Given the problem, we need to determine the range of possible values for $a$ such that the condition $|x - a| \leq 2a$ implies $-2 \leq x \leq 5$.

Step 1: Analyzing the Condition $|x - a| \leq 2a$

The inequality $|x - a| \leq 2a$ can be rewritten as: $-a \leq x - a \leq 2a$ Adding $a$ to all parts of the inequality: $0 \leq x \leq 3a$ So, $x$ is constrained by $0 \leq x \leq 3a$.

Step 2: Comparing with $-2 \leq x \leq 5$

For $|x - a| \leq 2a$ to imply $-2 \leq x \leq 5$, the interval $[0, 3a]$ must be contained within $[-2, 5]$.

Thus, we require: $0 \geq -2 \quad \text{(which is always true)}$ $3a \leq 5$ This inequality simplifies to: $a \leq \frac{5}{3}$

Step 3: Determining the Lower Bound of $a$

To satisfy the full inclusion of $[0, 3a]$ within $[-2, 5]$, we must also ensure that: $0 \geq -2 \quad \text{and} \quad 3a \geq 0$ Since $a$ is a positive real number, $3a \geq 0$ is always true.

Conclusion

The range of $a$ is: $0 < a \leq \frac{5}{3}$

Would you like more details on any part of this solution?

Related Questions:

1. What happens if $a = \frac{5}{3}$?
2. How would the solution change if the condition was $|x - a| < 2a$?
3. Can you graph the intervals to visualize the containment of $[0, 3a]$ in $[-2, 5]$?
4. What if $a$ were a negative real number? How would that affect the solution?
5. How do you solve inequalities involving absolute values in general?
6. What is the significance of the fact that $a$ is positive in this problem?
7. Can you derive an expression for $a$ if the condition was $|x - a| \geq 2a$?
8. How would the solution be different if the inequality was $|x - a| \leq a$?

Tip:

When solving inequalities involving absolute values, it's helpful to consider the definition of absolute value and break the inequality into multiple cases.