How many real solutions does the equation sin x = x/10 have?

Step 1. Where to look. $|\sin x| \le 1$, so a solution needs $\left|\dfrac{x}{10}\right| \le 1$, that is $-10 \le x \le 10$.

Step 2. The origin. $x = 0$ is a solution.

Step 3. Positive $x$. The line climbs slowly from $0$ to $1$. On $(0, \pi)$ the arch of the sine starts above the line and ends below it: one crossing. On $(2\pi, 3\pi)$, about $6.28$ to $9.42$, the arch rises to $1$ while the line is below $1$: two crossings. The next arch starts after $4\pi \approx 12.6$, too far.

Step 4. Negative $x$. By symmetry there are $3$ more.