Solve x-4/x-2-x/x+2=1/x+2-2x+5/x^2-4

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

Solve for x

$$x=1$$

Steps for Solving Linear Equation

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

$$\left(x+2\right)\left(x-4\right)-\left(x-2\right)x=x-2-\left(2x+5\right)$$

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

$$x^{2}-2x-8-\left(x-2\right)x=x-2-\left(2x+5\right)$$

Use the distributive property to multiply x-2 by x.

$$x^{2}-2x-8-\left(x^{2}-2x\right)=x-2-\left(2x+5\right)$$

To find the opposite of x^{2}-2x, find the opposite of each term.

$$x^{2}-2x-8-x^{2}+2x=x-2-\left(2x+5\right)$$

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

$$-2x-8+2x=x-2-\left(2x+5\right)$$

Combine -2x and 2x to get 0.

$$-8=x-2-\left(2x+5\right)$$

To find the opposite of 2x+5, find the opposite of each term.

$$-8=x-2-2x-5$$

Combine x and -2x to get -x.

$$-8=-x-2-5$$

Subtract 5 from -2 to get -7.

$$-8=-x-7$$

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

$$-x-7=-8$$

Add 7 to both sides.

$$-x=-8+7$$

Add -8 and 7 to get -1.

$$-x=-1$$

Multiply both sides by -1.

$$x=1$$