Answer
$ (2, \infty ) $
Work Step by Step
We break this into two different inequalities.
$$x+3 \lt 2x+1$$
$$and $$
$$ 2x+1 \lt 4x + 6 $$
Then, follow PEMDAS. In other words, first address parentheses, and then resolve exponents. Next, simplify division and multiplication, and lastly address subtraction and addition. This gives:
$$x \gt 2$$
$$and $$
$$ x \gt -5/2 $$
We are only considering the intersection of these two sets, so we know that $ (2, \infty ) $ is the correct answer.