Example 6 Let $a>0$ be a constant, solve the inequality $\sqrt{x^{2}+1}-a x \leqslant 1$.


Solve the inequality by transforming it into $\sqrt{x^{2}+1} \leqslant a x+1$. Let $y=\sqrt{x^{2}+1}(y \geqslant 0)$, which represents the upper branch of the hyperbola $y^{2}-x^{2}=1$. $y=a x+1$ represents a line passing through the fixed point $P(0,1)$ with a slope of $a$. See Figure 5-5.
When $0<a<1$, the line intersects the upper branch of the hyperbola at two points, i.e.,
$$
\sqrt{x^{2}+1}=a x+1 \text {, }
$$

Solving this yields $x_{1}=0, x_{2}=\frac{2 a}{1-a}$.
From Figure 5-6, the solution to the original inequality is
$$
0<x<\frac{2 a}{1-a} \text {. }
$$

When $a \geqslant 1$, the line intersects the hyperbola at only one point $P(0,1)$, in which case the solution to the inequality is $x \geqslant 0$.