Proving $|a| + |b| \leq |a+b| + |a-b|$ using the Triangle Inequality

Question

Prove $$|a| + |b| \leq |a+b| + |a-b|$$
using the Triangle Inequality.

Struggling a bit with this question, here's the little progress I made.

$|a+b| \leq |a| + |b|$

$|a-b| \leq |a| + |b|$

$|a+b| + |a-b| \leq 2|a|$

Answer 1

$$|a| = \left|\dfrac{a+b}{2} + \dfrac{a-b}{2}\right|\leq\dfrac{1}{2}(|a+b| + |a-b|)$$ and you can do a similar thing for $|b|$ to obtain your claim.