f.4 a.math 高手入

i)
For | 2 - 5x | - | x + 2 | = x
Case I : x<-2
| 2 - 5x | - | x + 2 | = x
(2-5x)-[-(x+2)]=x
4-4x=x
x=4/5 (rejected , as 4/5>-2)
Case II : -2<=x<=2/5
| 2 - 5x | - | x + 2 | = x
(2-5x)-(x+2)=x
-6x=x
x=0
Case III: x>2/5
| 2 - 5x | - | x + 2 | = x
-(2-5x)-(x+2)=x
4x-4=x
x=4/3
Combine the 3 cases , the solution is x>=-2
ii)
| |x| - 2 | = 3
|x| - 2=3 or |x| - 2=-3
|x|=5 or |x|=-1 (rejected as |x|>=0)
x=-5 or x=5


2009-07-30 23:24:01 補充：
Correction for question 1 ,
the solution is x=0 or x=4/3

2009-07-30 23:30:38 補充：
Consider three cases :

1: x<-2 => | 2 - 5x | =2-5x and | x + 2 |=-(x+2)
2: -2<=x<=2/5 => | 2 - 5x | =2-5x and | x + 2 |=x+2
3: x>2/5 => | 2 - 5x | =-(2-5x) and | x + 2 |=x+2

After considering three cases , there are three different combinations for | 2 - 5x | and | x + 2 | ~

(i).
| 2 - 5x | - | x + 2 | = x
Consider the following cases:(1) when x＜-2
　　　　　　　　　　　　　　(2) when -2＜＝x＜＝2/5
　　　　　　　　　　　　　　(3) when x＞2/5
Cases (1):the equation becomes
　　(2-5x) + (x+2) = x
　　　　　　4-4x = x
　　　　　　　x = 4/5　　　(rejected)　(x＜-2)
Cases (2):the equation becomes
　　　(2-5x) - (x+2) = x
　　　　　　　-6x = x
x is no answer
Cases (3):the equation becomes
　　-(2-5x) - (x+2) = x
　　　　　　4x-4 = x
　　　　　　　x = 4/3
Combining the three cases,x = 4/3

(ii).
| |x| - 2 | = 3
　| x | - 2 = 3　　　　　or 　　　| x | -2 = -3
　　| x | = 5　　　　　 or 　　　　| x | = -1　　(rejected)
　　　x = 5　or x = -5

for (i)
combine all 3 cases, the solution is x=0 or x=4/3