# real analysis – given \$ x \$ irrational, can you find \$ a, b in mathbb {Q} \$ such that \$ a + bx = r \$ for all \$ r in mathbb {R} \$

given $$x$$ irrational can you find $$a, b in mathbb {Q}$$ such as $$a + bx = r$$ for everyone $$r in mathbb {R}$$.

I'm trying to solve that. My attempt is to choose $$b$$ pretty close to $$bx$$ such as $$bx to 0$$ and $$a$$ pretty close to $$r$$ such as $$a to r$$.

Or in a rigorous sense:

to choose $$b in mathbb {Q}$$ such as $$bx = epsilon$$

to choose $$a in mathbb {Q}$$ such as $$a = r – epsilon$$

I can choose such $$a, b in mathbb {Q}$$ since $$Q$$ is dense in $$mathbb {R}$$