The number of solutions of the equation $9 x^2-18|x|+5=0$ belonging to the domain of definition of $\log _e\{(x+1)(x+2)\}$, is

We have, $9 x^2-18|x|+5=0$ $\begin{array}{lr}\Rightarrow & 9|x|^2-18|x|+5=0 \\ \Rightarrow & (3|x|-1)(3|x|-5)=0 \\ \Rightarrow & |x|=\frac{1}{3} \text { or }|x|=\frac{5}{3} \\ \Rightarrow & x= \pm \frac{1}{3} \text { or } x= \pm \frac{5}{3}\end{array}$ Now, $\log _\theta\{(x+1)(x+2)\}$ is defined, if $(x+1)(x+2)\gt0$ $\Rightarrow \quad x \lt -2$ or $x\gt-1$ Clearly, $x=-\frac{5}{3}$ does not satisfy this condition. But all other values of $x$ satisfy the above conditions. Hence, the number of solutions lying in the domain of definition of the given function is 3 .