Question

The inequality $\frac{2}{x} < 3$ is true, when $x$ belongs to

• A. $ \left[\frac{2}{3},\,\infty \right)$
• B. $ \left(-\infty ,\,\frac{2}{3} \right]$
• C. $\left(-\infty,\,0 \right)\cup \left(\frac{2}{3},\,\infty \right)$
• D. None of these

Answer 1

Answer: (C)

Case $I$ :
When $x > 0, \frac{2}{x} < 3 \Rightarrow 2 < 3x \Rightarrow \frac{2}{3} < x$ or $x >\frac{2}{3}$
Case $II$ :
When $x > 0$, $\frac{2}{x} < 3 \Rightarrow 2 < 3x \Rightarrow \frac{2}{3} > x$ or $x <\frac{2}{3}$
which is satisfied when $x < 0$.
$\therefore \quad x\in \left(-\infty,\,0\right) \cup \left(\frac{2}{3}, \,\infty\right)$.