Solve 3/x-4+3/x+y/frac{3x{x-4}-3x/x+4}

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

Evaluate

$$\frac{\left(x+4\right)\left(2x+y-4\right)}{8x\left(x+y\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-4 and x+y is \left(x-4\right)\left(x+y\right). Multiply \frac{3}{x-4} times \frac{x+y}{x+y}. Multiply \frac{3}{x+y} times \frac{x-4}{x-4}.

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

Since \frac{3\left(x+y\right)}{\left(x-4\right)\left(x+y\right)} and \frac{3\left(x-4\right)}{\left(x-4\right)\left(x+y\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(x+y\right)+3\left(x-4\right).

$$\frac{\frac{3x+3y+3x-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x}{x-4}-\frac{3x}{x+4}}$$

Combine like terms in 3x+3y+3x-12.

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x}{x-4}-\frac{3x}{x+4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-4 and x+4 is \left(x-4\right)\left(x+4\right). Multiply \frac{3x}{x-4} times \frac{x+4}{x+4}. Multiply \frac{3x}{x+4} times \frac{x-4}{x-4}.

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x\left(x+4\right)}{\left(x-4\right)\left(x+4\right)}-\frac{3x\left(x-4\right)}{\left(x-4\right)\left(x+4\right)}}$$

Since \frac{3x\left(x+4\right)}{\left(x-4\right)\left(x+4\right)} and \frac{3x\left(x-4\right)}{\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x\left(x+4\right)-3x\left(x-4\right)}{\left(x-4\right)\left(x+4\right)}}$$

Do the multiplications in 3x\left(x+4\right)-3x\left(x-4\right).

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x^{2}+12x-3x^{2}+12x}{\left(x-4\right)\left(x+4\right)}}$$

Combine like terms in 3x^{2}+12x-3x^{2}+12x.

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{24x}{\left(x-4\right)\left(x+4\right)}}$$

Divide \frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)} by \frac{24x}{\left(x-4\right)\left(x+4\right)} by multiplying \frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)} by the reciprocal of \frac{24x}{\left(x-4\right)\left(x+4\right)}.

$$\frac{\left(6x+3y-12\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+y\right)\times 24x}$$

Cancel out x-4 in both numerator and denominator.

$$\frac{\left(x+4\right)\left(6x+3y-12\right)}{24x\left(x+y\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(x+4\right)\left(2x+y-4\right)}{24x\left(x+y\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(x+4\right)\left(2x+y-4\right)}{8x\left(x+y\right)}$$

Expand the expression.

$$\frac{2x^{2}+xy+4x+4y-16}{8x^{2}+8xy}$$

Expand

$$\frac{2x^{2}+xy+4x+4y-16}{8x\left(x+y\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-4 and x+y is \left(x-4\right)\left(x+y\right). Multiply \frac{3}{x-4} times \frac{x+y}{x+y}. Multiply \frac{3}{x+y} times \frac{x-4}{x-4}.

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

Since \frac{3\left(x+y\right)}{\left(x-4\right)\left(x+y\right)} and \frac{3\left(x-4\right)}{\left(x-4\right)\left(x+y\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 3\left(x+y\right)+3\left(x-4\right).

$$\frac{\frac{3x+3y+3x-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x}{x-4}-\frac{3x}{x+4}}$$

Combine like terms in 3x+3y+3x-12.

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x}{x-4}-\frac{3x}{x+4}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-4 and x+4 is \left(x-4\right)\left(x+4\right). Multiply \frac{3x}{x-4} times \frac{x+4}{x+4}. Multiply \frac{3x}{x+4} times \frac{x-4}{x-4}.

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x\left(x+4\right)}{\left(x-4\right)\left(x+4\right)}-\frac{3x\left(x-4\right)}{\left(x-4\right)\left(x+4\right)}}$$

Since \frac{3x\left(x+4\right)}{\left(x-4\right)\left(x+4\right)} and \frac{3x\left(x-4\right)}{\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x\left(x+4\right)-3x\left(x-4\right)}{\left(x-4\right)\left(x+4\right)}}$$

Do the multiplications in 3x\left(x+4\right)-3x\left(x-4\right).

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{3x^{2}+12x-3x^{2}+12x}{\left(x-4\right)\left(x+4\right)}}$$

Combine like terms in 3x^{2}+12x-3x^{2}+12x.

$$\frac{\frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)}}{\frac{24x}{\left(x-4\right)\left(x+4\right)}}$$

Divide \frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)} by \frac{24x}{\left(x-4\right)\left(x+4\right)} by multiplying \frac{6x+3y-12}{\left(x-4\right)\left(x+y\right)} by the reciprocal of \frac{24x}{\left(x-4\right)\left(x+4\right)}.

$$\frac{\left(6x+3y-12\right)\left(x-4\right)\left(x+4\right)}{\left(x-4\right)\left(x+y\right)\times 24x}$$

Cancel out x-4 in both numerator and denominator.

$$\frac{\left(x+4\right)\left(6x+3y-12\right)}{24x\left(x+y\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(x+4\right)\left(2x+y-4\right)}{24x\left(x+y\right)}$$

Cancel out 3 in both numerator and denominator.

$$\frac{\left(x+4\right)\left(2x+y-4\right)}{8x\left(x+y\right)}$$

Expand the expression.

$$\frac{2x^{2}+xy+4x+4y-16}{8x^{2}+8xy}$$