Solve 5-12/x^2-16=2x+2/x+4-3x-16/4-x

$$5 - \frac { 12 } { x ^ { 2 } - 16 } = \frac { 2 x + 2 } { x + 4 } - \frac { 3 x - 16 } { 4 - x }$$

Solve for x

$$x=2$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -4,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x+4\right), the least common multiple of x^{2}-16,x+4,4-x.

$$\left(x-4\right)\left(x+4\right)\times 5-12=\left(x-4\right)\left(2x+2\right)-\left(-\left(4+x\right)\left(3x-16\right)\right)$$

Consider \left(x-4\right)\left(x+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$\left(x^{2}-16\right)\times 5-12=\left(x-4\right)\left(2x+2\right)-\left(-\left(4+x\right)\left(3x-16\right)\right)$$

Use the distributive property to multiply x^{2}-16 by 5.

$$5x^{2}-80-12=\left(x-4\right)\left(2x+2\right)-\left(-\left(4+x\right)\left(3x-16\right)\right)$$

Subtract 12 from -80 to get -92.

$$5x^{2}-92=\left(x-4\right)\left(2x+2\right)-\left(-\left(4+x\right)\left(3x-16\right)\right)$$

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

$$5x^{2}-92=2x^{2}-6x-8-\left(-\left(4+x\right)\left(3x-16\right)\right)$$

Use the distributive property to multiply -1 by 4+x.

$$5x^{2}-92=2x^{2}-6x-8-\left(-4-x\right)\left(3x-16\right)$$

Use the distributive property to multiply -4-x by 3x-16 and combine like terms.

$$5x^{2}-92=2x^{2}-6x-8-\left(4x+64-3x^{2}\right)$$

To find the opposite of 4x+64-3x^{2}, find the opposite of each term.

$$5x^{2}-92=2x^{2}-6x-8-4x-64+3x^{2}$$

Combine -6x and -4x to get -10x.

$$5x^{2}-92=2x^{2}-10x-8-64+3x^{2}$$

Subtract 64 from -8 to get -72.

$$5x^{2}-92=2x^{2}-10x-72+3x^{2}$$

Combine 2x^{2} and 3x^{2} to get 5x^{2}.

$$5x^{2}-92=5x^{2}-10x-72$$

Subtract 5x^{2} from both sides.

$$5x^{2}-92-5x^{2}=-10x-72$$

Combine 5x^{2} and -5x^{2} to get 0.

$$-92=-10x-72$$

Swap sides so that all variable terms are on the left hand side.

$$-10x-72=-92$$

Add 72 to both sides.

$$-10x=-92+72$$

Add -92 and 72 to get -20.

$$-10x=-20$$

Divide both sides by -10.

$$x=\frac{-20}{-10}$$

Divide -20 by -10 to get 2.

$$x=2$$