📚 IGCSE CIE Maths: Inequalities Key Points | IGCSE CIE 数学：不等式 考点精讲
Inequalities are statements that compare two expressions using symbols like <, >, ≤ and ≥. In IGCSE CIE Mathematics, you need to solve linear and quadratic inequalities, represent solutions on a number line using open or closed circles, express answer sets using interval notation, and apply inequalities to real-life problems. This article walks you through all the key concepts, common pitfalls and exam-style techniques, making sure you master every important detail for your revision.
不等式是使用 <、>、≤、≥ 等符号比较两个表达式的陈述。在 IGCSE CIE 数学考试中，你需要求解一元一次不等式和二次不等式、在数轴上用空心圆或实心圆表示解集、用区间记号表达答案，并将不等式应用到实际问题中。本文为你梳理所有核心概念、常见陷阱和应试技巧，帮助你彻底掌握每一个重要考点。
1. What are Inequalities? | 什么是不等式？
An inequality states that two values are not equal. While an equation claims that one side exactly balances the other, an inequality expresses a range of possible values that satisfy a condition. The basic forms you will encounter are linear inequalities, such as 2x + 3 > 7, and quadratic inequalities, such as x² − 5x + 6 < 0.
不等式表明两个值不相等。方程要求两边严格相等，而不等式表达的是一组满足某个条件的可能取值范围。你将遇到的基本形式有线性不等式，例如 2x + 3 > 7，以及二次不等式，例如 x² − 5x + 6 < 0。
2. Inequality Symbols and Their Meanings | 不等号及其含义
The four main symbols are: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). In IGCSE, you also sometimes see ≠ (not equal to). It is crucial to distinguish between strict inequalities (<, >) and inclusive inequalities (≤, ≥), because they affect whether the boundary point is included in the solution set and how you draw it on a number line.
四个主要不等号是：<（小于）、>（大于）、≤（小于或等于）和 ≥（大于或等于）。在 IGCSE 中，你有时也会见到 ≠（不等于）。区分严格不等式（<、>）和含等号的不等式（≤、≥）至关重要，因为这会影响边界点是否包含在解集中，以及如何在数轴上绘制。
- Strict inequality: x < 4 means 4 is not included; use an open circle on the number line. | 严格不等式：x < 4 表示 4 不包含在内；在数轴上用空心圆表示。
- Inclusive inequality: x ≤ 4 means 4 is included; use a closed (solid) circle. | 含等号不等式：x ≤ 4 表示 4 包含在内；用实心圆表示。
3. Representing Inequalities on a Number Line | 在数轴上表示不等式
You must be able to draw a simple number line and mark the solution set correctly. For x > −1, draw an open circle at −1 and shade the line to the right. For x ≤ 3, draw a solid circle at 3 and shade to the left. When an inequality has two boundaries, such as 2 < x ≤ 5, you put an open circle at 2, a solid circle at 5 and shade the region between them. Always label your number line with a few key values and an arrow to show the direction of the shading.
你必须能够画出简单的数轴并正确标示解集。对于 x > −1，在 −1 处画空心圆并向右描出阴影；对于 x ≤ 3，在 3 处画实心圆并向左描出阴影。当不等式有两个边界时，例如 2 < x ≤ 5，先在 2 处画空心圆，在 5 处画实心圆，然后描出中间的区域。一定要在数轴上标注几个关键数值，并用箭头标明阴影方向。
4. Solving Linear Inequalities | 解一元一次不等式
Solving a linear inequality follows the same steps as solving a linear equation: you can add, subtract, multiply or divide both sides by a number. However, there is one vital rule: if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. For example, solve −2x < 8. Dividing both sides by −2 gives x > −4. It is the most common error in inequalities, so double-check every negative operation.
解一元一次不等式的步骤与解一元一次方程相同：你可以两边加、减、乘、除同一个数。但有一条至关重要的规则：如果两边乘或除以一个负数，必须反转不等号的方向。例如，解 −2x < 8，两边除以 −2 得到 x > −4。这是不等式中最常见的错误，所以每次涉及负数运算时都要仔细检查。
For a more complex example, solve 5 − 3x ≤ 2x + 10. First bring like terms together: −3x − 2x ≤ 10 − 5 → −5x ≤ 5. Divide by −5 and reverse the sign: x ≥ −1. The solution set is x ≥ −1, drawn as a solid circle at −1 shading to the right.
更复杂的例子：解 5 − 3x ≤ 2x + 10。先移项：−3x − 2x ≤ 10 − 5 → −5x ≤ 5。除以 −5 并反转不等号：x ≥ −1。解集为 x ≥ −1，在数轴上用实心圆表示 −1，向右描阴影。
5. Solving Double Inequalities | 解双端不等式
A double inequality like −4 < 2x + 1 ≤ 7 can be solved by treating it as two separate inequalities or by performing the same operation on all three parts simultaneously. The simultaneous method is efficient: subtract 1 from all three parts to get −5 < 2x ≤ 6, then divide everything by 2 to obtain −2.5 < x ≤ 3. Always write the final answer with x in the middle and keep the symbols pointing the same way.
像 −4 < 2x + 1 ≤ 7 这样的双端不等式，可以视为两个单独的不等式来解，也可以同时对三部分进行相同的运算。同时运算更高效：三部分同时减去 1 得到 −5 < 2x ≤ 6，然后三部分同时除以 2 得到 −2.5 < x ≤ 3。最终的答案要用 x 在中间的形式书写，并保持不等号方向一致。
6. Quadratic Inequalities | 二次不等式
Quadratic inequalities, such as x² − 4x − 5 ≤ 0, require a slightly different approach. First solve the corresponding quadratic equation x² − 4x − 5 = 0 by factorising: (x − 5)(x + 1) = 0, giving roots x = 5 and x = −1. These roots divide the number line into three intervals: x < −1, −1 < x < 5, and x > 5. You then test a value from each interval in the original inequality to determine which region(s) satisfy it. Alternatively, sketch the graph of the quadratic function. Because the coefficient of x² is positive, the parabola opens upward, so the inequality ≤ 0 is satisfied between the roots: −1 ≤ x ≤ 5. Always include the equals part if the symbol is ≤ or ≥.
二次不等式，例如 x² − 4x − 5 ≤ 0，需要稍有不同的解法。首先解对应的二次方程 x² − 4x − 5 = 0，因式分解得 (x − 5)(x + 1) = 0，根为 x = 5 和 x = −1。这两个根将数轴分成三个区间：x < −1、−1 < x < 5 和 x > 5。接着从每个区间选取一个测试值代入原不等式，判断哪些区间满足条件。另一种方法是画出二次函数图像。由于 x² 系数为正，抛物线开口向上，所以不等式 ≤ 0 在两个根之间成立：−1 ≤ x ≤ 5。遇到 ≤ 或 ≥ 符号时，务必将等号部分包含进去。
For x² + 2x − 3 > 0, factorise to (x + 3)(x − 1) > 0. Roots at −3 and 1. The upward-opening parabola is above zero outside the roots, so the solution is x < −3 or x > 1. Use “or” to connect the two separate pieces, never “and” because x cannot be less than −3 and greater than 1 at the same time.
对于 x² + 2x − 3 > 0，因式分解为 (x + 3)(x − 1) > 0。根为 −3 和 1。开口向上的抛物线在根的两侧大于零，因此解为 x < −3 或 x > 1。要用 “或” 来连接两个独立的部分，绝不能用 “和”，因为 x 不可能同时小于 −3 又大于 1。
7. Graphical Method and Sign Tables | 图形法与符号表
While you can always sketch a parabolas to visualise a quadratic inequality, the sign table (or region-testing method) gives a reliable algebraic solution. After finding the roots in increasing order, create a table with intervals and test points. For (x + 2)(x − 4) ≥ 0, test x = −3, 0, 5. You find the product is positive for x ≤ −2 and x ≥ 4, giving the final answer. The graphical method helps you check: the curve cuts the x‑axis at −2 and 4, and sits above or on the axis outside those points.
虽然你总是可以通过画出抛物线来直观地理解二次不等式，但符号表（或区间测试法）能提供可靠的代数解法。按从小到大列出根后，创建一个包含区间和测试点的表格。对于 (x + 2)(x − 4) ≥ 0，测试 x = −3、0、5。你会发现乘积在 x ≤ −2 和 x ≥ 4 时非负，从而得到最终答案。图形法有助于检验：曲线与 x 轴交于 −2 和 4，在这两点之外位于 x 轴或之上。
8. Real-Life Applications of Inequalities | 不等式的实际应用
IGCSE questions often wrap inequalities in real‑world contexts. For example: “A mobile phone plan costs £15 per month plus £0.05 per minute of calls. You want to spend no more than £25. Write and solve an inequality to find the maximum minutes you can use.” Translate: 15 + 0.05m ≤ 25 → 0.05m ≤ 10 → m ≤ 200 minutes. Always define your variable clearly and check whether the final value makes sense in context.
IGCSE 题目常常把不等式包裹在实际情境中。例如：“某手机套餐每月收费 15 英镑，加上通话每分钟 0.05 英镑。你希望每月花费不超过 25 英镑。写出并求解不等式，找出最多可使用多少分钟。” 转化：15 + 0.05m ≤ 25 → 0.05m ≤ 10 → m ≤ 200 分钟。始终要清晰地定义变量，并检验最终结果在情境中是否合理。
Other common contexts include geometry constraints (e.g. perimeter of a rectangle must be less than a given value), temperature ranges, and scores needed to reach a target average. Make sure you interpret “at least”, “no more than”, “over” and “under” correctly and map them to the right symbol.
其他常见情境包括几何约束（如矩形的周长必须小于某给定值）、温度范围，以及达到目标平均分所需的分数等。务必正确解读 “至少”、“不超过”、“超过” 和 “低于” 这些词语，并将它们映射到正确的符号上。
9. Interval Notation | 区间记号
Interval notation is a concise way to write solution sets, and you may be asked to give answers in this form. A round bracket ( or ) means the endpoint is not included (strict inequality), while a square bracket [ or ] means the endpoint is included. The symbol ∞ (infinity) is always paired with a round bracket. For example, x > 3 is written as (3, ∞); x ≤ −1 becomes (−∞, −1]; and −2 ≤ x < 4 is [−2, 4). The union symbol ∪ is used for separate intervals: x < 0 or x ≥ 5 gives (−∞, 0) ∪ [5, ∞).
区间记号是书写解集的一种简洁方式，你可能会被要求以这种形式给出答案。圆括号 ( 或 ) 表示端点不包含在内（严格不等式），而方括号 [ 或 ] 表示端点包含在内。∞（无穷）符号总是与圆括号搭配。例如，x > 3 写作 (3, ∞)；x ≤ −1 写作 (−∞, −1]；−2 ≤ x < 4 写作 [−2, 4)。对于分离的区间则使用并集符号 ∪：x < 0 或 x ≥ 5 写作 (−∞, 0) ∪ [5, ∞)。
| Inequality | Interval Notation | Number line feature |
| x > 5 | (5, ∞) | Open circle at 5, arrow right |
| x ≤ 2 | (−∞, 2] | Solid circle at 2, arrow left |
| 0 ≤ x < 7 | [0, 7) | Solid at 0, open at 7, shade between |
10. Common Mistakes and Pitfalls | 常见错误与陷阱
Mistake 1: Forgetting to reverse the inequality sign when multiplying or dividing by a negative. This is the number one error. Always pause and ask: “Am I dividing by a negative?”
错误一：在乘或除以负数时忘记反转不等号。这是第一大错误。每次停下来问问自己：“我除以的是负数吗？”
Mistake 2: Incorrectly connecting the solution parts of a quadratic inequality. For an upward-opening parabola with inequality > 0, the solution is two outer regions connected by “or”. For < 0, it is the inner region. Never mix “and” and “or” incorrectly.
错误二：错误地连接二次不等式的解的部分。对于开口向上的抛物线，不等号 > 0 时，解是两侧的外部区域，用 “或” 连接；< 0 时，解是内部区域。绝不要混淆 “和” 与 “或”。
Mistake 3: Misreading the word problem keywords. “At least 10” means ≥ 10, not > 10. “More than 10” means > 10. Underline these words in the exam.
错误三：误读应用题关键词。“at least 10” 意味着 ≥ 10，而不是 > 10；“more than 10” 意味着 > 10。考试时把这些词划下来。
Mistake 4: Writing the interval notation with the larger number on the left, e.g. (5, 2) instead of (2, 5). Interval notation always goes from smallest to largest.
错误四：区间记号中将较大的数写在左边，比如 (5, 2) 而非 (2, 5)。区间记号总是从小到大书写。
11. Brief Introduction to Absolute Value Inequalities | 绝对值不等式简介
Some IGCSE extended papers may include simple absolute value inequalities. For example, |x − 3| < 2 means the distance between x and 3 is less than 2. This translates to the double inequality −2 < x − 3 < 2, which solves to 1 < x < 5. The inequality |x + 1| ≥ 4 splits into x + 1 ≤ −4 or x + 1 ≥ 4, giving x ≤ −5 or x ≥ 3. Remember: |expression| < a becomes −a < expression < a; |expression| > a becomes expression < −a or expression > a.
某些 IGCSE 扩展试卷可能会包含简单的绝对值不等式。例如，|x − 3| < 2 表示 x 到 3 的距离小于 2。这可以转化为双端不等式 −2 < x − 3 < 2，解得 1 < x < 5。不等式 |x + 1| ≥ 4 则拆分为 x + 1 ≤ −4 或 x + 1 ≥ 4，得到 x ≤ −5 或 x ≥ 3。记住：|表达式| < a 化为 −a < 表达式 < a；|表达式| > a 化为 表达式 < −a 或 表达式 > a。
12. Summary and Exam Tips | 总结与备考建议
To succeed with inequalities in IGCSE CIE Maths, master the reversal rule for negative multipliers, practise both number‑line sketches and interval notation until they are second nature, and always check your answers by substituting boundary values and a test point from each region. In the exam, read word problems carefully, define your variable, write the inequality, solve it step by step and then interpret the result back in the context. Keep your working neat; examiners award marks for correct method even if a sign error occurs later. With thorough practice, inequality questions become reliable marks.
要在 IGCSE CIE 数学不等式部分取得成功，请掌握负数乘除时的不等号反转规则，反复练习数轴草图和区间记号直到游刃有余，并总是通过代入边界值和每个区域的测试点来检验答案。考试时，仔细阅读应用题，定义变量，写出不等式，逐步求解，然后将结果带回情境中解读。保持卷面整洁；即使后来出现了符号错误，考官仍然会给正确的解法步骤分。通过充分练习，不等式题目会成为你稳定的得分来源。
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