Complex inequality with absolute values

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SUMMARY

The discussion focuses on solving the complex inequality |z+2| > 1 + |z-2|. The conclusion indicates that for real values, the inequality holds for z > 1/2. Participants suggest that while the triangle inequality does not yield significant insights, using tools like Wolfram Alpha reveals that the solution encompasses an area defined by two intersecting lines. Further exploration is needed to derive the equations of these lines.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with absolute value inequalities
  • Knowledge of the triangle inequality in mathematics
  • Basic skills in using computational tools like Wolfram Alpha
NEXT STEPS
  • Explore the derivation of the equations for the intersecting lines that define the solution area
  • Study the application of the triangle inequality in complex analysis
  • Learn how to graph complex inequalities using software tools
  • Investigate advanced techniques for solving inequalities involving absolute values
USEFUL FOR

Mathematics students, educators, and anyone interested in complex analysis or solving inequalities involving absolute values.

Grothard
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Homework Statement



Determine the values of [itex]z \in \mathbb{C}[/itex] for which [itex]|z+2| > 1 + |z-2|[/itex] holds.

Homework Equations



Nothing complicated I can think of.

The Attempt at a Solution



For real values this holds for anything greater than [itex]1/2[/itex]. If I could figure out the boundaries of the area I'd be set, but the triangle inequality doesn't return anything nontrivial here. Tedious expansion into real and imaginary terms could be a solution, but there's probably a better way.
 
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I've found out through wolfram alpha that the inequality holds for an area enclosed by two crossing lines. Not quite sure where to get the two lines from
 

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