Mba-day29 arithmetic - preliminary understanding of absolute value

1. Basic knowledge points

1.1 Definition

The absolute value |a| The law of undressing , Take off the symbol

|a| = {
    a  (a  > 0)
    0  (a = 0)
    -a (a < 0)
}

for example ：|x - 3|

1)  if  x > 3  when ,  be  |x - 3| = x -3
2)  if  x < 2  when ,  be  |x - 3| = 3 - x

1.2 Geometric meaning

|a| representative a The distance from the point to the origin , namely |a| >= 0, namely |a| Constant greater than 0

• for example |x - 2|, representative x To 2 Distance between
• for example |x + 4|, representative x To -4 Distance between

1.3 nature

1. symmetry
   |-a| = |a|

2. Equivalence
   Radical sign (a^2) = |a|

3. Self comparability
   x / |x| = |x| / x

x / |x| = |x| / x = {
     if x > 0,  Then for  1
     if x < 0,  Then for  -1
}

4. Nonnegativity
   |a| >= 0

 Examination site ： Common nonnegativity 
1.  Distance , |a| >= 0, Then it will always be greater than  0
2. a^2, a^4, a^8....a^2n >=0
3.  Radical sign a,  Radical sign 4 Power a

-> |a| + b^2 +  Radical sign c = 0
->  Implication ：a>=0  b>=0  c>=0
-> a = 0, b = 0, c = 0

2011 The real title of , seek |a-1| + (b-2)^2 + Radical sign （c -4）= 0, have to a = 1, b = 2, c = 4

5. inequality

• |f(x)| < a (a > 0) -> -a < f(x) < a

for example |x| < 2, Introduction -> -2 < x < 2

• |f(x)| > a (a > 0) -> f(x) > a or f(x) < -a

for example ： seek |x-1| > 2

x - 1 > 2  or  x - 1 < -2
 so  x > 3  or  x < -1