Verify if the equation 5/(x+5) - 5=0 has an 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.