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1. The problem statement, all variables and given/known data

What does the inequality pz + conjugate(pz) + c < 0 represent if |p|^2 >c ?

2. Relevant equations

p is a constant and a member of the set of complex numbers. c is a constant and a member of the set of real numbers.

3. The attempt at a solution

First, for the EQUALITY pz + conjugate(pz) + c = 0, where p = a+ib, z = x+iy :

pz + conjugate(pz) + c = 0 => (a+ib)(x+iy) + conjugate((a+ib)(x+iy)) + c = 0 => ax + iay + ibx - by + ax - iay -ixb -by + c = 0 => y=((a/b)*x)+ c/(2*b).

So, the inequality is y> ((a/b)*x)+ c/(2*b).

MY QUESTION IS: WHAT IS THE IMPORTANCE OF --> |p|^2 >c WHEN SHOWING THE INEQUALITY --> pz + conjugate(pz) + c < 0.

I know that |p| = sqrt((a^2) + (b^2)), so |p|^2 >c implies ((a^2) + (b^2)) > c but I don't understand how this helps to show the the inequality pz + conjugate(pz) + c < 0... or more so, the inequality y > ((a/b)*x)+ c/(2*b).

ALSO, if I were to graph this inequality, would it be a (diagonal) dotted line (I say dotted because points on the line are not a part of the solution) where the graph is shaded above the dotted line?

Any help would be greatly appreciated.

Thank you.

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# Homework Help: Solving Inequality With Complex Numbers Question

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