Final answer:
To solve the rational equation, we find a common denominator and multiply by the appropriate factors to eliminate the denominators. The solutions to the equation are x = 5.50 and x = -5.50. The domain of the equation is all real numbers except x = 5 and x = -5.
Step-by-step explanation:
To solve this rational equation, we need to find the common denominator, which is (x+5)(x-5)(x²-25). Then, we can multiply each term by the appropriate factor to get rid of the denominators. After simplifying, we get:
(4)(x-5)(x²-25) + (2)(x+5)(x²-25) = (32)(x+5)(x-5)
Simplifying further:
4(x-5)(x+5)(x²-25) + 2(x+5)(x-5)(x²-25) = 32(x+5)(x-5)
Cancelling out common factors:
4(x²-25) + 2(x²-25) = 32
Expanding and collecting like terms:
4x²-100 + 2x²-50 = 32
Combining like terms:
6x²-150 = 32
Adding 150 to both sides:
6x² = 182
Dividing by 6:
x² = 30.33
Taking the square root of both sides:
x ≈ ±5.50
Therefore, the solutions to this equation are approximately x = 5.50 and x = -5.50.
The domain of this equation is all real numbers except x = 5 and x = -5, which would make the denominators zero.