How do you solve x+3=-abs(3x-1)?

1 Answer
Alan P.
Apr 7, 2015

Consider two possibilities which are significant for the absolute value term

Possibility 1: (x<=1/3)
In this case (3x-1) <= 0
and the equation can be re-written
x+3 = - (-(3x-1))
x+3 = 3x-1
-2x = -4
x= 2
Note that this is an extraneous solution since it does not exist within the range of values for Possibility 1: (x<=1/3)

**Possibility 2: (x>1/3)
In this case (3x-1) > 0
and the equation can be re-written
x+3 = -(3x -1)
4x = -2
x = -1/2
Note that once again we have an extraneous solution sin it does not exist within the range of values for Possibility 2: (x>1/3)

To see why this happens consider a re-arrangement of the original equation into the form:
x+3 +abs(3x-1) = 0
The graph of the left side of this equation is shown below. Notice that it does not intersect the X-axis (that is it is never equal to 0).
graph{x+3+abs(3x-1) [-16.01, 16.02, -8.01, 8]}

Related questions
See all questions in Absolute Value Equations
Impact of this question
1775 views around the world
You can reuse this answer
Creative Commons License