用代数方法解不等式

题目要求用代数。分母 $x+2$ 的正负不固定，所以不要直接两边同乘。更稳妥的做法是移到同一边，通分后用临界点做符号分析。

Start with

Bring all terms to the left-hand side:

Expand the numerator:

So

The critical values are

Check the sign of $\frac{(2x+5)(x+1)}{x+2}$:

We need the expression to be negative. Since the original expression is undefined at $x=-2$, and the inequality is strict, the solution is

因为分母 $x+2$ 的正负号决定了不等号是否需要翻转，所以可以直接在不等式两边同乘分母的平方 $(x+2)^2$（因为 $x \ne -2$，所以 $(x+2)^2$ 恒正）。如此一来可以避免分区间讨论，直接将原分式不等式化简为三次整式不等式，提取公因式后求解。

Since $x \ne -2$, the term $(x+2)^2$ is strictly positive. Multiply both sides of the inequality by $(x+2)^2$:

Move all terms to one side:

Factor out the common term $(x+2)$:

The critical values are:

By testing the intervals (or using a sign chart), the product is negative on: