Solve x-2/2x+1/x+2/frac{3{2}-6/x^2+3x}

$$\frac { \frac { x - 2 } { 2 x } + \frac { 1 } { x + 2 } } { \frac { 3 } { 2 } - \frac { 6 } { x ^ { 2 } + 3 x } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x and x+2 is 2x\left(x+2\right). Multiply \frac{x-2}{2x} times \frac{x+2}{x+2}. Multiply \frac{1}{x+2} times \frac{2x}{2x}.

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

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

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

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

$$\frac{\frac{x^{2}+2x-2x-4+2x}{2x\left(x+2\right)}}{\frac{3}{2}-\frac{6}{x^{2}+3x}}$$

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

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3}{2}-\frac{6}{x^{2}+3x}}$$

Factor x^{2}+3x.

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3}{2}-\frac{6}{x\left(x+3\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and x\left(x+3\right) is 2x\left(x+3\right). Multiply \frac{3}{2} times \frac{x\left(x+3\right)}{x\left(x+3\right)}. Multiply \frac{6}{x\left(x+3\right)} times \frac{2}{2}.

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3x\left(x+3\right)}{2x\left(x+3\right)}-\frac{6\times 2}{2x\left(x+3\right)}}$$

Since \frac{3x\left(x+3\right)}{2x\left(x+3\right)} and \frac{6\times 2}{2x\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3x\left(x+3\right)-6\times 2}{2x\left(x+3\right)}}$$

Do the multiplications in 3x\left(x+3\right)-6\times 2.

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3x^{2}+9x-12}{2x\left(x+3\right)}}$$

Divide \frac{x^{2}+2x-4}{2x\left(x+2\right)} by \frac{3x^{2}+9x-12}{2x\left(x+3\right)} by multiplying \frac{x^{2}+2x-4}{2x\left(x+2\right)} by the reciprocal of \frac{3x^{2}+9x-12}{2x\left(x+3\right)}.

$$\frac{\left(x^{2}+2x-4\right)\times 2x\left(x+3\right)}{2x\left(x+2\right)\left(3x^{2}+9x-12\right)}$$

Cancel out 2x in both numerator and denominator.

$$\frac{\left(x+3\right)\left(x^{2}+2x-4\right)}{\left(x+2\right)\left(3x^{2}+9x-12\right)}$$

Use the distributive property to multiply x+3 by x^{2}+2x-4 and combine like terms.

$$\frac{x^{3}+5x^{2}+2x-12}{\left(x+2\right)\left(3x^{2}+9x-12\right)}$$

Use the distributive property to multiply x+2 by 3x^{2}+9x-12 and combine like terms.

$$\frac{x^{3}+5x^{2}+2x-12}{3x^{3}+15x^{2}+6x-24}$$

Expand

$$\frac{x^{3}+5x^{2}+2x-12}{3\left(x+2\right)\left(x^{2}+3x-4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x and x+2 is 2x\left(x+2\right). Multiply \frac{x-2}{2x} times \frac{x+2}{x+2}. Multiply \frac{1}{x+2} times \frac{2x}{2x}.

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

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

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

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

$$\frac{\frac{x^{2}+2x-2x-4+2x}{2x\left(x+2\right)}}{\frac{3}{2}-\frac{6}{x^{2}+3x}}$$

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

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3}{2}-\frac{6}{x^{2}+3x}}$$

Factor x^{2}+3x.

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3}{2}-\frac{6}{x\left(x+3\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and x\left(x+3\right) is 2x\left(x+3\right). Multiply \frac{3}{2} times \frac{x\left(x+3\right)}{x\left(x+3\right)}. Multiply \frac{6}{x\left(x+3\right)} times \frac{2}{2}.

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3x\left(x+3\right)}{2x\left(x+3\right)}-\frac{6\times 2}{2x\left(x+3\right)}}$$

Since \frac{3x\left(x+3\right)}{2x\left(x+3\right)} and \frac{6\times 2}{2x\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3x\left(x+3\right)-6\times 2}{2x\left(x+3\right)}}$$

Do the multiplications in 3x\left(x+3\right)-6\times 2.

$$\frac{\frac{x^{2}+2x-4}{2x\left(x+2\right)}}{\frac{3x^{2}+9x-12}{2x\left(x+3\right)}}$$

Divide \frac{x^{2}+2x-4}{2x\left(x+2\right)} by \frac{3x^{2}+9x-12}{2x\left(x+3\right)} by multiplying \frac{x^{2}+2x-4}{2x\left(x+2\right)} by the reciprocal of \frac{3x^{2}+9x-12}{2x\left(x+3\right)}.

$$\frac{\left(x^{2}+2x-4\right)\times 2x\left(x+3\right)}{2x\left(x+2\right)\left(3x^{2}+9x-12\right)}$$

Cancel out 2x in both numerator and denominator.

$$\frac{\left(x+3\right)\left(x^{2}+2x-4\right)}{\left(x+2\right)\left(3x^{2}+9x-12\right)}$$

Use the distributive property to multiply x+3 by x^{2}+2x-4 and combine like terms.

$$\frac{x^{3}+5x^{2}+2x-12}{\left(x+2\right)\left(3x^{2}+9x-12\right)}$$

Use the distributive property to multiply x+2 by 3x^{2}+9x-12 and combine like terms.

$$\frac{x^{3}+5x^{2}+2x-12}{3x^{3}+15x^{2}+6x-24}$$