What to conclude from $x y \le N$ and $x\le 1$?

Question

For real $x, y, N$, what can we conclude for $y$ from two inequalities:

$$x y \le N\qquad {\rm and} \qquad x\le 1?$$

Answer 1

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

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

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

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

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

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

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