x+15\right)}{\left(x-5\right)\left(x+15\right)}}{1-\frac{5}{x-5}}$$

Do the multiplications in \left(x-10\right)\left(x-5\right)+\left(x-10\right)\left(x+15\right).

$$\frac{\frac{x^{2}-5x-10x+50+x^{2}+15x-10x-150}{\left(x-5\right)\left(x+15\right)}}{1-\frac{5}{x-5}}$$

Combine like terms in x^{2}-5x-10x+50+x^{2}+15x-10x-150.

$$\frac{\frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)}}{1-\frac{5}{x-5}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-5}{x-5}.

$$\frac{\frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)}}{\frac{x-5}{x-5}-\frac{5}{x-5}}$$

Since \frac{x-5}{x-5} and \frac{5}{x-5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)}}{\frac{x-5-5}{x-5}}$$

Combine like terms in x-5-5.

$$\frac{\frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)}}{\frac{x-10}{x-5}}$$

Divide \frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)} by \frac{x-10}{x-5} by multiplying \frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)} by the reciprocal of \frac{x-10}{x-5}.

$$\frac{\left(2x^{2}-10x-100\right)\left(x-5\right)}{\left(x-5\right)\left(x+15\right)\left(x-10\right)}$$

Cancel out x-5 in both numerator and denominator.

$$\frac{2x^{2}-10x-100}{\left(x-10\right)\left(x+15\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x-10\right)\left(x+5\right)}{\left(x-10\right)\left(x+15\right)}$$

Cancel out x-10 in both numerator and denominator.

$$\frac{2\left(x+5\right)}{x+15}$$

Expand the expression.

$$\frac{2x+10}{x+15}$$

Expand

$$\frac{2\left(x+5\right)}{x+15}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+15 and x-5 is \left(x-5\right)\left(x+15\right). Multiply \frac{x-10}{x+15} times \frac{x-5}{x-5}. Multiply \frac{x-10}{x-5} times \frac{x+15}{x+15}.

$$\frac{\frac{\left(x-10\right)\left(x-5\right)}{\left(x-5\right)\left(x+15\right)}+\frac{\left(x-10\right)\left(x+15\right)}{\left(x-5\right)\left(x+15\right)}}{1-\frac{5}{x-5}}$$

Since \frac{\left(x-10\right)\left(x-5\right)}{\left(x-5\right)\left(x+15\right)} and \frac{\left(x-10\right)\left(x+15\right)}{\left(x-5\right)\left(x+15\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x-10\right)\left(x-5\right)+\left(x-10\right)\left(x+15\right)}{\left(x-5\right)\left(x+15\right)}}{1-\frac{5}{x-5}}$$

Do the multiplications in \left(x-10\right)\left(x-5\right)+\left(x-10\right)\left(x+15\right).

$$\frac{\frac{x^{2}-5x-10x+50+x^{2}+15x-10x-150}{\left(x-5\right)\left(x+15\right)}}{1-\frac{5}{x-5}}$$

Combine like terms in x^{2}-5x-10x+50+x^{2}+15x-10x-150.

$$\frac{\frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)}}{1-\frac{5}{x-5}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-5}{x-5}.

$$\frac{\frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)}}{\frac{x-5}{x-5}-\frac{5}{x-5}}$$

Since \frac{x-5}{x-5} and \frac{5}{x-5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)}}{\frac{x-5-5}{x-5}}$$

Combine like terms in x-5-5.

$$\frac{\frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)}}{\frac{x-10}{x-5}}$$

Divide \frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)} by \frac{x-10}{x-5} by multiplying \frac{2x^{2}-10x-100}{\left(x-5\right)\left(x+15\right)} by the reciprocal of \frac{x-10}{x-5}.

$$\frac{\left(2x^{2}-10x-100\right)\left(x-5\right)}{\left(x-5\right)\left(x+15\right)\left(x-10\right)}$$

Cancel out x-5 in both numerator and denominator.

$$\frac{2x^{2}-10x-100}{\left(x-10\right)\left(x+15\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x-10\right)\left(x+5\right)}{\left(x-10\right)\left(x+15\right)}$$

Cancel out x-10 in both numerator and denominator.

$$\frac{2\left(x+5\right)}{x+15}$$

Expand the expression.

$$\frac{2x+10}{x+15}$$