Example 2. Solve the inequality $|x-2|+|x+1|>5$.

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Solution: Let $P(x, 0)$ be a point on the 2-stretch, then the points satisfying the original inequality must be outside the ellipse
$$
\frac{\left(x-\frac{1}{2}\right)^{2}}{25 / 4}+\frac{y^{2}}{4}=1
$$

Let $y=0$, solving yields $x_{1}=-2, x_{2}=3$, hence the solution to the original inequality is $(-\infty,-2) \cup(3,+\infty)$.

Second, the application of hyperbolas can be used to solve equations of the form $\sqrt{\left(x-m_{1}\right)^{2}+n^{2}}-\sqrt{\left(x-m_{2}\right)+n^{2}}=k$ $\left(k<\left|m_{2}-m_{1}\right|\right), \sqrt{(x-m)+n_{1}^{2}}-$ $\sqrt{(x-m)^{2}+n_{2}^{2}}=k\left(k<\left|n_{2}-n_{1}\right|\right)$, $|x-m!-| x-n \mid=k(k<|m-n|)$ or similar inequalities.