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.