Here it works, not because of the possibility of -1 in the first step but because of the inversion in the third step.
1/5 < 1/(x+1) < 1/2. Well, you see if you can invert 1/(x+1), you can easily solve it. But before inverting inequality, you must check all parts of the inequality to ensure that none can be negative. Here 1/5, 1/2 cannot be negative. So, 1/(x+1) also cannot be. Hence, you can safely invert the inequality by reversing the sign:
5>(x+1)>2. If you find a variable portion can be negative or positive, you must stop inverting.