If that truly is the correct problem statement, then your answer is correct, and the book is wrong.
Textbooks can definitely be wrong. Some books acquire a reputation for how many mistakes they have. I have a friend who edits textbooks for a living, and he found a proof-by-example actually being promoted as a valid proof technique. Naturally, you can't expect every book to be perfect. But these days, I would expect a decent author to publish known errata somewhere. Authors, if they're generally competent, would want to know about an error. So you might pass that one along to him, if you can get a hold of him.
You're welcome. Have a good one!