Rejecting a Solution to a Modulus Question

Question

Why is the solution of $|1+3x|<6x$ only $x>1/3$? After applying the properties of modulus, I get $-6x<1+3x<6x$. And after solving each inequality, I get $x>-1/9$ and $x>1/3$, but why is $x>-1/9$ rejected?

Answer 1

$$6x>|1+3x|\ge0\implies x>0$$

Answer 2

You need $x$ to be both bigger than $1/3$ and bigger than $-1/9$. But clearly anything bigger than $1/3$ is automatically bigger than $-1/9$ as $1/3>-1/9$. So you only keep that solution.