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} $