To solve these inequalities, we will break them down one by one:

1) 2x - 1 < x + 3
First, we want to simplify the inequality by moving all terms to one side:
2x - x < 3 + 1
x < 4

2) 5x - 1 > 6 - 2x
Again, we simplify the inequality by moving terms to one side:
5x + 2x > 6 + 1
7x > 7
x > 1

3) x - 3 < 0
This inequality is already simplified, so we can solve for x:
x < 3

Therefore, the solution to the system of inequalities is x < 3 and x > 1, which means x is between 1 and 3.