Let a be a positive real number. If |x − a| ≤ 2a is a sufficient condition for −2 ≤ x≤ 5, then
find the range of possible values of a.

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.

Explore how to determine the range of possible values for a such that the condition |x - a| ≤ 2a implies -2 ≤ x ≤ 5. This problem involves inequalities, absolute values, and interval notation, suitable for high school students.

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