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

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

Evaluate

$$-\frac{2x+5}{\left(x+2\right)\left(x+3\right)}$$

Short Solution Steps

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

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

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

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

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

$$\frac{2}{\frac{2x+4+2}{x+2}}-\frac{2}{2-\frac{2}{x+3}}$$

Combine like terms in 2x+4+2.

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

Divide 2 by \frac{2x+6}{x+2} by multiplying 2 by the reciprocal of \frac{2x+6}{x+2}.

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

Factor the expressions that are not already factored in \frac{2\left(x+2\right)}{2x+6}.

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

Cancel out 2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x+3}{x+3}.

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

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

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

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

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

Combine like terms in 2x+6-2.

$$\frac{x+2}{x+3}-\frac{2}{\frac{2x+4}{x+3}}$$

Divide 2 by \frac{2x+4}{x+3} by multiplying 2 by the reciprocal of \frac{2x+4}{x+3}.

$$\frac{x+2}{x+3}-\frac{2\left(x+3\right)}{2x+4}$$

Factor the expressions that are not already factored in \frac{2\left(x+3\right)}{2x+4}.

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

Cancel out 2 in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+3 and x+2 is \left(x+2\right)\left(x+3\right). Multiply \frac{x+2}{x+3} times \frac{x+2}{x+2}. Multiply \frac{x+3}{x+2} times \frac{x+3}{x+3}.

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

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

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

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

$$\frac{x^{2}+2x+2x+4-x^{2}-3x-3x-9}{\left(x+2\right)\left(x+3\right)}$$

Combine like terms in x^{2}+2x+2x+4-x^{2}-3x-3x-9.

$$\frac{-2x-5}{\left(x+2\right)\left(x+3\right)}$$

Expand \left(x+2\right)\left(x+3\right).

$$\frac{-2x-5}{x^{2}+5x+6}$$

Factor

$$-\frac{2x+5}{\left(x+2\right)\left(x+3\right)}$$