How can I solve inequalities involving absolute values?

[carlosbgois]

Homework Statement

d) Show that $\left|x-y\right| \leq \left|x\right|+\left|y\right|$
e) Show that $\left|x\right|-\left|y\right| \leq \left|x-y\right|$

The Attempt at a Solution

For item d) I've tried some approaches but none was promising.

For item e), I tried squaring $\left|x-y\right|$ to get
$(\left|x-y\right|)^{2} \geq (\left|x\right|-\left|y\right|)^{2}$

But if I take the square roots, the right side may not always be positive, then I don't have a proof, right?

Thanks

 

[LCKurtz]

Homework Statement

d) Show that $\left|x-y\right| \leq \left|x\right|+\left|y\right|$
e) Show that $\left|x\right|-\left|y\right| \leq \left|x-y\right|$

The Attempt at a Solution

For item d) I've tried some approaches but none was promising.

For item e), I tried squaring $\left|x-y\right|$ to get
$(\left|x-y\right|)^{2} \geq (\left|x\right|-\left|y\right|)^{2}$

But if I take the square roots, the right side may not always be positive, then I don't have a proof, right?

Thanks

Do you have the ordinary triangle inequality to work with: |x+y| ≤ |x| + |y|? You can get these by using it. For example, what happens if you substitute -y for y in that?

 

[carlosbgois]

Yes I do, and I got the answer with your help:

Having the triangular inequality and substituting y for -y, we get:
$|x+(-y)|\leq|x|+|-y|$. As $|y|=|-y|$, then $|x-y|\leq|x|+|y|$.

But what about the second exercise? I now was able to do it using the first one,
but is there any way of getting it right with my previous attempt, which I posted up there?

Thanks

 

[206PiruBlood]

But what about the second exercise? I now was able to do it using the first one,
but is there any way of getting it right with my previous attempt, which I posted up there?

Thanks

This will be a common trend throughout Spivak. Many of the questions have multiple parts and the later parts often require the earlier parts to complete the proof; although, I think this occurs more frequently when you get to the real meat of the book beginning in chapter 9. It's common for previous, perhaps long forgotten, problems to be used in the proofs as well. Also Spivak has a lot of difficult problems so you might come to a point later in the book where you need to skip part A and just assume its true to solve part B - at least I occasionally did :)