Solve the following for the variable x.
(7/x-2)+(1/x+2)=(3x/x^2-4) Thank you.
To solve the equation for the variable x, let's start by simplifying the equation:
(7/(x-2)) + (1/(x+2)) = (3x/(x^2-4))
First, let's find a common denominator for the fractions on the left side of the equation. The common denominator is (x-2)(x+2) because multiplying the denominators results in the product of the two factors: (x-2)(x+2).
Now we can rewrite the equation with the common denominator:
((7(x+2) + (1(x-2))) / ((x-2)(x+2))) = (3x/(x^2-4))
Next, let's distribute and simplify:
((7x + 14 + x - 2) / ((x-2)(x+2))) = (3x/(x^2-4))
Combining like terms:
((8x + 12) / ((x-2)(x+2))) = (3x/(x^2-4))
Now, let's eliminate the denominators by multiplying both sides of the equation by ((x-2)(x+2)):
((8x + 12) / ((x-2)(x+2))) * ((x-2)(x+2)) = (3x/(x^2-4)) * ((x-2)(x+2))
Simplifying, we get:
8x + 12 = 3x
Now, let's isolate the variable x by moving all the terms containing x to one side of the equation:
8x - 3x = -12
Combining like terms:
5x = -12
Finally, solve for x by dividing both sides of the equation by 5:
x = -12/5
So the solution to the equation is x = -12/5.