prove that $x=ax+b$ has solution for all values of $a$ and $b$

Question

Let $\Bbb R_+$ be the set of positive real numbers (including $\infty$). $(\Bbb R_+, +, \cdot, \leq)$ is an ordered semiring endowed with usual addition and multiplication in $\Bbb R_+$. Then it is easy to verify that the equation $$a+x=b$$ has a solution in $\Bbb R_+$ if $a\leq b$. Now how to verify that the equation

$x=ax+b$ has solution for all values of $b$?

Answer 1

HINT: Solve for $x$ and see the condition under which it is nonnegative.