What is the solution to solving two inequalities?

[annoymage]

Homework Statement

$$||a|-|b||\leq{i don't know from here}\leq|a|-|b|\leq|a-b|$$

Homework Equations

n/a

The Attempt at a Solution

help

 

[hunt_mat]

You should know that:
$$|x+y|\leqslant |x|+|y|$$
So, we can write:
$$|x|=|x-y+y|\leqslant |x-y|+|y|$$
Likewise
$$|y|=|y-x+x|\leqslant |x-y|+|x|$$
These two inequalities should answer your question.

 

[annoymage]

i knoe, but that would only answer this
$$|a|-|b|\leq|a-b|$$

but this one, is it related? i can't see T_T
$$||a|-|b||\leq|a-b|$$

 

[hunt_mat]

it shows that:
$$|a|-|b|\leqslant |a-b|$$
and
$$-(|a|-|b|)\leqslant |a-b|$$

 

[hunt_mat]

finally the penny drops. This is why I recommended the analysis books!

 

[annoymage]

i still, like structure more than limit tough, maybe the reason i dislike analysis NOW(maybe later i love it more) because now I'm studying calculus and some methodS, too much memorizing, not vigorous at all, which despise me. ngahaha

 

[hunt_mat]

Give it time to internalise. Analysis was my first love in maths