Illustrate in the complex plane the following set of points and explain your answer
`|z-4| lt 1`

Answer

To illustrate the set of points defined by the inequality \( |z - 4| < 1 \) in the complex plane, we can follow these steps:

### Step 1: Understand the expression
We start with the expression \( |z - 4| < 1 \). Here, \( z \) is a complex number, which can be expressed as \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.

### Step 2: Rewrite the expression
Substituting \( z \) into the expression, we have:
\[ |z - 4| = |(x + iy) - 4| = |(x - 4) + iy| \]
This can be simplified to:
\[ |(x - 4) + iy| = \sqrt{(x - 4)^2 + y^2} \]

### Step 3: Set up the inequality
Now, we set up the inequality based on the original expression:
\[ \sqrt{(x - 4)^2 + y^2} < 1 \]

### Step 4: Square both sides
To eliminate the square root, we square both sides of the inequality:
\[ (x - 4)^2 + y^2 < 1^2 \]
This simplifies to:
\[ (x - 4)^2 + y^2 < 1 \]

### Step 5: Identify the geometric representation
The inequality \( (x - 4)^2 + y^2 < 1 \) represents the interior of a circle in the Cartesian plane. The center of this circle is at the point \( (4, 0) \) and the radius is \( 1 \).

### Step 6: Draw the circle
To illustrate this in the complex plane:
1. Draw a circle with center at the point \( (4, 0) \).
2. The radius of the circle is \( 1 \), meaning the circle extends from \( (3, 0) \) to \( (5, 0) \) along the real axis and from \( (4, 1) \) to \( (4, -1) \) along the imaginary axis.
3. Shade the area inside the circle to represent all points \( z \) such that \( |z - 4| < 1 \).

### Conclusion
The set of points satisfying the inequality \( |z - 4| < 1 \) is the interior of the circle centered at \( (4, 0) \) with a radius of \( 1 \).

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