Solve x+2+3/x-2/x-6+frac{7{x+2}}

$$\frac{ x+2+ \frac{ 3 }{ x-2 } }{ x-6+ \frac{ 7 }{ x+2 } }$$

Evaluate

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

Short Solution Steps

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

$$\frac{\frac{\left(x+2\right)\left(x-2\right)}{x-2}+\frac{3}{x-2}}{x-6+\frac{7}{x+2}}$$

Since \frac{\left(x+2\right)\left(x-2\right)}{x-2} and \frac{3}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x+2\right)\left(x-2\right)+3}{x-2}}{x-6+\frac{7}{x+2}}$$

Do the multiplications in \left(x+2\right)\left(x-2\right)+3.

$$\frac{\frac{x^{2}-2x+2x-4+3}{x-2}}{x-6+\frac{7}{x+2}}$$

Combine like terms in x^{2}-2x+2x-4+3.

$$\frac{\frac{x^{2}-1}{x-2}}{x-6+\frac{7}{x+2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x-6 times \frac{x+2}{x+2}.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{\left(x-6\right)\left(x+2\right)}{x+2}+\frac{7}{x+2}}$$

Since \frac{\left(x-6\right)\left(x+2\right)}{x+2} and \frac{7}{x+2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{\left(x-6\right)\left(x+2\right)+7}{x+2}}$$

Do the multiplications in \left(x-6\right)\left(x+2\right)+7.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{x^{2}+2x-6x-12+7}{x+2}}$$

Combine like terms in x^{2}+2x-6x-12+7.

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

Divide \frac{x^{2}-1}{x-2} by \frac{x^{2}-4x-5}{x+2} by multiplying \frac{x^{2}-1}{x-2} by the reciprocal of \frac{x^{2}-4x-5}{x+2}.

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

Factor the expressions that are not already factored.

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

Cancel out x+1 in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{2}+x-2}{x^{2}-7x+10}$$

Expand

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

Short Solution Steps

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

$$\frac{\frac{\left(x+2\right)\left(x-2\right)}{x-2}+\frac{3}{x-2}}{x-6+\frac{7}{x+2}}$$

Since \frac{\left(x+2\right)\left(x-2\right)}{x-2} and \frac{3}{x-2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x+2\right)\left(x-2\right)+3}{x-2}}{x-6+\frac{7}{x+2}}$$

Do the multiplications in \left(x+2\right)\left(x-2\right)+3.

$$\frac{\frac{x^{2}-2x+2x-4+3}{x-2}}{x-6+\frac{7}{x+2}}$$

Combine like terms in x^{2}-2x+2x-4+3.

$$\frac{\frac{x^{2}-1}{x-2}}{x-6+\frac{7}{x+2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x-6 times \frac{x+2}{x+2}.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{\left(x-6\right)\left(x+2\right)}{x+2}+\frac{7}{x+2}}$$

Since \frac{\left(x-6\right)\left(x+2\right)}{x+2} and \frac{7}{x+2} have the same denominator, add them by adding their numerators.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{\left(x-6\right)\left(x+2\right)+7}{x+2}}$$

Do the multiplications in \left(x-6\right)\left(x+2\right)+7.

$$\frac{\frac{x^{2}-1}{x-2}}{\frac{x^{2}+2x-6x-12+7}{x+2}}$$

Combine like terms in x^{2}+2x-6x-12+7.

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

Divide \frac{x^{2}-1}{x-2} by \frac{x^{2}-4x-5}{x+2} by multiplying \frac{x^{2}-1}{x-2} by the reciprocal of \frac{x^{2}-4x-5}{x+2}.

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

Factor the expressions that are not already factored.

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

Cancel out x+1 in both numerator and denominator.

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

Expand the expression.

$$\frac{x^{2}+x-2}{x^{2}-7x+10}$$