Including the Edge

With $\ge$ the fraction may also equal 0. Take $\frac{x-1}{x+3} \ge 0$. A fraction is 0 when its top is 0, so x = 1 gives 0 over 4, which is 0, and 0 is at least 0. The top's zero belongs in the answer.

The bottom's zero is different. At x = -3 the fraction is -4 over 0, which has no value at all, so -3 is never in the answer. That is why the sign cases keep the bottom strict: the cases are $x-1 \ge 0$ and $x+3 > 0$, or $x-1 \le 0$ and $x+3 < 0$.

The first case gives $x \ge 1$, since any x of at least 1 is also above -3. The second gives $x < -3$. So the answer is $x < -3$ or $x \ge 1$: a closed end at the top's zero, an open end at the bottom's.