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

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

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

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.

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

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

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

Use the distributive property to multiply x+2 by 4.

$$x^{2}-2x-\left(4x+8\right)=\left(x-2\right)\left(x+2\right)$$

To find the opposite of 4x+8, find the opposite of each term.

$$x^{2}-2x-4x-8=\left(x-2\right)\left(x+2\right)$$

Combine -2x and -4x to get -6x.

$$x^{2}-6x-8=\left(x-2\right)\left(x+2\right)$$

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.

$$x^{2}-6x-8=x^{2}-4$$

Subtract x^{2} from both sides.

$$x^{2}-6x-8-x^{2}=-4$$

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

$$-6x-8=-4$$

Add 8 to both sides.

$$-6x=-4+8$$

Add -4 and 8 to get 4.

$$-6x=4$$

Divide both sides by -6.

$$x=\frac{4}{-6}$$

Reduce the fraction \frac{4}{-6} to lowest terms by extracting and canceling out 2.

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