Solution: Clearly, $a \neq 0$. Without loss of generality, assume $a>0$ (otherwise, multiplying $a, b, c$ by -1 does not change the line). Since the inclination angle of the line is acute, $b<0$.

Therefore, the number of ways to choose $a, b, c$ is $3 \times 3 \times 5=45$.
However, there are repetitions: $2 x-2 y=0, 3 x-3 y=0, x-y=0$ represent the same line. Hence, there are $45-2=43$ lines that meet the requirements of the problem.
Example 1 In the line $a x+b y+c=0$, $a, b, c$ are 3 different elements chosen from the set $\{-3,-2,-1,0,1,2,3\}$, and the inclination angle of the line is acute. How many such lines are there?