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

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

Solve for x

$$x=14$$

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.

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

Multiply x-2 and x-2 to get \left(x-2\right)^{2}.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$x^{2}-2x-8=x^{2}-4x+4+x+2$$

Combine -4x and x to get -3x.

$$x^{2}-2x-8=x^{2}-3x+4+2$$

Add 4 and 2 to get 6.

$$x^{2}-2x-8=x^{2}-3x+6$$

Subtract x^{2} from both sides.

$$x^{2}-2x-8-x^{2}=-3x+6$$

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

$$-2x-8=-3x+6$$

Add 3x to both sides.

$$-2x-8+3x=6$$

Combine -2x and 3x to get x.

$$x-8=6$$

Add 8 to both sides.

$$x=6+8$$

Add 6 and 8 to get 14.

$$x=14$$