APPENDIX A - Numbers, Inequalities, and Absolute Values - A Exercises - Page A9: 38

$x \lt -\frac{1}{3}~~~$ or $~~~x \geq 1$ The solution set is $~~~(-\infty, -\frac{1}{3}) \cup [1, \infty)$

$-3 \lt \frac{1}{x} \leq 1$ Case 1: $x \lt 0$ $-3 \lt \frac{1}{x}$ $-\frac{1}{3} \gt x$ Case 2: $x \gt 0$ $\frac{1}{x} \leq 1$ $x \geq 1$ Solution: $x \lt -\frac{1}{3}~~~$ or $~~~x \geq 1$ The solution set is $~~~(-\infty, -\frac{1}{3}) \cup [1, \infty)$