|3x−1|+|1−3x|=6
|3x−1|+|−3x+1|=6
Step 1: Add -|-3x+1| to both sides.
|3x−1|+|−3x+1|+−|−3x+1|=6+−|−3x+1|
|3x−1|=−|−3x+1|+6
Either 3x−1=−|−3x+1|+6 or 3x−1=−(−|−3x+1|+6)

Part 1: 3x−1=−|−3x+1|+6
(Flip the equation)
−|−3x+1|+6=3x−1
−|−3x+1|+6+−6=3x−1+−6(Add -6 to both sides)
−|−3x+1|=3x−7
−|−3x+1|
−1
=
3x−7
−1
(Divide both sides by -1)
|−3x+1|=−3x+7
We know either−3x+1=−3x+7or−3x+1=−(−3x+7)

−3x+1=−3x+7(Possibility 1)
−3x+1+3x=−3x+7+3x(Add 3x to both sides)
1=7
1−1=7−1(Subtract 1 from both sides)
0=6

−3x+1=−(−3x+7)(Possibility 2)
−3x+1=3x−7(Simplify both sides of the equation)
−3x+1−3x=3x−7−3x(Subtract 3x from both sides)
−6x+1=−7
−6x+1−1=−7−1(Subtract 1 from both sides)
−6x=−8
−6x/ -6 = -8/-6
(Divide both sides by -6)
x= 4/3
Check answers. (Plug them in to make sure they work.)
x= 4/3
(Works in original equation)
Part 2: 3x−1=−(−|−3x+1|+6)
(Flip the equation)
|−3x+1|−6=3x−1
|−3x+1|−6+6=3x−1+6(Add 6 to both sides)
|−3x+1|=3x+5
We know either−3x+1=3x+5or−3x+1=−(3x+5)
−3x+1=3x+5(Possibility 1)
−3x+1−3x=3x+5−3x(Subtract 3x from both sides)
−6x+1=5
−6x+1−1=5−1(Subtract 1 from both sides)
−6x=4
−6x/ - 6 = 4/ -6
(Divide both sides by -6)
x= -2/3
−3x+1=−(3x+5)(Possibility 2)
−3x+1=−3x−5(Simplify both sides of the equation)
−3x+1+3x=−3x−5+3x(Add 3x to both sides)
1=−5
1−1=−5−1(Subtract 1 from both sides)
0=−6

Check answers. (Plug them in to make sure they work.)
x=
−2
3
(Works in original equation)

Answer:
x=
4
3
 or x=
−2
3