Solve |3x+1| = 1

Firstly square both sides. This way, we now know that both sides have to be positive

(3x+1)^2 = 1

(3x+1)(3x+1) = 1

Expand the quadratic...

9x^2 + 12x + 4 = 1

9x^2 + 12x + 3 = 0

Now complete the square. Firstly divide everything by 12 to make this simpler

x^2 + (4/3)x + 1/3 = 0

(x + (2/3))^2 -4/9 + 1/3 = 0

(x + (2/3))^2 = 1/9

Square root both sides, be sure to have give the negative value on RHS too

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

x=-1/3 x=-1