Module Question

|3x - 1|  =  |2x + 6|

Either     3x - 1  =  2x + 6     or     3x - 1  =  -(2x + 6)

Can you finish it from here?

In response to geno3141 (Idk how to reply to your comment. Sorry)

I do know how to finish from there, but as of my understanding, I needed to do:

+ and +   

+ and -

- and -

- and +

You gave me only these solutions:

+ and +             

+ and -

Could you explain why?

Note, guest that  the  - /-  solution is

- (3x - 1)  = - (2x + 6)

But multiplying through by -1 would produce the first solution that geno presented

And the  -/+   solution is

- (3x - 1)  = (2x + 6)

Again, mutiplying through by -1 produces geno's second solution

So.....these other two are superfluous........

I'll give the explanation a go :)

|3x-1|=|2x+6|

+ and +           3x-1 = 2x+6      (1)

+ and -           3x-1 = - ( 2x+2)    (2) 

- and -           -  (3x-1) = - ( 2x+2)    (3) 

- and +           - (3x-1) = + ( 2x+2)    (4) 

If you look at these 4 options you can see that  

1 and 3 are really the same

and

2 and 4 are really the same.   :)

So there are really only 2 distict options :)

Does that explain it good enough for you?

|3x-1|=|2x+6|

\(\begin{array}{|rcll|} \hline |3x-1| &=& |2x+6| \quad & | \quad \text{square both sides} \\ (3x-1)^2 &=& (2x+6)^2 \\ 9x^2-6x+1 &=& 4x^2+24x+36 \\ 9x^2-4x^2-6x-24x+1-36 &=& 0 \\ 5x^2-30x-35 &=& 0 \\ 5x^2-30x-35 &=& 0 \quad & | \quad : 5 \\ x^2-6x-7 &=& 0 \\ (x-7)(x+1) &=& 0 \\ \hline \end{array} \)

x = 7 or x = -1

check:

\(\begin{array}{|rcll|} \hline |3x-1| &=& |2x+6| \quad & | \quad x=-1 \\ |3\cdot(-1)-1| &=& |2\cdot(-1)+6| \\ |-4| &=& |4| \\ 4 &=& 4 \ \checkmark \\\\ |3x-1| &=& |2x+6| \quad & | \quad x=7 \\ |3\cdot7-1| &=& |2\cdot7+6| \\ |20| &=& |20| \\ 20 &=& 20 \ \checkmark \\ \hline \end{array}\)

Thank you so much everyone for answering. I'm very grateful.