It’s easy to get caught up in the words here. The trick I think was realising that cost is not the same as price and that addition is associative (ie. 1 + 2 = 2 + 1). So if you reword the statement to “a book costs half it’s price plus $1,” it becomes a little easier to see. This can then be put into an equation:
x = x/2 + 1
which solves to
x = 2 dollars

When I saw this I immediately thought $1.50…