Understanding the Inequality of Complex Numbers: |z+w|=|z-w|?

[mynameisfunk]

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|.

How do we know that?
is |z+w|=|z-w|?? Note that z and w are complex numbers.

 

[Dick]

No, |z+w| is NOT equal to |z-w|. Your first inequality is true for all w. Therefore it also must be true for -w. Substitute '-w' everywhere you see 'w' in the first inequality.

 

[mynameisfunk]

No, |z+w| is NOT equal to |z-w|. Your first inequality is true for all w. Therefore it also must be true for -w. Substitute '-w' everywhere you see 'w' in the first inequality.

Thanks a lot Dick, again. You rock