To solve the problem described in the image, we break the inequality into its key components and analyze it step by step in the context of the complex number $z = x + yi$, where $x = \text{Re}(z)$ (real part) and $y = \text{Im}(z)$ (imaginary part).

Problem Restatement

The inequality to solve is:

$(\text{Im}(2z + i))^2 - 1 \leq 4|z|^2 - 8\text{Re}(z) < -(z - \overline{z})^2$

where $|z|^2 = x^2 + y^2$, $\text{Im}(z) = y$, and $\text{Re}(z) = x$.

Step 1: Expand and Simplify the Terms

Term 1: $\text{Im}(2z + i)$

$2z + i = 2(x + yi) + i = 2x + 2yi + i = 2x + (2y + 1)i$ $\text{Im}(2z + i) = 2y + 1$ $(\text{Im}(2z + i))^2 = (2y + 1)^2 = 4y^2 + 4y + 1$

Term 2: $4|z|^2$

$|z|^2 = x^2 + y^2$ $4|z|^2 = 4(x^2 + y^2)$

Term 3: $-(z - \overline{z})^2$

$z - \overline{z} = (x + yi) - (x - yi) = 2yi$ $(z - \overline{z})^2 = (2yi)^2 = -4y^2$ $-(z - \overline{z})^2 = 4y^2$

Step 2: Plug into the Inequality

Now substitute the simplified terms back into the inequality: $(4y^2 + 4y + 1) - 1 \leq 4(x^2 + y^2) - 8x < 4y^2$ Simplify each part of the inequality:

Left Side:

$4y^2 + 4y \leq 4(x^2 + y^2) - 8x$ $0 \leq 4x^2 + 4y^2 - 8x - 4y^2$ $0 \leq 4x^2 - 8x$ $0 \leq 4x(x - 2)$ $x(x - 2) \geq 0$

This implies: $x \leq 0 \quad \text{or} \quad x \geq 2$

Right Side:

$4(x^2 + y^2) - 8x < 4y^2$ $4x^2 + 4y^2 - 8x < 4y^2$ $4x^2 - 8x < 0$ $4x(x - 2) < 0$

This implies: $0 < x < 2$

Combined Solution

From the two parts, the solution is: $x \in (0, 2)$

Step 3: Sketch the Solution in the Complex Plane

1. For $x \in (0, 2)$, the real part of $z$ is restricted to the interval $(0, 2)$.
2. The imaginary part $y$ can vary freely, but the conditions on $y$ should ensure $(\text{Im}(2z + i))^2 - 1 \leq 4|z|^2 - 8\text{Re}(z) < -(z - \overline{z})^2$.

Would you like a detailed graph or further elaboration? Let me know!

Here are five questions related to this solution:

1. How can you interpret $z - \overline{z}$ geometrically in the complex plane?
2. Why does $x(x - 2) \geq 0$ result in two disjoint regions for $x$?
3. What additional constraints might $y$ impose in this solution?
4. How does the inequality structure determine the boundaries of the solution set?
5. Could you generalize this problem for a different transformation of $z$?

Tip: Always verify both real and imaginary parts when solving inequalities involving complex numbers!