1/(X-2)=3/(X+2)-(6X/(X^2-4)

To solve the equation 1/(x-2) = 3/(x+2) - (6x/(x^2-4)), we need to follow a series of steps, which I will explain in detail below:

Step 1: Check for restricted values
The equation involves fractions, so we need to make sure there are no values of x that make the denominators equal to zero. In this case, we have two denominators: (x-2) and (x^2-4). Let's first look at the denominator (x^2-4). We can see that (x^2-4) can be factored as (x-2)(x+2). So, we have (x-2)(x+2) = 0. This equation has two solutions: x = 2 and x = -2. Therefore, x = 2 and x = -2 are restricted values, where the equation is undefined.

Step 2: Simplify the equation
To simplify the equation, we need to get rid of the fractions. We can do this by multiplying every term by the least common multiple (LCM) of the denominators. In this case, the LCM of (x-2) and (x+2) is (x-2)(x+2), which is the same as (x^2-4).

Multiply every term by (x^2-4):

(x^2-4) * (1/(x-2)) = (x^2-4) * (3/(x+2)) - (x^2-4) * (6x/(x^2-4))

Simplifying:

1 = 3(x+2) - 6x

Step 3: Solve for x
Now, we can solve the simplified equation for x. Distribute 3 to (x+2):

1 = 3x + 6 - 6x

Combine like terms:

1 = -3x + 6

Subtract 6 from both sides:

-5 = -3x

Divide by -3:

x = 5/3

Step 4: Check for extraneous solutions
Since we multiplied by (x^2-4), we need to make sure that x = 2 and x = -2 are not solutions in the simplified equation. Plugging them into x, we can see that neither x = 2 nor x = -2 satisfy the original equation.

Therefore, the solution to the equation is x = 5/3.