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In the picture it is Complex Variables and Transforms, now slove the equation as given in picture

The problem in the image asks to define the Argand diagram for a complex number and then to find the region in the Argand diagram for the inequality \( |z| < 2 \), where \( z = x + iy \) is a complex number, with \( -2 < x, y < 2 \).

### Argand Diagram Explanation:
An **Argand diagram** is a graphical representation of complex numbers on a plane, where:
- The x-axis (real axis) represents the real part of the complex number.
- The y-axis (imaginary axis) represents the imaginary part.

A complex number \( z = x + iy \) is plotted as the point \( (x, y) \) on the Argand plane.

### Solving the Inequality \( |z| < 2 \):
1. The modulus of a complex number \( z = x + iy \) is given by:
   \[
   |z| = \sqrt{x^2 + y^2}
   \]
   The inequality \( |z| < 2 \) becomes:
   \[
   \sqrt{x^2 + y^2} < 2
   \]
   Squaring both sides:
   \[
   x^2 + y^2 < 4
   \]
   This represents the interior of a circle with radius 2, centered at the origin \( (0, 0) \) on the Argand plane.

2. The additional condition \( -2 < x, y < 2 \) implies that we are considering the region where both the real and imaginary parts of the complex number are between -2 and 2.

### Interpretation:
- The inequality \( |z| < 2 \) gives a **disk** (open circle) of radius 2 centered at the origin.
- The condition \( -2 < x, y < 2 \) means we are further restricting the solution to a **square region** with side length 4 (from -2 to 2 on both axes) that inscribes the circle.

Thus, the solution is the part of the disk (circle of radius 2) that lies within the square \( -2 < x < 2 \) and \( -2 < y < 2 \).

Would you like a more detailed breakdown of the Argand diagram or this concept? Here are some related questions you might find interesting:

1. How can the Argand diagram be used to represent addition of complex numbers geometrically?
2. What is the geometric interpretation of complex conjugation on the Argand plane?
3. How does the inequality \( |z - a| < r \) change the position of the circle on the Argand diagram?
4. How do we represent polar coordinates in the Argand diagram?
5. What is the significance of the real and imaginary axes in operations involving complex numbers?

**Tip:** When graphing on an Argand diagram, always start by identifying the real and imaginary components of the complex numbers. This helps in visualizing transformations like rotation and scaling.

Define Argand diagram for a complex number. Find region in Argand diagram of |z| < 2, -2 < x, y < 2.

This problem covers the solution to finding the region in the Argand diagram for the inequality |z| < 2, where z = x + iy is a complex number. The solution involves understanding the Argand diagram, modulus of a complex number, and the geometry of a circle.

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