We are tasked with finding all real numbers $a$ such that $|a| = r$, where $r \geq 0$.

Step-by-step analysis:

1. Absolute value definition:
   The absolute value of $a$, denoted $|a|$, is defined as:

   \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$$

2. Equating absolute value:
   We want $|a| = r$, where $r \geq 0$. This means: $\text{If } a \geq 0, \quad a = r$ $\text{If } a < 0, \quad -a = r \quad \Rightarrow \quad a = -r$

3. Solutions:

   • If $a \geq 0$, then $a = r$.
   • If $a < 0$, then $a = -r$. Hence, the two possible solutions for $a$ are: $a = r \quad \text{or} \quad a = -r.$

4. Conclusion: Therefore, for $r \geq 0$, the two real numbers $a$ that satisfy $|a| = r$ are $a = r$ and $a = -r$.

5. How many solutions?
   There are two possible values for $a$ unless $r = 0$, in which case both solutions coincide, $a = 0$.

Summary:

For any $r \geq 0$, the two real numbers $a$ such that $|a| = r$ are $a = r$ and $a = -r$. Thus, there are two solutions unless $r = 0$, in which case there is only one solution (i.e., $a = 0$).

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Would you like more details on absolute values or the solution process? Here are some related questions for you:

1. What happens if $r < 0$?
2. How does the solution change if $r = 0$?
3. What is the geometric interpretation of $|a| = r$ on the number line?
4. How does this problem extend to complex numbers?
5. What is the significance of absolute value in optimization problems?

Tip: Remember that the absolute value function $|a|$ always returns a non-negative value, which means $|a| = r$ always results in two possible values for $a$ unless $r = 0$.