Prove that if a≥0 then |x|≤a if and only if -a ≤x≤a

We are tasked with proving that for \( a \geq 0 \), the statement \( |x| \leq a \) is true if and only if \( -a \leq x \leq a \).

### Part 1: Proving \( |x| \leq a \implies -a \leq x \leq a \)

#### Step 1: Definition of Absolute Value
By definition, the absolute value of \( x \) is given as:
\[
|x| =
\begin{cases}
x, & \text{if } x \geq 0 \\
-x, & \text{if } x < 0
\end{cases}
\]
So, \( |x| \leq a \) means that the magnitude of \( x \) is no larger than \( a \).

#### Step 2: Consider Two Cases Based on \( x \)
1. **Case 1**: If \( x \geq 0 \), then \( |x| = x \). The condition \( |x| \leq a \) becomes:
   \[
   x \leq a
   \]
   Since \( x \geq 0 \), combining these gives:
   \[
   0 \leq x \leq a
   \]

2. **Case 2**: If \( x < 0 \), then \( |x| = -x \). The condition \( |x| \leq a \) becomes:
   \[
   -x \leq a \implies x \geq -a
   \]
   Since \( x < 0 \), combining these gives:
   \[
   -a \leq x < 0
   \]

#### Step 3: Combine Both Cases
From both cases, we conclude that \( |x| \leq a \) implies:
\[
-a \leq x \leq a
\]

### Part 2: Proving \( -a \leq x \leq a \implies |x| \leq a \)

Assume that \( -a \leq x \leq a \).

#### Step 1: Definition of Absolute Value
We want to show that \( |x| \leq a \). Again, by the definition of absolute value:
\[
|x| =
\begin{cases}
x, & \text{if } x \geq 0 \\
-x, & \text{if } x < 0
\end{cases}
\]

#### Step 2: Consider Two Cases Based on \( x \)
1. **Case 1**: If \( x \geq 0 \), then \( |x| = x \). Since \( -a \leq x \leq a \), we have \( x \leq a \), so:
   \[
   |x| = x \leq a
   \]

2. **Case 2**: If \( x < 0 \), then \( |x| = -x \). Since \( -a \leq x \leq a \), we have \( x \geq -a \), which implies:
   \[
   -x \leq a \implies |x| = -x \leq a
   \]

#### Step 3: Conclusion
In both cases, we find that \( |x| \leq a \). Thus, \( -a \leq x \leq a \) implies \( |x| \leq a \).

### Final Conclusion:
Since both directions have been proven, we conclude that:
\[
|x| \leq a \iff -a \leq x \leq a
\]
for \( a \geq 0 \).

Would you like more details or have any further questions?

### Related Questions:
1. How does the definition of absolute value affect inequalities?
2. Can the inequality \( |x| \leq a \) have negative solutions?
3. What happens when \( a < 0 \) in this context?
4. How would this proof change for \( |x| < a \)?
5. Can similar reasoning be applied to functions of \( x \), not just \( x \) itself?

### Tip:
When solving absolute value inequalities, it's helpful to always break the problem into cases based on the definition of the absolute value function.

Explore the proof that for a ≥ 0, |x| ≤ a is equivalent to -a ≤ x ≤ a. This solution breaks down the concept of absolute value, covering key mathematical concepts such as inequalities and real numbers. Suitable for students in Grades 9-12.

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