Your logic is wrong. Yes, because x can't be 0 on the left, we must have x> 0 or x< 0. Since x> 0 does not satisfy the inequality, we must have x< 0.

NOT necessarily

**all** x< 0.

You haven't used the full argument.

if x< 0, multiplying by x reverses the inequality:]

Subtracting x+ 1 from both sides

In order that the product of two numbers be negative the two numbers must have opposite signs. Either

x-1> 0 and x+ 1< 0 or x-1< 0 and x+ 1> 0.

x- 1> 0 gives x>1 but remember that we are assuming x< 0 so this is not a valid solution.

x-1< 0 gives x< 1 but since x< 0, that gives nothing new. x+1> 0 gives x> -1. -1< x< 0 satisfies both conditions.

The solution set to the inequality is -1< x< 0.