How Can You Simplify the Inequality |Im(z^2 - z̅ + 6)| < 12 Given |z| < 3?

[jav]

Homework Statement

Know: modulus(z) < 3
WTS: |Im(z² - z_bar + 6)| <12

where z_bar is the complex conjugate

Homework Equations

z = x + iy

The Attempt at a Solution

|Im(z² - z_bar + 6)|
= |Im(x² + 2i*x*y - y² - x + iy + 6)|
= |2xy + y|

So I want to show |2xy + y|< 12

I already proved it using maximization and Lagrange multipliers, but it seems like overkill, and I think there is some kind of arithmetic trick I am missing. Anyone see it?

Thanks

 

[╔(σ_σ)╝]

|Im(z)|$$\leq$$ |z|
Then use the triangle inequality to derive an upper bound.

 

[hunt_mat]

One thing to be careful about is to use:
$$|z_{1}-z_{2}|\leqslant ||z_{1}|-|z_{2}||$$