Question

The solution set of the inequation, $\left|2x+3\right|-\left|x-1\right|=6$, is

• A. $x\in \left(-10,2\right)$
• B. $x\in \left[-10,2\right)$
• C. $x\in \left[-10,2\right]$
• D. $x\in \left\{-10,2\right\}$

Answer 1

The correct option is C $x\in \left[-10,2\right]$

The solution set of the equation:

$|2x+3|-|x-1|=6$

$|x-1|=x-1$ if $x>1$

$=-(x-1)$ if $x<1$

$|2x+3|=2x+3$ if $x>\frac{-3}{2}$

$=-2x-3$ if $x<\frac{-3}{2}$

When $x<\frac{-3}{2}$

$-2x-3+x-1=6$

or, $-x-4=6$ or $x=-10$

When, $\frac{-3}{2}<x<1$

$=2x+3+x-1=6$

or, $x=\frac{4}{3}$

When $x>1,2x+3-x+1=6$

$x=2$

$x\in \{-10,\frac{4}{3},2\}$

$\therefore x\in \{-10,2\}$.....................$\because -10<\frac{4}{3}<2$

The answer is option ${}^{\prime }{C}^{\prime }$