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

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

Solve for x

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

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,x^{2}-4.

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

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

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

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

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

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

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

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

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

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

$$2x-8-4x-4=4x-2$$

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

$$-2x-8-4=4x-2$$

Subtract 4 from -8 to get -12.

$$-2x-12=4x-2$$

Subtract 4x from both sides.

$$-2x-12-4x=-2$$

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

$$-6x-12=-2$$

Add 12 to both sides.

$$-6x=-2+12$$

Add -2 and 12 to get 10.

$$-6x=10$$

Divide both sides by -6.

$$x=\frac{10}{-6}$$

Reduce the fraction \frac{10}{-6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{5}{3}$$