Proving the Openness of the Left Half Plane

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In summary, the conversation is about a problem that involves determining whether x is negative or not. The person is struggling with the use of minus signs and is seeking help to move forward. They mention that x_0 is negative but x is unknown. The problem is represented by a PDF document and an image link is provided.
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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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http://img149.imageshack.us/img149/1692/problembj.jpg
 
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What does it mean for a left half plane to be "open"?

In mathematics, an open set is a set that does not contain its boundary. In the context of a left half plane, this means that the line dividing the left half plane from the right half plane is not included in the set.

Why is proving that the left half plane is open important?

In mathematics, proving that a set is open is important because it allows for the application of various theorems and properties that only hold for open sets. In the context of a left half plane, proving that it is open allows for the use of these theorems and properties in further mathematical calculations.

How is the left half plane typically represented mathematically?

The left half plane is typically represented as the set of all complex numbers with a negative real part. This can be written as {(x+yi) | x < 0} where x is a real number and y is an imaginary number.

What is the process for proving that the left half plane is open?

To prove that the left half plane is open, we need to show that for any point in the set, we can find a small enough open ball around that point that is also contained within the set. This can be done by choosing an arbitrary point in the left half plane and showing that it is possible to find a ball of a certain radius around that point that is completely contained in the left half plane.

What are some examples of open sets in the left half plane?

Some examples of open sets in the left half plane are {(x+yi) | x < -1}, {(x+yi) | -3 < x < 2}, and {(x+yi) | x < -5 and y > 2}. These sets all contain points with negative real parts and do not include their boundaries, making them open sets in the left half plane.

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