Solve 2x/x-2-x+2/x-3=1

$$\frac { 2 x } { x - 2 } - \frac { x + 2 } { x - 3 } = 1$$

Solve for x

$$x=-2$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right), the least common multiple of x-2,x-3.

$$\left(x-3\right)\times 2x-\left(x-2\right)\left(x+2\right)=\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-3 by 2.

$$\left(2x-6\right)x-\left(x-2\right)\left(x+2\right)=\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 2x-6 by x.

$$2x^{2}-6x-\left(x-2\right)\left(x+2\right)=\left(x-3\right)\left(x-2\right)$$

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.

$$2x^{2}-6x-\left(x^{2}-4\right)=\left(x-3\right)\left(x-2\right)$$

To find the opposite of x^{2}-4, find the opposite of each term.

$$2x^{2}-6x-x^{2}+4=\left(x-3\right)\left(x-2\right)$$

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

$$x^{2}-6x+4=\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-3 by x-2 and combine like terms.

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

Subtract x^{2} from both sides.

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

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

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

Add 5x to both sides.

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

Combine -6x and 5x to get -x.

$$-x+4=6$$

Subtract 4 from both sides.

$$-x=6-4$$

Subtract 4 from 6 to get 2.

$$-x=2$$

Multiply both sides by -1.

$$x=-2$$