In this problem |-3x+1| represents the absolute value of -3x+1. This means |-3x+1|= -3x+1 if -3x+1 is positive and |-3x+1|= -(-3x+1)= 3x-1 if -3x+1 is negative.

To solve 2*|-3x+1|=8:

First let us assume -3x+1 is positive, this gives -3x+1>0 or 3x<1 or x< 1/3.

Now 2*|-3x+1|=8

=> 2*(-3x+1)=8

=> -6x+2=8

=> -6x=6

=>x=-1.

As -1< (1/3), this is valid. **Therefore x=-1**

Next we assume -3x+1 is negative, or -3x+1<0 or 3x>1 or x>(1/3)

Now 2*|-3x+1|=8

=>2*[-(-3x+1)]=8

=>2*(3x-1)=8

=>6x-2=8

=>6x=10

=>x= (10/6 )= (5/3)

As in this case x>(1/3), x=5/3 is also valid.

**Therefore the valid values are -1 and 5/3.**

To solve 2|-3x+1 = 8

Solution:

When -3x +1> 0 or x < -1/3., |-3x+1| is equal to -3x+1.

So the equation is:

2( -3x+1 )= 8.

Subtrat 2 from both sides:

-6x = 8-2 = 6

Divide by -6:

-6x/-6 = 6/-6

x= -1

When -3x+1 < 0, or when x > 1/3,

|-3x+1| = 3x-1.

So the given equation becomes:

2(3x-1) = 8

Add 2 to both sides:

6x = 8+2 = 10

Divide by 6:

6x/6 = 10/6 = 5/3

x = 3.