Changing sign on inequalities I have a question as to the below.  Why does the sign for $(x + 1)$ change from less than or equal to, to greater than and NOT greater than or equal to.  The instructor here rules out $-1$ as an integer, but I would have included it. Also why is Case II not a possible scenario?
The question is:

What are the integer values of $x$ satisfying
  $$\frac{2x - 3}{x + 1} \le 0 ?$$

Explanation:
We have $\frac{2x - 3}{x + 1} \le 0$, thus we have:
Case I:
$2x - 3 \le 0$ AND $x + 1 > 0\Rightarrow x \le 3/2$ AND $x > -1$. So $-1 < x \le 3/2$
The integer values of $x$ are: $0$ and $1$.
Case II:
$2x - 3 \ge 0$ AND $x + 1 < 0$, so $x \ge 3/2$ AND $x < -1$ which is not a possible scenario.
Thus, the only possible integer values of $x$ are: $0$ and $1$.

The value of  $ (x+1)$ is less or greater than zero, in the two cases. It can never be equal to zero, since  $ (x+1)$  is in the denominator, so for $ (x+1) =0$ the expression is not defined. Therefore, $x = -1$ would not be a solution even if you allow for it.
As for your second question: this case is ruled out since a number cannot simultaneously be $x\ge 3/2$ (positive) AND $x<−1$ (negative).
