Thread: Inequantity be/tn absolute value of difference and the difference of absolute values

1. Inequantity be/tn absolute value of difference and the difference of absolute values

The appendix of my calculus book lists Absolute Value Properties:
1. |-a| = |a|
2. |ab| = |a||b|
3. |a/b| = |a|/|b|
4. |a+b| <= |a|+|b|

For 4., it is also mentioned that if 'a' and 'b' differ in sign, then |a+b| is less than |a|+|b|. In "all other cases", |a+b| equals |a|+|b|. I verified that 4. holds true a<0, b<0 (with a few examples).

I questioned about |a-b| and whether it is <,>,=, <=, or greater than or equal to |a|-|b|. I decided to see whether the signs mattered (whether a<0 or a>0 and b<0 and b>0 mattered) and whether the order of "a" and "b" matterred.

With a couple of examples, I found that |a-b| is greater than or equal to |a|-|b|. I found that if 'a' and 'b' differ in signs, then |a-b| > |a|-|b|. However, whether the inequality was "equal" or "greater than" varied with the order in which 'a' and 'b' is subsituted:

When a,b are positive numbers and a<b, the inequality statement is true when |a-b| > |a|-|b|. However, if a and b were reversed in the inequality statement, the statement is true when |b-a| = |b|-|a|.

When a,b are negative numbers and a<b, the inequality statement is true when |a-b| = |a|-|b|. However, if a and b were reversed in the inequality statement, the statement is true when |b-a| > |b|-|a|.

So am I right to conclude that:
(1) |a-b|=>|a|-|b|
(2)

When a,b are positive numbers and a<b, the inequality statement is true when |a-b| > |a|-|b|. However, if a and b were reversed in the inequality statement, the statement is true when |b-a| = |b|-|a|.

When a,b are negative numbers and a<b, the inequality statement is true when |a-b| = |a|-|b|. However, if a and b were reversed in the inequality statement, the statement is true when |b-a| > |b|-|a|.

2. Re: Inequantity be/tn absolute value of difference and the difference of absolute val

$|a+b|\le|a|+|b|$ holds for all $a,b\in\mathbb R.$

On the other hand, we have $|a|=|a-b+b|\le|a-b|+|b|$ so $|a-b|\ge |a|-|b|$ and you can do the same by writing $|b|=|b-a+a|.$

3. Re: Inequantity be/tn absolute value of difference and the difference of absolute val

What do you mean? I asked about whether |a-b|>= |a|-|b| holds true for all real a,b.

4. Re: Inequantity be/tn absolute value of difference and the difference of absolute val

It does, as I showed above.