Question

The number of solutions of $||x-4|-1|=2$ is

• A. four
• B. two
• C. three
• D. one

Answer 1

The correct option is B two

$||x-4|-1|=2$

For $x\ge 4$ we get

$|x-4-1|=2$

$|x-5|=2$

now for $x\ge 4$ and $x<5$ we get

$5-x=2$

$x=3$

For $x\ge 5$ we get

$x-5=2$

$x=7$

For $x<4$

$|3-x|=2$

For $x>3$ and $x\le 4$ we get

$x-3=2$

$x=5$

For $x\le 3$ and $x>0$ we get

$3-x=2$

$x=1$

For $x<0$

There will be no solution since $|x-4|>4$.

Hence in total four solutions.

However, $5$ and $3$ do not satisfy the equation and we are left with only $2$ solutions, $7$ and $1.$