QJEEM24Ma4800: JEE Main 2024 Mathematics - Class 11 JEEM24Ma4800 Solution: The number of distinct real roots of the equation is __________.

$|x| \quad|x+2|-5|x+1|-1=0$

$\begin{aligned}& \text { (I) if } x<-2 \\& x^2+2 x+5 x+5-1=0 \\& x^2+7 x+4=0 \Rightarrow \text { one root satisfying } x<-2\end{aligned}$

$\begin{aligned}& \text { (II) if }-2 \leq x<-1 \\& -x^2-2 x+5 x+5-1=0 \\& x^2-3 x-4=0 \Rightarrow \text { not root satisfying }-2 \leq x<-1\end{aligned}$

$\begin{aligned}& \text { (III) if }-1 \leq x<0 \\& -x^2-2 x-5 x-5-1=0 \\& x^2+7 x+6=0 \\& x=-1 \text { is only root satisfying }-1 \leq x<0\end{aligned}$

(IV) if $x \geq 0$

$\begin{aligned}& x^2+2 x-5 x-5-1=0 \\& x^2-3 x-6=0\end{aligned}$

one root satisfying $x \geq 0$

$\Rightarrow$ The number of distinct real roots are three.