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 $x<0$.
$\therefore x\in \left(-\infty ,0\right)\cup \left(\frac{2}{3},\infty \right)$.