Solve ((x+6)-5/(x-5)-(x+6)/x)/(1+frac{(x+6)-5{x-5}*x+6/x)}

$$\frac { ( \frac { ( x + 6 ) - 5 } { ( x - 5 ) } - \frac { ( x + 6 ) } { x } ) } { ( 1 + \frac { ( x + 6 ) - 5 } { x - 5 } \cdot \frac { x + 6 } { x } ) }$$

Evaluate

$$\frac{15}{x^{2}+x+3}$$

Short Solution Steps

Subtract 5 from 6 to get 1.

$$\frac{\frac{x+1}{x-5}-\frac{x+6}{x}}{1+\frac{x+6-5}{x-5}\times \frac{x+6}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-5 and x is x\left(x-5\right). Multiply \frac{x+1}{x-5} times \frac{x}{x}. Multiply \frac{x+6}{x} times \frac{x-5}{x-5}.

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

Since \frac{\left(x+1\right)x}{x\left(x-5\right)} and \frac{\left(x+6\right)\left(x-5\right)}{x\left(x-5\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+1\right)x-\left(x+6\right)\left(x-5\right).

$$\frac{\frac{x^{2}+x-x^{2}+5x-6x+30}{x\left(x-5\right)}}{1+\frac{x+6-5}{x-5}\times \frac{x+6}{x}}$$

Combine like terms in x^{2}+x-x^{2}+5x-6x+30.

$$\frac{\frac{30}{x\left(x-5\right)}}{1+\frac{x+6-5}{x-5}\times \frac{x+6}{x}}$$

Subtract 5 from 6 to get 1.

$$\frac{\frac{30}{x\left(x-5\right)}}{1+\frac{x+1}{x-5}\times \frac{x+6}{x}}$$

Multiply \frac{x+1}{x-5} times \frac{x+6}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{30}{x\left(x-5\right)}}{1+\frac{\left(x+1\right)\left(x+6\right)}{\left(x-5\right)x}}$$

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

$$\frac{\frac{30}{x\left(x-5\right)}}{\frac{\left(x-5\right)x}{\left(x-5\right)x}+\frac{\left(x+1\right)\left(x+6\right)}{\left(x-5\right)x}}$$

Since \frac{\left(x-5\right)x}{\left(x-5\right)x} and \frac{\left(x+1\right)\left(x+6\right)}{\left(x-5\right)x} have the same denominator, add them by adding their numerators.

$$\frac{\frac{30}{x\left(x-5\right)}}{\frac{\left(x-5\right)x+\left(x+1\right)\left(x+6\right)}{\left(x-5\right)x}}$$

Do the multiplications in \left(x-5\right)x+\left(x+1\right)\left(x+6\right).

$$\frac{\frac{30}{x\left(x-5\right)}}{\frac{x^{2}-5x+x^{2}+6x+x+6}{\left(x-5\right)x}}$$

Combine like terms in x^{2}-5x+x^{2}+6x+x+6.

$$\frac{\frac{30}{x\left(x-5\right)}}{\frac{2x^{2}+2x+6}{\left(x-5\right)x}}$$

Divide \frac{30}{x\left(x-5\right)} by \frac{2x^{2}+2x+6}{\left(x-5\right)x} by multiplying \frac{30}{x\left(x-5\right)} by the reciprocal of \frac{2x^{2}+2x+6}{\left(x-5\right)x}.

$$\frac{30\left(x-5\right)x}{x\left(x-5\right)\left(2x^{2}+2x+6\right)}$$

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

$$\frac{30}{2x^{2}+2x+6}$$

Expand

$$\frac{30}{2x^{2}+2x+6}$$

Short Solution Steps

Subtract 5 from 6 to get 1.

$$\frac{\frac{x+1}{x-5}-\frac{x+6}{x}}{1+\frac{x+6-5}{x-5}\times \frac{x+6}{x}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-5 and x is x\left(x-5\right). Multiply \frac{x+1}{x-5} times \frac{x}{x}. Multiply \frac{x+6}{x} times \frac{x-5}{x-5}.

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

Since \frac{\left(x+1\right)x}{x\left(x-5\right)} and \frac{\left(x+6\right)\left(x-5\right)}{x\left(x-5\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+1\right)x-\left(x+6\right)\left(x-5\right).

$$\frac{\frac{x^{2}+x-x^{2}+5x-6x+30}{x\left(x-5\right)}}{1+\frac{x+6-5}{x-5}\times \frac{x+6}{x}}$$

Combine like terms in x^{2}+x-x^{2}+5x-6x+30.

$$\frac{\frac{30}{x\left(x-5\right)}}{1+\frac{x+6-5}{x-5}\times \frac{x+6}{x}}$$

Subtract 5 from 6 to get 1.

$$\frac{\frac{30}{x\left(x-5\right)}}{1+\frac{x+1}{x-5}\times \frac{x+6}{x}}$$

Multiply \frac{x+1}{x-5} times \frac{x+6}{x} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{30}{x\left(x-5\right)}}{1+\frac{\left(x+1\right)\left(x+6\right)}{\left(x-5\right)x}}$$

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

$$\frac{\frac{30}{x\left(x-5\right)}}{\frac{\left(x-5\right)x}{\left(x-5\right)x}+\frac{\left(x+1\right)\left(x+6\right)}{\left(x-5\right)x}}$$

Since \frac{\left(x-5\right)x}{\left(x-5\right)x} and \frac{\left(x+1\right)\left(x+6\right)}{\left(x-5\right)x} have the same denominator, add them by adding their numerators.

$$\frac{\frac{30}{x\left(x-5\right)}}{\frac{\left(x-5\right)x+\left(x+1\right)\left(x+6\right)}{\left(x-5\right)x}}$$

Do the multiplications in \left(x-5\right)x+\left(x+1\right)\left(x+6\right).

$$\frac{\frac{30}{x\left(x-5\right)}}{\frac{x^{2}-5x+x^{2}+6x+x+6}{\left(x-5\right)x}}$$

Combine like terms in x^{2}-5x+x^{2}+6x+x+6.

$$\frac{\frac{30}{x\left(x-5\right)}}{\frac{2x^{2}+2x+6}{\left(x-5\right)x}}$$

Divide \frac{30}{x\left(x-5\right)} by \frac{2x^{2}+2x+6}{\left(x-5\right)x} by multiplying \frac{30}{x\left(x-5\right)} by the reciprocal of \frac{2x^{2}+2x+6}{\left(x-5\right)x}.

$$\frac{30\left(x-5\right)x}{x\left(x-5\right)\left(2x^{2}+2x+6\right)}$$

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

$$\frac{30}{2x^{2}+2x+6}$$