In general it is very difficult to just "inspect" your equation $Ax = b$ by hand and tell if there is a solution that will have all entries in $x$ that satisfy certain inequalities. What you CAN do, in this case, is formulate a linear programming feasibility problem:
$$Ax = b, \,\,\,\,\, 0 \leq x_i \leq 1, \,\,\,\, \forall i$$.
And this is possible because all the inequalities on the $x_i$ are linear combinations of the $x_i$ set to be less than or equal or greater than or equal to (or strictly equal to) some constant. There is a lot of free software available online to find a feasible solution to a linear programming problem, and you can even find an "optimal" feasible solution that maximizes $c \cdot x$ for any vector $c$ that you want.
If your vector $x$ is parametrized by an argument $t$ that defines a curve, then the problem is basically impossible to decide without explicitly knowing and being able to work with the form of $x(t)$ as a function of $t$. Just imagine the feasibility region: You can define infinitely many curves $x(t)$ that pass through the feasibility region and infinitely many curves that don't. And they can basically look like anything. How are you going to decide whether an arbitrary given curve $x(t)$ intersects the feasibility region?