Evaluating sign of inequality under constraint

Question

Suppose for real numbers $b$ and $c$ $$|b+c|-|b|<0.$$

Can I infer for any real $a$ that

$$|a+b+c|-|a+b|<0?$$

And if so, how should I formally derive this?

Edit: Additional question after comments, for which range of $a$ does the inequality hold?

for some real $a$? Try $a=0$ :-). Do you mean for any real $a$? — gt6989b, Jan 02 '14 at 13:58

score 1 · Answer 1 · answered Jan 02 '14 at 14:00

1

for some real $a$? Try $a=0$ :-).

Do you mean for any real $a$? That's unlikely, pick $a = -b$ to guarantee $|a+b+c|-|a+b| = |c| \geq 0$.

answered Jan 02 '14 at 14:00

gt6989b

OK thank you, that helps! – tomka Jan 02 '14 at 14:02
@tomka What is more interesting is to find the entire range of values of $a$ where it is true and false... – gt6989b Jan 02 '14 at 14:03
Yes, I just thought so too. I will edit the question to include this. – tomka Jan 02 '14 at 14:05

1 Answers1