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

2 Answers

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  • Alan
    Lv 7
    4 weeks ago

    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 

    answer is : 

    -3 < x<  2  

     

  • Anonymous
    4 weeks ago

    I’m posting my answer as a photo since it won’t post otherwise. Anyways I get in case 1 that x is greater than -3 (and x can of course not exceed 1). In case 2 there is no solution and in case 3 that x is less than 2 (and of course x can’t be below 1).

    Attachment image
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