Outside, with a Fraction

Take $\frac{\left|x+1\right|}{2} \ge 1$. The absolute value is divided by 2, so multiply both sides by 2 first to get $\left|x+1\right| \ge 2$. The sign is "greater than or equal to", so there are two cases.

$$x+1 \ge 2$$

$$x \ge 1$$

or

$$x+1 \le -2$$

$$x \le -3$$

The solution is $x \le -3$ or $x \ge 1$.

Get the bars alone, split into two cases, solve each case, and keep the word "or" between them. A fraction outside the bars is one more step to do. Remove it first, before the split, by multiplying both sides.