How do you solve x+3=|3x1|?

1 Answer
Apr 7, 2015

Consider two possibilities which are significant for the absolute value term

Possibility 1: (x13)
In this case (3x1)0
and the equation can be re-written
x+3=((3x1))
x+3=3x1
2x=4
x=2
Note that this is an extraneous solution since it does not exist within the range of values for Possibility 1: (x13)

**Possibility 2: (x>13)
In this case (3x1)>0
and the equation can be re-written
x+3=(3x1)
4x=2
x=12
Note that once again we have an extraneous solution sin it does not exist within the range of values for Possibility 2: (x>13)

To see why this happens consider a re-arrangement of the original equation into the form:
x+3+|3x1|=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]}