$|a|−|b|$ is less than or equal to $|a−b|$

Question

Prove: $|a|−|b|$ is less than or equal to $a−b|$

I sloved this problem by assuming it was right, than checking if $a-b>0$, $a$ positive\negative b$$ positive\negative than doing the same thing with $a-b<0$.

I have a feeling it has a more simple way

Answer 1

Hint: $|a| = |b + (a-b)| \le |a-b| + ?$

Answer 2

We can prove a stronger result, called the "reverse triangle inequality": $$ \bigl||a|-|b|\bigr|\le |a-b| \, . $$ The regular triangle inequality states that $|a+b|\le|a|+|b|$:

• If we replace $a$ with $a-b$, the triangle inequality becomes $|a|\le|a-b|+|b|$. So $|a|-|b|\le|a-b|$, which is the result you wanted to prove.
• If we replace $b$ with $b-a$, the triangle inequality becomes $|b|\le|a|+|b-a|$, so $|b|-|a|\le|b-a|=|a-b|$.

Now what can you conclude by combining the results $|a|-|b|\le|a-b|$, and $|b|-|a|\le|a-b|$?