Already posted are several practical algebraic methods for dealing with the problems. What follows is less practical, but may be a useful idea in other settings.
In general, $|x-a|$ is the distance between $x$ and $a$. Now let's look for example at Problem $2$.
We are looking at $|x-(-1)| + |x-3|$. Draw a "number line" and put dots at $-1$ and at $3$. Then $|x-(-1)| + |x-3|$ is the sum of the distances from $x$ to the two numbers $-1$ and $3$.
This sum is clearly $4$ if $x$ is between $-1$ and $3$. Now let us start at $x=3$ and imagine $x$ moving slowly to the right. Then the sum of the distances is at first $4$, and then steadily increases, becoming large after a while. Now start at $-1$ and move slowly to the left. Again, the sum of the distances is at first $4$, and steadily increases.
It follows that the correct answer for ($2$) is $[4,\infty)$.
The idea also works for Problem $3$, but becomes geometrically less natural. Start at $x=4$ and move steadily to the right. Then at first $x+|x-4|$ is $4$, then steadily increases. Now start at $4$ and move steadily to the left. Every increase in the quantity $|x-4|$ is exactly compensated by a decrease in $x$, so the sum $x+|x-4|$ remains unchanged at $4$.
It follows that the correct answer for ($3$) is $[4,\infty)$.