The formula of absolute inequality in senior high school mathematics is ||||| A |-| B || ≤| AB |≤| A |+| B |. |a| is the absolute value of the number A when the distance from point A on the exponential axis to the origin is called. When the signs of A and B are the same, 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 a and b are different symbols, they are located on both sides of the origin, and the distance between a and b is less than the sum of their distances to the origin. ?

Two important properties of absolute value inequality;

1、ab|=|a||b|

|a/b|=|a|/|b|(b≠0)[ 1]？

2. | a | & lt|b| Reversible push |b| >|a|

||||| A |-| B |||||≤| A+B |≤| A |+B |, the left equal sign holds if and only if ab≤0, and the right equal sign holds if ab≥0.

? Deduction process of inequality of absolute value of extended data | | | a |-| b | |≤| a b |≤| a |+| b |；;

We know that |x|={x, (x >；; 0); x，(x = 0)； -x，(x & lt0);

Therefore, there are:

-|a|≤a≤|a|......①

-|b|≤b≤|b|......②

-|b|≤-b≤|b|......③

From ①+②:

-(|a|+|b|)≤a+b≤|a|+|b|

That is | a+b |≤| a |+b |...④

From ①+③:

-(|a|+|b|)≤a-b≤|a|+|b|

That is | a-b |≤| a |+| b |...⑤

Another one:

|a|=|(a+b)-b|=|(a-b)+b|

|b|=|(b+a)-a|=|(b-a)+a|

Learn from ④:

| a | = |(a+b)-b |≤| a+b |+|-b | = & gt； |a|-|b|≤|a+b|.......⑥

| b | = |(b+a)-a |≤| b+a |+|-a | = & gt； |a|-|b|≥-|a+b|.......⑦

| a | = |(a-b)+b |≤| a-b |+| b | = & gt； |a|-|b|≤|a-b|.......⑧

| b | = |(b-a)+a |≤| b-a |+| a | = & gt； |a|-|b|≥-|a-b|.......⑨

From ⑥ and ⑥:

||a|-|b||≤|a+b|......⑩

Starting from ⑧ and ⑨:

||a|-|b||≤|a-b|......？

Comprehensive ④ ⑤ ⑩? An important inequality about absolute value is obtained: | a |-| b |≤| a b |≤| a |+b |.

References:

Baidu Encyclopedia-Absolutely unequal