# Solving Complex Inequality: t > (1/2) + a / |w|^2

• eckiller
In summary, the conversation discusses an inequality involving a complex number and a condition to be imposed in order for all values of the complex number to satisfy the inequality. The condition involves finding a sector around the negative real axis and determining the appropriate value of t for this sector.
eckiller
Hello,

I have the inequality

t > (1/2) + a / |w|^2

where w is a complex number, w = a + bi. So the a in the inequality is the
real part.

So I need to find t such that all w are in a sector around the negative real
axis. Note t in [0, 1].

I am having trouble figuring out the condition to impose.

For example, before I wanted to find t such that the entire negative half of the complex plane satisfied the above inequality. t > 1/2 clearly satisfied this. Now I want to find t such that a sector around the negative real
axis satisfies the above inequality.

I do not understand your description but ##t>\dfrac{1}{2}+\dfrac{a}{a^2+b^2}## which means for ##t\in [0,1]##that ##-\dfrac{1}{2} < \dfrac{a}{a^2+b^2}< \dfrac{1}{2}\,.## Now you can go on with whatever your condition on ##w## is.

## What is the meaning of t > (1/2) + a / |w|^2 in solving complex inequality?

In mathematics, t > (1/2) + a / |w|^2 represents an inequality where the value of t is greater than the sum of 1/2 and the quotient of a divided by the absolute value of w squared. This inequality is used to compare two complex numbers and determine which one is larger.

## What is the process for solving complex inequality t > (1/2) + a / |w|^2?

The process for solving this type of complex inequality involves isolating the variable t on one side of the inequality symbol and all other terms on the other side. Then, the inequality can be solved by using properties of complex numbers, such as comparing the real and imaginary parts of the numbers.

## How can complex numbers be represented graphically to solve the inequality t > (1/2) + a / |w|^2?

Complex numbers can be represented graphically as points on a complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. The inequality t > (1/2) + a / |w|^2 can then be solved by comparing the position of the points on the plane and determining which one is located further to the right.

## What are the key properties of complex numbers that are used in solving the inequality t > (1/2) + a / |w|^2?

The key properties of complex numbers used in solving this inequality include the commutative, associative, and distributive properties, as well as the ability to compare the real and imaginary parts of the numbers. Additionally, knowledge of the modulus and argument of complex numbers is crucial in solving complex inequalities.

## What are some real-world applications of solving complex inequality t > (1/2) + a / |w|^2?

Solving complex inequalities has many real-world applications in fields such as engineering, physics, and economics. For example, in electrical engineering, this type of inequality can be used to determine the stability of a control system. In economics, it can be used to analyze the relationship between variables such as supply and demand.

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