Prove that $|x'| + |x| + 1 = 0$

Question

Studying a field of analysis of differential equations within Dynamical Systems, I came upon the following problem which I do not know how to proceed :

Prove that the differential equation : $$|x'| + |x| + 1 = 0$$ has no solutions.

I have handled some problems like : prove that the solutions of $x'' + x + x^3 =0$ have solutions that are set over $\mathbb R$, which are handled by multiplying and integrating or by handling them as duffing equations, but I cannot seem how to proceed in this one. Any thorough help will be appreciated !

Answer 1

For any differentiable function $x$, $|x'|+|x|+1\geqslant1>0$.

Answer 2

This equation doesn't need to be "differential" to be impossible. No non-negative quantity added to $1$ can yield $0$.