1. **State the problem:** Solve the inequality $$\frac{3}{x} - 5 < 16$$ for $$x$$. 2. **Add 5 to both sides:** $$\frac{3}{x} - 5 + 5 < 16 + 5$$ $$\frac{3}{x} < 21$$ 3. **Multiply both sides by $$x$$:** We must consider the sign of $$x$$ because multiplying by a negative number reverses the inequality. - If $$x > 0$$: $$3 < 21x$$ Divide both sides by 21: $$\frac{3}{21} < x$$ $$\frac{1}{7} < x$$ - If $$x < 0$$: Multiplying by $$x$$ (negative) reverses inequality: $$3 > 21x$$ Divide both sides by 21: $$\frac{1}{7} > x$$ 4. **Check domain:** Since $$x$$ is in the denominator, $$x \neq 0$$. 5. **Combine results:** - For $$x > 0$$, $$x > \frac{1}{7}$$ - For $$x < 0$$, $$x < 0$$ but also $$x < \frac{1}{7}$$ is always true since $$\frac{1}{7} > 0$$. 6. **Test values to confirm:** - For $$x = 1$$ (positive and > $$\frac{1}{7}$$): $$\frac{3}{1} - 5 = -2 < 16$$ true. - For $$x = 0.1$$ (positive but < $$\frac{1}{7}$$): $$\frac{3}{0.1} - 5 = 30 - 5 = 25 \not< 16$$ false. - For $$x = -1$$ (negative): $$\frac{3}{-1} - 5 = -3 - 5 = -8 < 16$$ true. 7. **Final solution:** $$x < 0$$ or $$x > \frac{1}{7}$$. **Answer:** $$x < 0 \text{ or } x > \frac{1}{7}$$