# How do I solve this WITHOUT the graphical method. 2-|x-1| > |x+2|-3?

Relevance
• map out when each absolute value will have negative arguments

| x -1|  =  x-1  for x>= 1

|x-1|  = -(x-1)  for x< 1

| x+ 2|  =  (x+2) for x>= -2

|x+2|  = -(x+2) for x< -2

so to be over cautious

I will take

these 5 sections

x< -2

x= -2

-2< x< 1

x = 1

x> 1

so you must take in sections

for x< -2

2 - - (x-1) >  -(x+2)  -  3

5>  -(x+2) - (x-1)

5 > -2x -2 +1

5 > -2x -1

6- > -2x

6/2 > -x

multiply both side by -1 and flip ">"

x> -3

so x> -3

so combine  x< -2 and x> -3

so far, we have  -3<x<  -2

for x = -2

-1>    -3   which is true for all x

so far, we have

-3<x<= -2

next,    -2<x< 1

2- - (x+1)  >  (x+2) -3

3  +x >  x -1

3>  - 1

so this is true  the whole area

from -2<x< 1

so , so far

-3< x< 1

Next , x = 1

2-  | 1-1| >  |1+2| - 3

2> 0 , yes,

so far,  we have

-3< x<= 1

now, the last section

x> 1

2- (x-1) >  (x+2) -3

3 -x  > x -1

4 > 2x

2> x

so the combination of

2> x and  x> 1

1< x< 2

so combining it all 