Proving the Openness of the Left Half Plane

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SUMMARY

The discussion centers on proving that the variable x is negative, thereby demonstrating that it lies within the left half plane in a mathematical context. The user expresses confusion regarding the management of negative signs in their inequalities. They acknowledge that x_0 is defined as negative but struggle with the implications for the unknown variable x. The user seeks clarification on their approach and the correctness of their inequalities.

PREREQUISITES
  • Understanding of complex analysis and the concept of the left half plane.
  • Familiarity with inequalities and their manipulation in mathematical proofs.
  • Basic knowledge of variable definitions and their implications in mathematical contexts.
  • Experience with mathematical problem-solving techniques, particularly in proofs.
NEXT STEPS
  • Review the properties of the left half plane in complex analysis.
  • Study techniques for manipulating inequalities in mathematical proofs.
  • Learn about variable definitions and their roles in mathematical arguments.
  • Practice solving similar problems involving negative variables and inequalities.
USEFUL FOR

Mathematics students, educators, and anyone involved in complex analysis or mathematical proofs who seeks to clarify their understanding of inequalities and the left half plane.

MidnightR
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Please see question 4 on the PDF document below to see the question & my attempt at a solution. I'm getting really muddled up with the minus signs at the moment, and what is, and what is not negative. Could you please help me to move forward, I believe I'm going in the right direction... I'm trying to show that x<0 therefore is inside the left half plane...

Thanks!

EDIT: I know x_0 is negative as that's how I've defined it, however x is unknown. The inequalities at the end must go wrong somehow I think?
 

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