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

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

Solve for x

$$x=4$$

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.

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

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

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

Consider \left(x-2\right)\left(x+2\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 2.

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

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

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

Subtract 4 from 4 to get 0.

$$2x^{2}-5x=2\left(x-2\right)\left(x-1\right)$$

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

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

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

$$2x^{2}-5x=2x^{2}-6x+4$$

Subtract 2x^{2} from both sides.

$$2x^{2}-5x-2x^{2}=-6x+4$$

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

$$-5x=-6x+4$$

Add 6x to both sides.

$$-5x+6x=4$$

Combine -5x and 6x to get x.

$$x=4$$