1. Key Concepts and Formulas

• A complex number $z$ can be represented as $z = x + iy$, where $x$ and $y$ are real numbers, and $i$ is the imaginary unit ($i^2 = -1$).
• The real part of $z$ is denoted by $\text{Re}(z) = x$.
• The modulus of $z$ is denoted by $|z| = \sqrt{x^2 + y^2}$.

2. Step-by-Step Solution

Step 1: Substitute the definitions into the inequality We are given the inequality $|z| - \text{Re}(z) \le 1$. Substitute $z = x + iy$, $|z| = \sqrt{x^2 + y^2}$, and $\text{Re}(z) = x$ into the inequality. $\sqrt{x^2 + y^2} - x \le 1$ Reasoning: This step replaces the complex number notation with real variables, allowing us to manipulate the inequality algebraically.

Step 2: Isolate the square root To prepare for squaring both sides, isolate the square root term. Add $x$ to both sides of the inequality. $\sqrt{x^2 + y^2} \le x + 1$ Reasoning: Isolating the square root simplifies the squaring process and avoids cross-terms that could complicate the inequality.

Step 3: Establish the condition for squaring both sides Before squaring, ensure that both sides of the inequality are non-negative. Since $\sqrt{x^2 + y^2}$ is always non-negative, we need $x + 1 \ge 0$, which means $x \ge -1$. $x+1 \ge 0 \implies x \ge -1$ Reasoning: Squaring both sides of an inequality is only valid if both sides are non-negative. This condition ensures that the direction of the inequality is preserved after squaring.

Step 4: Square both sides of the inequality Now that we have the condition $x \ge -1$, we can square both sides of the inequality: $(\sqrt{x^2 + y^2})^2 \le (x + 1)^2$ $x^2 + y^2 \le x^2 + 2x + 1$ Reasoning: Squaring eliminates the square root, resulting in a simpler polynomial inequality.

Step 5: Simplify the inequality Subtract $x^2$ from both sides of the inequality: $y^2 \le 2x + 1$ Reasoning: Simplifying by canceling terms makes the inequality easier to compare to the provided options.

Step 6: Rewrite the inequality in the desired form Factor out a 2 from the right side of the inequality: $y^2 \le 2\left(x + \frac{1}{2}\right)$ Reasoning: This manipulation puts the inequality into the form presented in option (A), making it directly comparable.

3. Common Mistakes & Tips

• Forgetting the non-negativity condition: Always check that both sides of an inequality are non-negative before squaring. Otherwise, you might introduce extraneous solutions.
• Geometric Interpretation: Recognize that $y^2 \le 2(x + 1/2)$ represents the region inside (or on) a parabola opening to the right with vertex at $(-1/2, 0)$. The condition $x \ge -1$ is inherently included in the solution because the inequality implies that the right side is non-negative.
• Careful Algebra: Pay attention to signs and exponents when expanding and simplifying algebraic expressions.

4. Summary

We started with the complex inequality $|z| - \text{Re}(z) \le 1$, substituted the definitions of modulus and real part, isolated the square root, and squared both sides after ensuring non-negativity. After simplification, we arrived at the inequality $y^2 \le 2\left(x + \frac{1}{2}\right)$, which represents the region inside or on a parabola.

5. Final Answer

The final answer is $y^2 \le 2\left(x + \frac{1}{2}\right)$, which corresponds to option (A). The final answer is \boxed{A}.