We have to determine if the equation 5/(x+5) - 5=0 has a unique root.
5/(x+5) - 5=0
=> 5/(x+5) = 5
=> 5 = 5x + 25
=> 5x = -20
=> x = -20/5
=> x = -4
Actually we don't need to solve the equation to determine that the root is unique.
A linear equation always has a unique root. The equation has a unique root.
Since an one to one function has an unique solution, we'll have to prove that the function is one to one function.
For this reason, we'll have to prove that the function is monotonous.
The monotony of a function could be demonstrated using the 1st derivative.
We'll calculate the first derivative using quotient rule for the term 5/(x+5):
f'(x) = -5/(x+5)^2<0 for any x over R set
Since the derivative is negative, the function f(x) is decreasing then it is an one to one function.
If the function is one to one, any parallel line to x axis will intercept the graph of the function once, so the equation will have an unique root.