A trigonometric equation and a polynomial

As $\sin x \leq 1$ so there will be solution if $\dfrac{x}{!0} \leq 1 \implies x \leq 10=3 \pi +\epsilon$ .It will give solution upto $3 \pi$ .After $3 \pi$ sine curve will go negative and $x/10$ will remain positive.This line will cut every concave part twice.Between $3 \pi$ there are three concave part where two parts are above X-axis and one is below X-axis.It will cut only positive X-axis.So, there will be $\text{No.of concave part} \times \text{no. of cuts}=2 \times 2=4$ solutions .As sine is symmetric then for the negative part there will be another four solutions.Total $4+4=8$ solutions.But we have calculated $x=0$ twice.Then number of distinct solutions are $8-1=7$ solutions.

Draw the graphs of y=sinx and y=x/10 .Easily you can obtain 7 solutions