Question

$|2x - 3| < |x + 5|$, then $x$ belongs to

• A. $(-3, 5)$
• B. $(5, 9)$
• C. $\left(-\frac{2}{3}, 8\right)$
• D. $\left(-8, \frac{2}{3}\right)$

Answer 1

Answer: (C)

$\left(-\frac{2}{3}, 8\right)$

Answer 2

We have, $|2x - 3| < |x + 5|$ $\Rightarrow |2x - 3| - |x + 5| < 0$ $\Rightarrow \begin{cases} 3-2x+x+5<0, & \quad \text x \le -5 \\\ 3-2x-x-5<0, & \quad \text -5, < x \le \frac{3}{2} \\\ 2x-3-x<0, & \quad \text x >\frac{3}{2} \end{cases}$ $\Rightarrow \begin{cases} x>8, x\le-5 & \quad \text{ } \\\ x>\- \frac{2}{3}, & \quad \text{} -5 < x \le \frac{3}{2} \\\ x<8, & \quad \text{} x > \frac{3}{2} \end{cases}$ $\Rightarrow x\in\left(-\frac{2}{3}, \frac{3}{2}\right)\cup\left(\frac{3}{2}, 8\right)$ $\Rightarrow x\in \left(-\frac{2}{3}, 8\right)$