Step 1: Treat it as two curves meeting.
Solutions of $|\sin(\pi x)| = \frac{1}{50}(x^2 + 1)$ are the points where the wavy curve $y = |\sin(\pi x)|$ meets the upward parabola $y = \frac{1}{50}(x^2 + 1)$.
Step 2: Use the symmetry.
Both sides are even functions (unchanged when $x$ becomes $-x$), so the picture is mirror-symmetric about the $y$-axis. We count solutions for $x \gt 0$, handle $x = 0$ separately, then double the positive count.
Step 3: Find where the parabola exceeds 1.
Since $|\sin(\pi x)| \le 1$, no solution exists once $\frac{1}{50}(x^2 + 1) \gt 1$, that is $x^2 \gt 49$, so $x \gt 7$. All solutions lie within $-7 \le x \le 7$.
Step 4: Check $x = 0$.
At $x = 0$: left side $|\sin 0| = 0$, right side $\frac{1}{50}(1) = 0.02$. Not equal, so $x = 0$ is not a solution.
Step 5: Count crossings on each sine arch for $0 \lt x \le 7$.
Over each unit interval the curve $|\sin(\pi x)|$ makes one arch from 0 up to 1 and back to 0. The small parabola value (staying well below 1 until near $x = 7$) cuts each such arch twice. There are 7 arches over $(0, 7]$, giving about $2 \times 7 = 14$, but the final arch near $x = 7$ where the parabola reaches 1 yields a single tangential-type meeting, trimming the count to 13 on the positive side.
Step 6: Double for symmetry.
With 13 solutions for $x \gt 0$ and the same 13 for $x \lt 0$, and none at $x = 0$, the total is $13 + 13 = 26$.
\[ \boxed{26} \]