You need to break the equation into 2 equations and solve them separately.

The following equation:

|6x|=|9x+1|

Breaks into two equations according to the definition for absolute value:

|9x+1|​ = 6x​
and
|9x+1|​ = - 6x​​

Now, use the solver to solve:

|9x+1|​ = 6x​
........
Click here to see the step by step solution
........
This equation has no solutions.

|9x+1|​ = - 6x​
..........
Click here to see the step by step solution
..........
x1 = -1/3
x2 = -1/15

The two solutions for the equation are -1/3 and -1/15.

You can also graph:

y = |6x| and y = |9x + 1|

and find the solution in the intercept:

Click here to see the graph.