Need a simple absolute value question explained

[sch]

Hi, it was 6 years since my last math course, and even then, it wasnt much since I am a political science major. In any case, as I am going through MGMAT guides (im on 3rd one now) I have a question on an example of an absolute value that MGMAT guide 3 (third edition).

The question is if |x-2|=|2x-3|, what are the possible values for x? There are two possibilities for x where the equations have the same signs and other is where one sign is negative. I have no problems when it is a same sign, nor do I have problems solving for x. What throws me off is when the negative value of the equation, which is equal to x=5/3 gets plugged in to check the answer. So the whole problem goes like that:
Case B.
(x-2)=-(2x-3) -----> 3x=5-----> x=5/3
Checking validity by plug in:
|5/3-2|=|2(5/3)-2| both sides equal 1/3 How can that be? Left hand side is ok because it is equal to |-1/3|. But when plugging in, how did they get -2 instead of -3 on the right side when original equations clearly states |2x-3|? I need that to be explained; and even so, wouldnt rest of it go like that: 2(5/3)=10/3 thus 10/3-2(1/3)= 8/3? or if there was -3 then equation would equal 7/3, and both sides are not equal to eachother?

[iamseer]

how did they get -2 instead of -3 on the right side when original equations clearly states |2x-3|?

A printing error which skipped the eye of the review team. It should be 3 and NOT 2.

Checking validity by plug in:
|5/3-2|=|2(5/3)-2| both sides equal 1/3 How can that be?

RHS = |-1/3|=1/3
LHS = |2(5/3)-3|=|10/3-3|=|1/3|=1/3

if there was -3 then equation would equal 7/3

not really. It would be 1/3.

absolute value is nothing but distance of that point on the number line from zero. so |-1/3|=|1/3|=1/3 since both points on the number line are at distance 1/3 from zero.

So, when 2 absolute values are equal then either
(number on right hand side) =(number on left hand side) or
(number on right hand side) =-(number on left hand side)

HTH

[kstv]

|x-2|=|2x-3|
squaring both sides
(x-2)²=(2x-3)²
x²-4x+4 = 4x²-12x+9
3x²-8x+5=0
3x²-3x-5x+5=0
3x(x-1)-5(x-1)=0
x = 1, 5/3