The formula of absolute inequality is ||a|-| b |≤| ab |≤| a |+b |, where | a | indicates that the distance from point A on the number axis to the origin is called the absolute value of number A. In the application of inequality, it often involves mass, area and volume, and also involves the size or absolute value of some mathematical objects (real numbers and vectors). Are measured by non-negative numbers.
The derivation process of absolute value important inequality;
We know that |x|={x, (x >；; 0); x，(x = 0)； -x，(x & lt0);
Therefore, there are:
①-|a|≤a≤|a|
②-|b|≤b≤|b|
③-|b|≤-b≤|b|
From ①+②:
④-(|a|+|b|)≤a+b≤|a|+|b|, that is, |a+b|≤|a|+|b|
From ①+③:
⑤-(| A |+| B|) ≤ A-B ≤||| | A |+| B |, that is, |a-b|≤|a|+|b|
In addition: |a|=|(a+b)-b|=|(a-b)+b|, |b|=|(b+a)-a|=|(b-a)+a|
Learn from ④:
⑥|a|=|(a+b)-b|≤|a+b|+|-b|= >|a|-|b|≤|a+b|
⑦|b|=|(b+a)-a|≤|b+a|+|-a|= >|a|-|b|≥-|a+b|
⑧|a|=|(a-b)+b|≤|a-b|+|b|= >|a|-|b|≤|a-b|
⑨|b|=|(b-a)+a|≤|b-a|+|a|= >|a|-|b|≥-|a-b|
From ⑥, ⑥: ||| A |-| B | ≤| A+B | ... Attending
From ⑧, ⑨: ||| A | | B || ≤| A-B | ...
Comprehensive ④ ⑤ ⑩? An important inequality about absolute value is obtained: | a |-| b |≤| a b |≤| a |+| b |.
Attention should be paid to the conditions under which the equal sign holds (especially for the maximum value), namely:
|a-b|=|a|+|b|→ab≤0
|a|-|b|=|a+b|→b(a+b)≤0
|a|-|b|=|a-b|→b(a-b)≥0
Similarly |a|-|b|=|a-b|→b(a-b)≥0.
Geometric significance of absolute value inequality
When a and b have the same sign, they are on the same side of the origin. At this time, the distance between a and b is equal to the sum of their distances to the origin. When the signs of A and B are different, they are located on both sides of the origin. At this time, the distance between A and B is less than the sum of their distances to the origin. (|a-b| means the distance from a-b to the origin, and also means the distance from A to B).
| a | & lt|b| Reversible push |b| >|a|, || a |-| b |||≤| a+b |≤| a |+| b |, the left equal sign is true if and only if ab≤0 and the right equal sign is true if ab≥0. In addition, there are: | a-b |≤| a | +|-b | =| a |+|-1| | b | = | a |+| b |, | | A |-| B | |≤A B | | | B | | | A B | | B | | A B | | | B | | B | | A B | | | B | | A B | | | B | | B | | B | | B | | A B | | | B | | B | | B | | B | | | B | | B | | B | | B | | B | | B | | B | | B | | B | | B | | B | | B