Solve 5+96/x^2-16=2x-1/x+4+3x-1/x-4

$$5 + \frac { 96 } { x ^ { 2 } - 16 } = \frac { 2 x - 1 } { x + 4 } + \frac { 3 x - 1 } { x - 4 }$$

Solve for x

$$x=8$$

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,x-4.

$$\left(x-4\right)\left(x+4\right)\times 5+96=\left(x-4\right)\left(2x-1\right)+\left(x+4\right)\left(3x-1\right)$$

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.

$$\left(x^{2}-16\right)\times 5+96=\left(x-4\right)\left(2x-1\right)+\left(x+4\right)\left(3x-1\right)$$

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

$$5x^{2}-80+96=\left(x-4\right)\left(2x-1\right)+\left(x+4\right)\left(3x-1\right)$$

Add -80 and 96 to get 16.

$$5x^{2}+16=\left(x-4\right)\left(2x-1\right)+\left(x+4\right)\left(3x-1\right)$$

Use the distributive property to multiply x-4 by 2x-1 and combine like terms.

$$5x^{2}+16=2x^{2}-9x+4+\left(x+4\right)\left(3x-1\right)$$

Use the distributive property to multiply x+4 by 3x-1 and combine like terms.

$$5x^{2}+16=2x^{2}-9x+4+3x^{2}+11x-4$$

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

$$5x^{2}+16=5x^{2}-9x+4+11x-4$$

Combine -9x and 11x to get 2x.

$$5x^{2}+16=5x^{2}+2x+4-4$$

Subtract 4 from 4 to get 0.

$$5x^{2}+16=5x^{2}+2x$$

Subtract 5x^{2} from both sides.

$$5x^{2}+16-5x^{2}=2x$$

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

$$16=2x$$

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

$$2x=16$$

Divide both sides by 2.

$$x=\frac{16}{2}$$

Divide 16 by 2 to get 8.

$$x=8$$