1. Statement of the problem: We want to find all real numbers $x$ satisfying the inequality $|x - 2| < 3$ and represent this set on a number line. 2. Understanding the absolute value inequality: Recall that $|x - a| < b$ means the distance between $x$ and $a$ is less than $b$. This implies $$-b < x - a < b.$$ 3. Apply this definition to $|x - 2| < 3$: $$-3 < x - 2 < 3.$$ 4. Add 2 to all parts of the inequality to isolate $x$: $$-3 + 2 < x < 3 + 2,$$ which simplifies to $$-1 < x < 5.$$ 5. Interpretation: The solution set is all real numbers between $-1$ and $5$, not including $-1$ and $5$ themselves (since the inequality is strict). 6. Representation on number line: Draw an open interval from $-1$ to $5$ indicating all $x$ values between these two points. Final answer: $$\boxed{-1 < x < 5}.$$