Final answer:
To solve the equation (x/x²-16) - (1/x+4) = (2/x-4), we find a common denominator and simplify. After reducing and setting the equation to zero, we find that x = -4 is not valid as it would make a denominator zero. A reassessment of the equation indicates there is no solution if the equation is correctly set up.
Step-by-step explanation:
The given equation is (x/x²-16) - (1/x+4) = (2/x-4). To solve the equation, we need to find a common denominator, which in this case would be the product of the denominators (x²-16), (x+4), and (x-4). The common denominator is (x+4)(x-4), which simplifies to x² - 16, as this is a difference of squares. Now, we can rewrite the equation with a single denominator:
(x) - (x-4)(1) = 2(x+4), all over (x²-16).
Next, we distribute the numerator terms and consolidate them:
x - (x - 4) = 2(x + 4)
After simplifying, we set the equation equal to zero and solve for x:
x - x + 4 = 2x + 8
4 = x + 8
x = -4
However, we must check to ensure that x is not equal to -4 or 4, as those values would make the original denominators zero, which is not permissible. Therefore, x = -4 is not a valid solution, leaving us to find any other potential solutions.
Since at this point the equation seems to cancel itself out, this indicates that this problem might be a trick question or that it's not properly set up. So, it's important to reassess the initial equation and make sure there are no mistakes in the original formula or during the solving process. Without any other values of x, we must conclude that there is no solution.