Solve 4(x+1)/x^2+4x:(9x/x^2-16-x+4/x^2-4x)

$$\frac { 4 ( x + 1 ) } { x ^ { 2 } + 4 x } : ( \frac { 9 x } { x ^ { 2 } - 16 } - \frac { x + 4 } { x ^ { 2 } - 4 x } )$$

Evaluate

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

Short Solution Steps

Factor x^{2}-16. Factor x^{2}-4x.

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{9x}{\left(x-4\right)\left(x+4\right)}-\frac{x+4}{x\left(x-4\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x+4\right) and x\left(x-4\right) is x\left(x-4\right)\left(x+4\right). Multiply \frac{9x}{\left(x-4\right)\left(x+4\right)} times \frac{x}{x}. Multiply \frac{x+4}{x\left(x-4\right)} times \frac{x+4}{x+4}.

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{9xx}{x\left(x-4\right)\left(x+4\right)}-\frac{\left(x+4\right)\left(x+4\right)}{x\left(x-4\right)\left(x+4\right)}}$$

Since \frac{9xx}{x\left(x-4\right)\left(x+4\right)} and \frac{\left(x+4\right)\left(x+4\right)}{x\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{9xx-\left(x+4\right)\left(x+4\right)}{x\left(x-4\right)\left(x+4\right)}}$$

Do the multiplications in 9xx-\left(x+4\right)\left(x+4\right).

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{9x^{2}-x^{2}-4x-4x-16}{x\left(x-4\right)\left(x+4\right)}}$$

Combine like terms in 9x^{2}-x^{2}-4x-4x-16.

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{8x^{2}-8x-16}{x\left(x-4\right)\left(x+4\right)}}$$

Divide \frac{4\left(x+1\right)}{x^{2}+4x} by \frac{8x^{2}-8x-16}{x\left(x-4\right)\left(x+4\right)} by multiplying \frac{4\left(x+1\right)}{x^{2}+4x} by the reciprocal of \frac{8x^{2}-8x-16}{x\left(x-4\right)\left(x+4\right)}.

$$\frac{4\left(x+1\right)x\left(x-4\right)\left(x+4\right)}{\left(x^{2}+4x\right)\left(8x^{2}-8x-16\right)}$$

Factor the expressions that are not already factored.

$$\frac{4x\left(x-4\right)\left(x+1\right)\left(x+4\right)}{8x\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Cancel out 4x\left(x+1\right)\left(x+4\right) in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Factor x^{2}-16. Factor x^{2}-4x.

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{9x}{\left(x-4\right)\left(x+4\right)}-\frac{x+4}{x\left(x-4\right)}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-4\right)\left(x+4\right) and x\left(x-4\right) is x\left(x-4\right)\left(x+4\right). Multiply \frac{9x}{\left(x-4\right)\left(x+4\right)} times \frac{x}{x}. Multiply \frac{x+4}{x\left(x-4\right)} times \frac{x+4}{x+4}.

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{9xx}{x\left(x-4\right)\left(x+4\right)}-\frac{\left(x+4\right)\left(x+4\right)}{x\left(x-4\right)\left(x+4\right)}}$$

Since \frac{9xx}{x\left(x-4\right)\left(x+4\right)} and \frac{\left(x+4\right)\left(x+4\right)}{x\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{9xx-\left(x+4\right)\left(x+4\right)}{x\left(x-4\right)\left(x+4\right)}}$$

Do the multiplications in 9xx-\left(x+4\right)\left(x+4\right).

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{9x^{2}-x^{2}-4x-4x-16}{x\left(x-4\right)\left(x+4\right)}}$$

Combine like terms in 9x^{2}-x^{2}-4x-4x-16.

$$\frac{\frac{4\left(x+1\right)}{x^{2}+4x}}{\frac{8x^{2}-8x-16}{x\left(x-4\right)\left(x+4\right)}}$$

Divide \frac{4\left(x+1\right)}{x^{2}+4x} by \frac{8x^{2}-8x-16}{x\left(x-4\right)\left(x+4\right)} by multiplying \frac{4\left(x+1\right)}{x^{2}+4x} by the reciprocal of \frac{8x^{2}-8x-16}{x\left(x-4\right)\left(x+4\right)}.

$$\frac{4\left(x+1\right)x\left(x-4\right)\left(x+4\right)}{\left(x^{2}+4x\right)\left(8x^{2}-8x-16\right)}$$

Factor the expressions that are not already factored.

$$\frac{4x\left(x-4\right)\left(x+1\right)\left(x+4\right)}{8x\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Cancel out 4x\left(x+1\right)\left(x+4\right) in both numerator and denominator.

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

Expand the expression.

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