≠ 不等式求解器（一次、二次、绝对值，区间表示）Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation)

使用说明与常见问题How-to & FAQ

「不等式求解器（一次、二次、绝对值，区间表示）」是什么？What is the Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation) tool?

求解三类不等式并用区间表示解集：一次不等式 ax + b ? c（系数为负时自动变号）、二次不等式 ax² + bx + c ? 0（按根和开口方向判断），以及绝对值不等式 |ax + b| ? c。同时给出解题步骤。本地处理。Solve three kinds of inequality and write the answer in interval notation: linear ax + b ? c (flipping the sign for a negative coefficient), quadratic ax² + bx + c ? 0 (using the roots and the direction of the parabola) and absolute value |ax + b| ? c, with the steps shown. Processed locally.

如何使用「不等式求解器（一次、二次、绝对值，区间表示）」？How do I use the Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation) tool?

在「不等式求解器（一次、二次、绝对值，区间表示）」中，填写「类型」、「a」、「b」、「c」、「关系」等参数，结果会随输入实时计算并显示在下方。In the Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation): Fill in "Type", "a", "b", "c", "Relation"; results are recalculated instantly as you type.

为什么除以负数要变号？Why does dividing by a negative flip the sign?

两边同时乘或除以负数时，数的大小顺序会反过来，例如 2 < 3 但 −2 > −3，所以不等号必须改变方向。Multiplying or dividing both sides by a negative reverses the order of numbers: 2 < 3 but −2 > −3, so the inequality sign must flip.

二次不等式怎么解？How is a quadratic inequality solved?

先求 ax² + bx + c = 0 的根。a > 0 时抛物线开口向上，“< 0”的解在两根之间，“> 0”的解在两根之外；a < 0 时正好相反。没有实根时要么全体实数、要么无解。Find the roots of ax² + bx + c = 0. If a > 0 the parabola opens up: < 0 holds between the roots and > 0 outside them; a < 0 reverses this. With no real roots the answer is all reals or nothing.

绝对值不等式怎么拆？How is an absolute value inequality split?

|u| < c（c > 0）等价于 −c < u < c；|u| > c 等价于 u < −c 或 u > c。c 为负数时，|u| < c 无解，|u| > c 恒成立。|u| < c with c > 0 means −c < u < c; |u| > c means u < −c or u > c. For negative c, |u| < c has no solution and |u| > c always holds.

能举个计算实例吗？Can you show a worked example?

例如在「不等式求解器（一次、二次、绝对值，区间表示）」中填 类型 = 一次 ax + b ? c，a = 2，b = -4，c = 0，关系 = <，结果区显示：解集：(−∞, 2)；步骤：2x < 4 → x < 2；即 x < 2For example, entering Type = Linear ax + b ? c, a = 2, b = -4, c = 0, Relation = < in the Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation) gives: Solution: (−∞, 2); Steps: 2x < 4 → x < 2; That is x < 2

哪些输入会被检查？Which inputs are validated?

「不等式求解器（一次、二次、绝对值，区间表示）」会检查这些输入——「a」：所选方式用到时必须填写；「b」、「c」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation) checks its inputs — "a": required when the chosen option uses it; "b", "c": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.

有哪些选项可以调整？What options can I adjust?

「不等式求解器（一次、二次、绝对值，区间表示）」的设置：「类型」可选 一次 ax + b ? c、二次 ax² + bx + c ? 0、绝对值 ∣ax + b∣ ? c（默认 一次 ax + b ? c）；「a」默认 2；「b」默认 -4；「c」默认 0；「关系」可选 ≤、>、≥（默认 ≤）。每次生成结果都按当前设置处理。Settings in the Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation) — "Type": Linear ax + b ? c, Quadratic ax² + bx + c ? 0, Absolute value ∣ax + b∣ ? c (default Linear ax + b ? c); "a" default 2; "b" default -4; "c" default 0; "Relation": ≤, >, ≥ (default ≤). The result always uses the current settings.

计算精度如何？How precise are the results?

「不等式求解器（一次、二次、绝对值，区间表示）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.

「不等式求解器（一次、二次、绝对值，区间表示）」免费吗？需要注册吗？Is the Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation) free? Do I need an account?

「不等式求解器（一次、二次、绝对值，区间表示）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Inequality Solver (Linear, Quadratic, Absolute Value, Interval Notation) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.