Question

The negative integer value(s) of $x$ satisfying $||x-|x-2|+3|-4|=3$ is/are

• A. $-1$
• B. $-2$
• C. $-3$
• D. $-4$

Answer 1

Answer: (A), (D)

$-4$

$||x-|x-2|+3|-4|=3$
Case $1:$ $x<2$
$\Rightarrow ||x+x-2+3|-4|=3$
$\Rightarrow ||2x+1|-4|=3$

If $x\ge -\frac{1}{2}$ or If $x<-\frac{1}{2}$
$|2x-3|=3$ or $|-2x-5|=3$
$\Rightarrow 2x-3=\pm 3$ or $-2x-5=\pm 3$
$\Rightarrow x=0(3\text{rejected as}x<2)\text{or}x=-4,-1$
$\therefore x=-4,-1,0$

Case $2:$ $x\ge 2$
$||x-x+2+3|-4|=3$
$\Rightarrow 1=3$ No solution

$\therefore$ The solutions are $x=-4,-1,0$