|6 - 2x|= 3x + 1

Clear the fraction by multiplying both sides by 2

  |6 - 2x| = 6x + 2

There are two cases to consider:

Case 1:  When the value of 6 - 2x has the same value as the right side.

  6 - 2x = 6x + 2
     -8x = -4
       x = 

Case 2:  When the value of 6 - 2x has -1 times the value of the right side.

  6 - 2x = -(6x + 2)
  6 - 2x = -6x - 2
      4x = -8
       x = -2

There appear to be two solutions  and -2. 

However we must check them to see that they satisfy the
original equation.

Checking 

|6 - 2x|= 3x + 1

|6 - 2()|= 3() + 1

|6 - 1|= 3() + 1

|5| = () + 1

(5) =  + 

 = 

That checks, so  is a solution.

Checking -2

|6 - 2x|= 3x + 1

|6 - 2(-2)|= 3(-2) + 1

|6 + 4|= -6 + 1

|10| = -5

(10) = -5

   5 = -5

So -2 is not a solutionh and we discard it.

The only solution is 

Edwin