Question

The number of real solution(s) of $||x-2|-2|-2|x|=|x-3|$ is

• A. $1$
• B. $0$
• C. $2$
• D. $3$

Answer 1

The correct option is B $0$

Given : $||x-2|-2|-2|x|=|x-3|$
The sign of mod changes at $0,2,3$

When $x<0$
$|-x+2-2|+2x=-x+3\Rightarrow |x|=-3x+3\Rightarrow -x=-3x+3\Rightarrow x=\frac{3}{2}$
No solution.

When $0\le x\le 2$
$|-x+2-2|-2x=-x+3\Rightarrow |x|=x+3\Rightarrow x=x+3\Rightarrow 0=3$
Which is not possible.

When $2<x<3$
$|x-2-2|-2x=-x+3\Rightarrow |x-4|=x+3\Rightarrow -x+4=x+3\Rightarrow x=\frac{1}{2}$
No solution.

When $x\ge 3$
$|x-2-2|-2x=x-3\Rightarrow |x-4|=3x-3$
If $x\ge 4$, we get
$\Rightarrow x-4=3x-3\Rightarrow x=-\frac{1}{2}$
No solution.
If $x<4$, we get
$\Rightarrow -x+4=3x-3\Rightarrow x=\frac{7}{4}$
No solution as $x\ge 3$

Hence, there is no solution of the given equation.