Expert: Sherry Wallin - 4/30/2010

Question
QUESTION: [x-1] + [2x+3] = 2
find x

[] stands for modulus

ANSWER: What mod are you using? If x = 1 and the mod is 3 then you have [1-1] + [2(1) + 3] = 2 mod 3

Math Prof

---------- FOLLOW-UP ----------

QUESTION: It is not 2 mod 3 ,it is just 2
The mod i am refering is the one in which we take the absolute value of the number
e.g. [-2]=2

Answer
Ok, here in America we just use vertical bars to talk about absolute value, so the problem would read like:  
| x - 1 | + | 2x + 3 | =  2.
This problem has no solution but I imagine your teacher wants to know how you know this to be true so the following will show your equation has no solution:

Rewrite the equation like so: |x-1| = 2 - |2x+3|


First find where each absolute value is zero: x-1=0 -> x = 1
2x+3 = 0 -> x = -3/2

Take the number line and break it up into intervals with these 'break' points:

intervals         (-00,-3/2)  (-3/2, 1)   (1, 00)
 
interval numbers     #1          #2        #3

In interval #1 both absolute value parts will be negative so remove the vertical bars and take the negative of the expression (or the opposite means the same thing):
-(x-1) = 2 - (2x+3)  **it is important and absolutely mandatory to understand this will only give
         you results for the specific interval you are in.

-x+1 = 2 - [-(2x -3)]
-x + 1 = 2 + 2x + 3
-x + 1 = 2x + 5
+x       +x
1 = 3x + 5
-5     -5
-4 = 3x
/3   /3
-4/3 = x          **NOTE -4/3 is NOT in the interval (-00, -3/2) so it can't be a solution

In interval #2 |x-1| will be negative and |2x+3| will be positive so remove the vertical bars and take the opposite of the first -(x-1) and the positive of -(2x+3)
-(x-1) = 2 -(2x+3)
-x + 1 = 2 -2x -3
-x +1 = -1 -2x
+2x        +2x
x + 1 = -1
   -1  -1
x = -2          **NOTE -2 is not in the interval (-3/2, 1) so it can't be a solution

In interval #3 both absolute value parts will be positive so just remove the vertical bars:
x - 1 = 2 - (2x+3)
x - 1 = 2 - 2x -3
x -1 = -1 -2x
+2x       +2x
3x -1 = -1
  +1   +1
3x = 0
/3   /3
x = 0          **NOTE 0 is NOT in the interval (1,00) thus there is no solution to the
         original absolute value equation. There is no value of x that makes the
         statement true.

Math Prof