Q:

How to solve an inequality with absolute values on both sides? 

I have the following inequality:
$$|x+3| \geq |x-1| $$
Following this answer I get:
$$
|x+3|=\left\{ \begin{align}
x+3 & \text{   , if }x\geq -3 \\
-x-3 & \text{   , if }x <-3 
\end{align}
\right\}
$$
$$
|x-1|=\left\{ \begin{align}
x-1 & \text{   , if }x\geq 1 \\
-x+1 & \text{   , if }x < 1 
\end{align}
\right\}
$$
Putting those together I get 3 sets of equations:
For $x<-3$:
$$-x-3\ge-x+1$$
For $-3 \le x< 1$:
$$x+3\ge-x+1$$
For $x\ge1$:
$$x+3\ge x-1$$
The first inequality however gives me: 
$$-3 \ge 1$$
What am I doing wrong here?  The answer to the problem is $x \ge -1$ by the way. 

A:

You haven't done anything wrong.  Getting $-3\ge 1$ in the first case means that the first case cannot occur, so there is no solution in the interval $(-\infty, -3)$.

