Question

Number of real solutions of the equation $|||x^2-1|-1|+3|=1$ is

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is D $0$

$|||x^2-1|-1|+3|=1$
$\Rightarrow ||x^2-1|-1|+3=+1$ or $||x^2-1|-1|+3=-1$

case $-1,||x^2-1|-1|+3=+1$
$\Rightarrow ||x^2-1|-1|=-2$,
That is not possible

case $-2,||x^2-1|-1|+3=-1$
$\Rightarrow ||x^2-1|-1|=-4$,
That is not possible
Hence, there is no solution for the given equation.
So, number of real solutions is $0$

Alternate :
$|||x^2-1|-1|+3|=1$
Since, we know that $||x^2-1|-1|\ge 0$ (always)
$\therefore |||x^2-1|-1|+3|\ge 3$ (always)
$\therefore |||x^2-1|-1|+3|=1$ has no real solution