What to conclude from $x y \le N$ and $x\le 1$? For real $x, y, N$, what can we conclude for $y$ from two inequalities:
$$x y \le N\qquad {\rm and} \qquad x\le 1?$$

Let $x \ne 0$. Then you have six cases following from $x y \le N$:

*

*For $0 < x \le 1$ and $N \ge 0$ you have $0  \le y \le \frac{N}{x}$ .


*For $0 < x \le 1$ and $N < 0$ you have $  y \le \frac{N}{x} < N < 0 $ .


*For $-1 \le x <0$ and $N \ge 0$ you have $0  \ge y \ge \frac{N}{x}$ .


*For $-1 \le x <0$ and $N < 0$ you have $0 \le -N  \le \frac{N}{x} \le y$ .


*For $x < -1 $ and $N \ge 0$ you have $0 \ge y \ge\frac{N}{x} > -N$ .


*For $x < -1 $ and $N < 0$ you have $  y \ge  \frac{N}{x}  \ge 0$ .
