The inequality \( |a - b| \geq |a| - |b| \) is a valid mathematical statement and can be proved using the **triangle inequality**, which is a fundamental concept in absolute values and metric spaces.

### Proof:

1. **Start with the triangle inequality**:
   \[
   |a| = |(a - b) + b| \leq |a - b| + |b|
   \]
   This simplifies to:
   \[
   |a| - |b| \leq |a - b|
   \]
   Rearranging this gives:
   \[
   |a - b| \geq |a| - |b|
   \]
   Which is the desired inequality.

Thus, we have shown that \( |a - b| \geq |a| - |b| \) holds true.

Would you like further details or examples? 

Here are 5 related questions for deeper understanding:
1. How is the triangle inequality applied to complex numbers?
2. Can the equality hold in \( |a - b| = |a| - |b| \)?
3. What happens when \( a \) or \( b \) equals 0 in the inequality?
4. How does this inequality relate to vector norms?
5. How does the reverse inequality behave and when does it hold?

**Tip:** Always remember, absolute values measure "distance" and inequalities often compare such distances.

Understand how to prove the inequality |a - b| ≥ |a| - |b| using the triangle inequality, a fundamental concept in algebra and metric spaces. This step-by-step guide explains the proof and its application in absolute value problems, making it ideal for students in Grades 9-12.

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