A nice method to find an approximate solution is to successively cut intervals in half, as follows: let's first rewrite this as $f(x) = 3x + \sin x - e^x = 0$ Now pick two values, $a$ and $b$, such that $f(a) < 0$ and $f(b) > 0$. (You might have to make a few guesses before finding such values!) In this case, let's choose $a = 0$ and $b = 1$: $f(a) = 3(0)+\sin(0)-e^{0} = -1 < 0$ $f(b) = 3(1)+\sin(1)-e^{1} = 1.12... > 0$
Now, because our function $f(x)$ is "smooth", there must be a solution somewhere between $a$ and $b$. Find the point halfway in between them, $\frac{0+1}{2} = 0.5$, and check to see whether it makes $f(x)$ positive or negative: $f(0.5) =3(0.5)+\sin(0.5)-e^{0.5} = 0.33... > 0$ So, since $f(x)$ is positive here, set $0.5$ as the new value for $b$. Once again calculate the midpoint of $a$ and $b$ (in this case, $\frac{0+0.5}{2} = 0.25$), evaluate it in $f$, and so on. Continue until you have the precision you are looking for.