The binary operation is defined on the set of real numbers, R, by x y = x + y + 10.

An element $e$ is the identity element if for any $x \in R$, $x \Delta e = x$ and $e \Delta x = x$.

Using $x \Delta e = x$:

Subtract $x$ from both sides:

Solve for $e$:

The identity element is $\boxed{-10}$.

The inverse of an element $x$, denoted $x^{-1}$, is such that $x \Delta x^{-1} = e$ and $x^{-1} \Delta x = e$. We found the identity element $e = -10$.

Let $3^{-1}$ be the inverse of 3. Using $3 \Delta 3^{-1} = e$:

Subtract 13 from both sides:

The inverse of 3 is $\boxed{-23}$.

Let $(-5)^{-1}$ be the inverse of -5. Using $(-5) \Delta (-5)^{-1} = e$:

Subtract 5 from both sides:

The inverse of -5 is $\boxed{-15}$.