Solve x-4/x*(x+4)-16/(4-x)*(4+x)

$$\frac { x - 4 } { x \times ( x + 4 ) } - \frac { 16 } { ( 4 - x ) \times ( 4 + x ) }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+4\right) and \left(4-x\right)\left(4+x\right) is x\left(x+4\right)\left(-x+4\right). Multiply \frac{x-4}{x\left(x+4\right)} times \frac{-x+4}{-x+4}. Multiply \frac{16}{\left(4-x\right)\left(4+x\right)} times \frac{x}{x}.

$$\frac{\left(x-4\right)\left(-x+4\right)}{x\left(x+4\right)\left(-x+4\right)}-\frac{16x}{x\left(x+4\right)\left(-x+4\right)}$$

Since \frac{\left(x-4\right)\left(-x+4\right)}{x\left(x+4\right)\left(-x+4\right)} and \frac{16x}{x\left(x+4\right)\left(-x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-4\right)\left(-x+4\right)-16x}{x\left(x+4\right)\left(-x+4\right)}$$

Do the multiplications in \left(x-4\right)\left(-x+4\right)-16x.

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

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

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

Factor the expressions that are not already factored in \frac{-x^{2}-8x-16}{x\left(x+4\right)\left(-x+4\right)}.

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

Extract the negative sign in -4-x.

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

Cancel out x+4 in both numerator and denominator.

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

Expand x\left(-x+4\right).

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

To find the opposite of x+4, find the opposite of each term.

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

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x+4\right) and \left(4-x\right)\left(4+x\right) is x\left(x+4\right)\left(-x+4\right). Multiply \frac{x-4}{x\left(x+4\right)} times \frac{-x+4}{-x+4}. Multiply \frac{16}{\left(4-x\right)\left(4+x\right)} times \frac{x}{x}.

$$\frac{\left(x-4\right)\left(-x+4\right)}{x\left(x+4\right)\left(-x+4\right)}-\frac{16x}{x\left(x+4\right)\left(-x+4\right)}$$

Since \frac{\left(x-4\right)\left(-x+4\right)}{x\left(x+4\right)\left(-x+4\right)} and \frac{16x}{x\left(x+4\right)\left(-x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x-4\right)\left(-x+4\right)-16x}{x\left(x+4\right)\left(-x+4\right)}$$

Do the multiplications in \left(x-4\right)\left(-x+4\right)-16x.

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

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

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

Factor the expressions that are not already factored in \frac{-x^{2}-8x-16}{x\left(x+4\right)\left(-x+4\right)}.

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

Extract the negative sign in -4-x.

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

Cancel out x+4 in both numerator and denominator.

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

Expand x\left(-x+4\right).

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

To find the opposite of x+4, find the opposite of each term.

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