Solve 3-4x/4-x+1+3x/4+x-x(x+22)/x^2-16

$$\frac { 3 - 4 x } { 4 - x } + \frac { 1 + 3 x } { 4 + x } - \frac { x ( x + 22 ) } { x ^ { 2 } - 16 }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4-x and 4+x is \left(x+4\right)\left(-x+4\right). Multiply \frac{3-4x}{4-x} times \frac{x+4}{x+4}. Multiply \frac{1+3x}{4+x} times \frac{-x+4}{-x+4}.

$$\frac{\left(3-4x\right)\left(x+4\right)}{\left(x+4\right)\left(-x+4\right)}+\frac{\left(1+3x\right)\left(-x+4\right)}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{x^{2}-16}$$

Since \frac{\left(3-4x\right)\left(x+4\right)}{\left(x+4\right)\left(-x+4\right)} and \frac{\left(1+3x\right)\left(-x+4\right)}{\left(x+4\right)\left(-x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(3-4x\right)\left(x+4\right)+\left(1+3x\right)\left(-x+4\right)}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{x^{2}-16}$$

Do the multiplications in \left(3-4x\right)\left(x+4\right)+\left(1+3x\right)\left(-x+4\right).

$$\frac{3x+12-4x^{2}-16x+4-x-3x^{2}+12x}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{x^{2}-16}$$

Combine like terms in 3x+12-4x^{2}-16x+4-x-3x^{2}+12x.

$$\frac{-2x+16-7x^{2}}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{x^{2}-16}$$

Factor x^{2}-16.

$$\frac{-2x+16-7x^{2}}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{\left(x-4\right)\left(x+4\right)}$$

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 \left(x-4\right)\left(x+4\right) is \left(x-4\right)\left(x+4\right). Multiply \frac{-2x+16-7x^{2}}{\left(x+4\right)\left(-x+4\right)} times \frac{-1}{-1}.

$$\frac{-\left(-2x+16-7x^{2}\right)}{\left(x-4\right)\left(x+4\right)}-\frac{x\left(x+22\right)}{\left(x-4\right)\left(x+4\right)}$$

Since \frac{-\left(-2x+16-7x^{2}\right)}{\left(x-4\right)\left(x+4\right)} and \frac{x\left(x+22\right)}{\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(-2x+16-7x^{2}\right)-x\left(x+22\right)}{\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in -\left(-2x+16-7x^{2}\right)-x\left(x+22\right).

$$\frac{2x-16+7x^{2}-x^{2}-22x}{\left(x-4\right)\left(x+4\right)}$$

Combine like terms in 2x-16+7x^{2}-x^{2}-22x.

$$\frac{-20x-16+6x^{2}}{\left(x-4\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{-20x-16+6x^{2}}{\left(x-4\right)\left(x+4\right)}.

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

Cancel out x-4 in both numerator and denominator.

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

Use the distributive property to multiply 2 by 3x+2.

$$\frac{6x+4}{x+4}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4-x and 4+x is \left(x+4\right)\left(-x+4\right). Multiply \frac{3-4x}{4-x} times \frac{x+4}{x+4}. Multiply \frac{1+3x}{4+x} times \frac{-x+4}{-x+4}.

$$\frac{\left(3-4x\right)\left(x+4\right)}{\left(x+4\right)\left(-x+4\right)}+\frac{\left(1+3x\right)\left(-x+4\right)}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{x^{2}-16}$$

Since \frac{\left(3-4x\right)\left(x+4\right)}{\left(x+4\right)\left(-x+4\right)} and \frac{\left(1+3x\right)\left(-x+4\right)}{\left(x+4\right)\left(-x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(3-4x\right)\left(x+4\right)+\left(1+3x\right)\left(-x+4\right)}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{x^{2}-16}$$

Do the multiplications in \left(3-4x\right)\left(x+4\right)+\left(1+3x\right)\left(-x+4\right).

$$\frac{3x+12-4x^{2}-16x+4-x-3x^{2}+12x}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{x^{2}-16}$$

Combine like terms in 3x+12-4x^{2}-16x+4-x-3x^{2}+12x.

$$\frac{-2x+16-7x^{2}}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{x^{2}-16}$$

Factor x^{2}-16.

$$\frac{-2x+16-7x^{2}}{\left(x+4\right)\left(-x+4\right)}-\frac{x\left(x+22\right)}{\left(x-4\right)\left(x+4\right)}$$

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 \left(x-4\right)\left(x+4\right) is \left(x-4\right)\left(x+4\right). Multiply \frac{-2x+16-7x^{2}}{\left(x+4\right)\left(-x+4\right)} times \frac{-1}{-1}.

$$\frac{-\left(-2x+16-7x^{2}\right)}{\left(x-4\right)\left(x+4\right)}-\frac{x\left(x+22\right)}{\left(x-4\right)\left(x+4\right)}$$

Since \frac{-\left(-2x+16-7x^{2}\right)}{\left(x-4\right)\left(x+4\right)} and \frac{x\left(x+22\right)}{\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-\left(-2x+16-7x^{2}\right)-x\left(x+22\right)}{\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in -\left(-2x+16-7x^{2}\right)-x\left(x+22\right).

$$\frac{2x-16+7x^{2}-x^{2}-22x}{\left(x-4\right)\left(x+4\right)}$$

Combine like terms in 2x-16+7x^{2}-x^{2}-22x.

$$\frac{-20x-16+6x^{2}}{\left(x-4\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{-20x-16+6x^{2}}{\left(x-4\right)\left(x+4\right)}.

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

Cancel out x-4 in both numerator and denominator.

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

Use the distributive property to multiply 2 by 3x+2.

$$\frac{6x+4}{x+4}$$