Question

If $|2x-3|<|x+5|$, then $x$ lies in the interval

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

Answer 1

The correct option is C

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

Explanation for the correct option:

There will be three different cases to find the interval of given equation:

$|2x-3|<|x+5|$

Case 1. If $–\infty <x<–5$

$\Rightarrow$ $–2x+3<–x–5$

$\Rightarrow$ $–x<–8$

$\Rightarrow$ $x>8$ — (1)

Case 2. If $–5<x<\frac{3}{2}$

$\Rightarrow$$–2x+3<–x–5$

$\Rightarrow$ $–3x<2$

$\Rightarrow$ $x>–\frac{2}{3}$

$\Rightarrow$$-\frac{2}{3}<x<\frac{3}{2}$ — (2)

Case 3. If $\frac{3}{2}<x<\infty$

$\Rightarrow$$2x–3<x+5$

$\Rightarrow$ $x<8$

$\Rightarrow$$\frac{3}{2}<x<8$ — (3)

Now, from (1), (2) and (3), we get

$\mathit{x}\mathbf{}\mathbf{\in }\mathbf{}\left(-\frac{2}{3},8\right)$

Hence, Option ‘C’ is Correct.