Question 781249: 4-x=|2-3x|
Answer by DrBeeee(684)   (Show Source):
You can put this solution on YOUR website! Please refer to other solutions using absolute value.
The "trick" is to make the following substitution;
(1) |y| = y when y>=0 and
(2) |y| = -y when y<=0
For your problem, (1) becomes
(3) 4 - x = +(2 - 3x) when 2 - 3x > 0 or
(4) 4 - x = +(2 - 3x) when 2 > 3x or
(5) 4 - x = +(2 - 3x) when x < 2/3
Now solve (5) for x and get
(6) 4 - x = 2 - 3x or
(7) 2 = -2x or
(8) x = -1
And since -1 < 2/3 the solution is x = -1.
For the other condition given by (2) we have
(9) 4 - x = -(2 - 3x) when 2 - 3x < 0 or
(10) 4 - x = -(2 - 3x) when 2 < 3x or
(11) 4 - x = -(2 - 3x) when x > 2/3
Now solve (11) for x and get
(12) 4 - x = -2 + 3x or
(13) 6 = 4x or
(14) x = 3/2
And since 3/2 > 2/3 the solution is x = 3/2.
Answer: x = -1 when x < 2/3, and x = 3/2 when x > 2/3
How about when x = 2/3? The answer can be -1 or 3/2 depending on the "side" of 2/3 that x is an infinitesimal distance from 2/3. If x was greater than 2/3 before it equaled 2/3 the answer is 3/2. If it was less than 2/3 before it equaled 2/3 the answer is -1. The point x = 2/3 is called a point of discontinuity. x "jumps from -1 to 3/2 instantaneously (zero distance) as x increases from <2/3 to x>2/3.
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