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

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

Solve for x

$$x=8$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+1\right), the least common multiple of x-2,x+1.

$$\left(x+1\right)\left(x-4\right)=\left(x-2\right)\left(x-2\right)$$

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

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

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

$$x^{2}-3x-4=\left(x-2\right)^{2}$$

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

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

Subtract x^{2} from both sides.

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

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

$$-3x-4=-4x+4$$

Add 4x to both sides.

$$-3x-4+4x=4$$

Combine -3x and 4x to get x.

$$x-4=4$$

Add 4 to both sides.

$$x=4+4$$

Add 4 and 4 to get 8.

$$x=8$$