Math Problem Statement

|(z - 2) / z| ≥ 1

Solution

The inequality in the image is:

z2z1\left| \frac{z - 2}{z} \right| \geq 1

This inequality involves a complex variable zz, and we want to find the set of all complex numbers zz that satisfy it.

Step-by-Step Solution:

  1. Rewrite the Inequality: z2z1\left| \frac{z - 2}{z} \right| \geq 1

    This implies that the magnitude (or modulus) of z2z\frac{z - 2}{z} is at least 1.

  2. Interpret the Fraction: Rewrite z2z\frac{z - 2}{z} as: z2z=12z\frac{z - 2}{z} = 1 - \frac{2}{z}

  3. Magnitude Condition: We are given that 12z1\left|1 - \frac{2}{z}\right| \geq 1.

  4. Further Analysis: Since z=x+yiz = x + yi where xx and yy are real numbers, we can break down this inequality into real and imaginary parts or use polar form if required. This will typically involve isolating real and imaginary parts to explore conditions on xx and yy (the real and imaginary parts of zz).

Do you need further steps in complex analysis to isolate zz in a particular region on the complex plane, or details on transforming this inequality?

Related Questions:

  1. What does it mean geometrically if f(z)1|f(z)| \geq 1 for a complex function f(z)f(z)?
  2. How can polar coordinates help simplify inequalities involving complex numbers?
  3. What set of complex numbers would satisfy zaR|z - a| \leq R for some aCa \in \mathbb{C} and R>0R > 0?
  4. How is the modulus z|z| of a complex number z=x+yiz = x + yi calculated?
  5. What other techniques can be used to solve inequalities involving complex numbers?

Tip:

For complex inequalities, sketching the regions or interpreting them in polar form can often clarify the solution set.

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Math Problem Analysis

Mathematical Concepts

Complex Analysis
Inequalities
Modulus of Complex Numbers

Formulas

Modulus of complex number |z|
Inequality involving complex modulus |(z - 2) / z| ≥ 1

Theorems

Triangle Inequality
Properties of Modulus in Complex Analysis

Suitable Grade Level

Undergraduate Level