Proving the Openness of the Left Half Plane

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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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