91Ó°ÊÓ

Inequalities are a fundamental concept in mathematics, including when dealing with complex numbers. An inequality relates two expressions using inequality signs such as \(<, >, \leq,\) or \(\geq\). In the context of complex numbers, inequalities often describe sets or regions within the complex plane. For example, the inequality \( |z+4| \leq 3 \) describes the set of all complex numbers \(z\) whose distance from the point \(-4\) is 3 or less. Here, the modulus \( |z+4| \) represents the distance of the complex number \(z = x + yi\) from \(-4, 0\) on the complex plane. This is an example of how inequalities can be used to define geometric shapes!
The inequality is essentially a property that restricts \(z\) to a specific circular region. When translating into an equation by squaring both sides \((x+4)^2 + y^2 \leq 9\), it confirms that the region is a circle centered at \(-4, 0\) with radius 3.

A circle in the complex plane is a collection of points that maintain a constant distance, known as the radius, from a fixed center. The circle defined by the inequality \( |z + 4| \leq 3 \) is crucial to solving this problem. Here, it means all points within or on the boundary of a defined circle.
In the complex plane, a circle centered at a point \((a, b)\) is represented by an equation such as \( |z - a - bi| = r \), where \(r\) is the radius. For \( |z+4| \leq 3 \), \(-4\) acts as the center (\