complex analysis - Did I misunderstand it or my book is wrong?

From my book, Prove that if $z \in \mathbb C$ where $|z|\leq 1$ then $|Im(1+\bar z+z^2)|\lt 3$

but, I have $|Im(1+\bar z+z^2)|\leq 3$

From $|Im(1+\bar z+z^2)|$ , I have

$$|Im(1+\bar z+z^2)|\leq|1+\bar z+z^2|\leq|1|+|\bar z|+|z|^2\leq1+|1|+|1|=3$$

Please check my solution, Thank you.