Why is $|a - b| \geq|a| - |b|$?
It's sometimes called the reverse triangle inequality. The proper form is $$\left| a - b \right| \ge \big||a| - |b|\big|$$ For the proof, consider $$|a| = |a - b + b| \le |a - b| + |b|$$ $$|b| = |b - a + a| \le |a - b| + |a|$$ so that we have $$-|a-b|\le|a|-|b| \le |a - b|$$
The length of any side of a triangle is greater than the absolute difference of the lengths of the other two sides:
Here is a proof:
(1) $$|a-b|\geq |a|-|b|$$
Interchanging $a$ and $b$, we get also
(2) $$|a-b|\geq |b|-|a|$$
Combining (1) and (2) we get our desired result.