You need to solve 

|x+1|<=|x+2|

The definition for absolute value is:

|x|=x if x>=0

|x|=-x if x<0

When x + 1 >= 0 and x+2>=0, that is x >= -1 we can write

x+1 <= x + 2 and this is valid for all x >= -1 

When x + 1 < 0 and x + 2 >= 0 that is -2<=x<-1 we can write

-(x+1)<=x+2 and this is valid for -1 1/2<=x<=-1 

When x+1<0 and x+2<0 that is x < - 2

-(x+1)<=-(x+2) we get 1>=2 which is contradiction

So the solutions are x >= -1 and -1 1/2<=x<=-1 or written as interval:

[-1 1/2, -1] U [-1, ~) = [-1 1/2, ~)