Prove that if a≥0 then |x|≤a iff -a ≤x≤a

We are tasked with proving that if \( a \geq 0 \), then the statement \( |x| \leq a \) is equivalent to \( -a \leq x \leq a \).

### Proof:

#### 1. **Implication \( |x| \leq a \implies -a \leq x \leq a \):**

Start by assuming that \( |x| \leq a \), where \( a \geq 0 \). The absolute value \( |x| \) is defined as:
\[
|x| =
\begin{cases}
x, & \text{if } x \geq 0, \\
-x, & \text{if } x < 0.
\end{cases}
\]

The inequality \( |x| \leq a \) means that the magnitude (or distance) of \( x \) from zero is less than or equal to \( a \). Thus, we have two cases to consider:

- If \( x \geq 0 \), then \( |x| = x \). Therefore, \( x \leq a \).
- If \( x < 0 \), then \( |x| = -x \). Therefore, \( -x \leq a \), which implies \( x \geq -a \).

Combining both cases, we conclude that \( -a \leq x \leq a \).

#### 2. **Implication \( -a \leq x \leq a \implies |x| \leq a \):**

Now, assume that \( -a \leq x \leq a \). We will show that this implies \( |x| \leq a \).

- If \( x \geq 0 \), then \( |x| = x \), and since \( x \leq a \), we have \( |x| \leq a \).
- If \( x < 0 \), then \( |x| = -x \), and since \( x \geq -a \), this means \( -x \leq a \), or \( |x| \leq a \).

Thus, in both cases, we conclude that \( |x| \leq a \).

### Conclusion:

Since both directions of the implication are true, we have shown that:
\[
|x| \leq a \iff -a \leq x \leq a.
\]
This completes the proof.

---

Would you like further clarification on any step?

Here are 5 related questions to deepen your understanding:
1. How is the absolute value function defined for real numbers?
2. What is the geometric interpretation of \( |x| \leq a \)?
3. How can you visualize \( |x| \leq a \) on a number line?
4. What happens if \( a < 0 \)? Does the proof still hold?
5. Can you generalize this result for other inequalities involving absolute value?

**Tip:** Always remember that inequalities involving absolute values are usually two-sided and can be broken down into simpler inequalities.

Prove that if a ≥ 0 then |x| ≤ a iff -a ≤ x ≤ a

This proof demonstrates that for any non-negative value of a, the statement |x| ≤ a is equivalent to -a ≤ x ≤ a. The solution involves understanding absolute value, inequalities, and the properties of absolute value functions. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation