Linear inequality problem: $2x + 1 > 10$ $2x + 1 > 10$
$2x > 9$
$x > 4.5$
The answer in the book says: $x\lt 4.5$.
Am I doing it wrong?

$$2x+1>10 \iff x\gt 4.5$$
In other words, your solution to the posted inequality is correct!
I suspect the book's (wrong) solution must be a typo/misprint, either in the solution, or in the book's statement of the problem. 
The book's solution satisfies the following inequality $$-2x + 1 \gt 10 \iff -2x \gt 9 \iff x\lt -4.5,$$
but does not satisfy the posted inequality. (Plug in, say $1<4.5$ into the original inequality and test it for yourself: Certainly, claiming that $2(1) + 1 = 3\gt 10$ is absurd.)
