📚 IGCSE CCEA Maths: Mastering Inequalities (Full Guide) | IGCSE CCEA 数学：不等式 考点精讲
Inequalities are a fundamental part of IGCSE Mathematics, allowing us to describe ranges of values rather than exact numbers. In CCEA examinations, you will be required to solve linear inequalities, represent solutions on number lines, tackle compound and quadratic inequalities, and work with graphical inequalities that involve shading regions. This guide builds your understanding step by step, ensuring you can handle standard questions as well as the more challenging problem‑solving tasks. We will use the symbols <, >, ≤, ≥ throughout, and every method is explained to match CCEA’s marking style.
不等式是 IGCSE 数学的基础内容，它帮助我们描述值的范围而非精确数字。在 CCEA 考试中，你需要会解线性不等式、在数轴上表示解集、处理复合不等式和二次不等式，以及绘制含阴影区域的不等式图形。本指南逐步构建你的理解，确保你能应对标准题目和更具挑战性的问题解决任务。全文将使用符号 <、>、≤、≥，每种方法均按 CCEA 评分风格进行解释。
1. What Are Inequalities? | 不等式是什么？
An inequality compares two expressions and states that one side is less than, greater than, less than or equal to, or greater than or equal to the other side. For example, x + 3 > 7 means that the value of x plus 3 is strictly larger than 7. Unlike equations, inequalities do not give a single answer; they give a set of values that make the statement true. In CCEA IGCSE, you will encounter the four inequality symbols regularly, and you must understand the difference between open and closed intervals when presenting solutions.
不等式比较两个表达式，并指出一边小于、大于、小于等于或大于等于另一边。例如，x + 3 > 7 表示 x 加 3 的值严格大于 7。与方程不同的是，不等式不给出单一答案，而是给出一组使陈述成立的值。在 CCEA IGCSE 中，你会经常遇到这四种不等号，并且必须理解呈现解集时开区间与闭区间的区别。
For instance, 2x ≤ 10 tells us that x can be 5 or any number less than 5. This set of solutions can be written in different forms: as an inequality x ≤ 5, in interval notation (−∞, 5], or displayed on a number line with a closed circle at 5. CCEA often expects you to translate fluently between these representations.
例如，2x ≤ 10 告诉我们 x 可以是 5 或任何小于 5 的数。该解集可以用不同形式表示：写成不等式 x ≤ 5、区间记号 (−∞, 5]，或在数轴上 5 处画一个实心圆点表示。CCEA 通常要求你能在这些表示方法之间流利转换。
2. Solving Linear Inequalities | 解线性不等式
Solving a linear inequality uses the same balancing method as solving a linear equation, with one critical exception: when you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. For example, solve 3x − 4 < 11. Add 4 to both sides to get 3x < 15, then divide by 3 to obtain x < 5. The inequality sign stays the same because we divided by a positive 3.
解线性不等式使用的平衡方法与解线性方程相同，但有一个关键例外：当你将两边同时乘以或除以一个负数时，必须反转不等号的方向。例如，解 3x − 4 < 11。两边加 4 得到 3x < 15，然后除以 3 得到 x < 5。因为除以的是正数 3，所以不等号方向不变。
If the inequality involves brackets, expand them first. Consider 2(x + 5) ≥ 18. Expand to 2x + 10 ≥ 18, subtract 10 from both sides to get 2x ≥ 8, and divide by 2 to find x ≥ 4. Always present your final answer with the variable on the left, e.g. x ≥ 4, as this is the expected form in CCEA mark schemes.
如果不等式中含有括号，要先展开。考虑 2(x + 5) ≥ 18。展开得 2x + 10 ≥ 18，两边减 10 得 2x ≥ 8，除以 2 得到 x ≥ 4。总是将变量放在左边来表示最终答案，如 x ≥ 4，这是 CCEA 评分方案中要求的形式。
3. The Negative Coefficient Rule | 负系数法则
The most common mistake in inequalities is forgetting to flip the sign when dividing or multiplying by a negative. Take −2x > 8. To isolate x, we divide both sides by −2. Because we are dividing by a negative number, the > sign changes to <. This gives x < −4. It is easy to check: pick a number less than −4, say −5, and substitute back: −2(−5) = 10, which is indeed greater than 8, confirming the solution is correct.
不等式中最常见的错误是忘记在做乘除负数运算时调转不等号。以 −2x > 8 为例。要解出 x，两边需除以 −2。因为除以的是负数，所以 > 号变为 <。这样得到 x < −4。检验很容易：选一个小于 −4 的数，比如 −5，代回去：−2(−5) = 10，确实大于 8，证实解集正确。
Similarly, solve 5 − 3x ≤ 14. Subtract 5 from both sides: −3x ≤ 9. Divide both sides by −3 and reverse the inequality sign: x ≥ −3. In CCEA, you may also be asked to write the solution set using set notation, for example {x : x ≥ −3}. Practise rewriting answers in this form as well.
类似地，解 5 − 3x ≤ 14。两边减 5：−3x ≤ 9。两边除以 −3 并反转不等号：x ≥ −3。在 CCEA 中，你可能还需要用集合符号写出解集，例如 {x : x ≥ −3}。也要练习用这种形式改写答案。
4. Number Line Representation | 数轴表示
CCEA questions often ask you to illustrate an inequality on a number line. For a strict inequality like x > 2, draw an open circle at 2 and an arrow pointing to the right. For x ≤ −1, use a closed (solid) circle at −1 and an arrow to the left. When the inequality is written as a range, such as −3 < x ≤ 4, you place an open circle at −3, a closed circle at 4, and draw a line segment connecting them.
CCEA 的题目常要求你在数轴上表示不等式。对于严格不等式如 x > 2，在 2 处画一个空心圆，并向右画箭头。对于 x ≤ −1，在 −1 处使用实心圆并向左画箭头。当不等式以范围形式给出时，比如 −3 < x ≤ 4，在 −3 处画空心圆，在 4 处画实心圆，然后画线段连接它们。
Remember that the variable must be on the left to read the arrow direction intuitively: 2 < x means the same as x > 2, but it is easier to sketch x > 2 on a number line. Always label key numbers clearly and use a ruler. Some mark schemes award marks for neatness and correct arrow direction.
记住，变量在左边时箭头方向才好直观阅读：2 < x 与 x > 2 意思相同，但在数轴上画 x > 2 更容易。始终清晰标注关键数字并使用直尺。有些评分方案会因整洁度和正确箭头方向而给分。
5. Compound Inequalities (Union and Intersection) | 复合不等式（并集与交集）
A compound inequality consists of two or more inequalities joined by the word ‘and’ or ‘or’. When ‘and’ is used, the solution must satisfy both conditions simultaneously – this is the intersection of the two solution sets. For example, x > 1 and x < 5 gives the interval 1 < x < 5. On a number line, this is the overlap region.
复合不等式由两个或多个用“且”或“或”连接的不等式组成。当使用“且”时，解必须同时满足两个条件——这是两个解集的交集。例如，x > 1 且 x < 5 得到区间 1 < x < 5。在数轴上，这是重叠区域。
When ‘or’ connects the inequalities, the solution is the union – any value that satisfies at least one of the conditions. For x < 2 or x > 6, the number line shows two separate rays pointing outward from open circles at 2 and 6. CCEA might ask you to solve 3x − 1 < 5 or 2x + 3 ≥ 11 and display the combined solution. Solve each part independently, then merge the regions.
当“或”连接不等式时，解集为并集——即满足至少一个条件的任何值。对于 x < 2 或 x > 6，数轴上显示两条从 2 和 6 的空心圆向外指的射线。CCEA 可能会要求你解 3x − 1 < 5 或 2x + 3 ≥ 11 并显示合并后的解集。分别解每一部分，然后合并区域。
6. Solving Double Inequalities | 解双重不等式
A double inequality like 4 ≤ 2x + 6 < 14 is an ‘and’ compound inequality written compactly. To solve it, perform the same operation on all three parts at once. Subtract 6 from each part: −2 ≤ 2x < 8. Then divide every part by 2: −1 ≤ x < 4. The solution is a continuous interval. It is crucial to keep the variable in the middle throughout the process; reversing signs when multiplying by a negative number applies to all three parts.
如 4 ≤ 2x + 6 < 14 这样的双重不等式是“且”型复合不等式的紧凑写法。解它时，要同时对三个部分进行相同运算。每部分减 6：−2 ≤ 2x < 8。然后每部分除以 2：−1 ≤ x < 4。解集是一个连续区间。整个过程里保持变量在中间至关重要；如果乘以负数需反转符号，三个部分都要反转。
Take −3 < 5 − 2x ≤ 7. Subtract 5 from all parts: −8 < −2x ≤ 2. Divide by −2, remembering to reverse both inequality signs: 4 > x ≥ −1. Rewriting this in standard form with the variable first gives −1 ≤ x < 4. Always rewrite the final answer so the smaller number is on the left; this matches the number line convention.
以 −3 < 5 − 2x ≤ 7 为例。所有部分减 5：−8 < −2x ≤ 2。除以 −2，记得两个不等号都要反转：4 > x ≥ −1。用标准形式改写，变量在前，得到 −1 ≤ x < 4。最终答案务必把小数写在左边，这符合数轴惯例。
7. Introduction to Quadratic Inequalities | 二次不等式入门
CCEA IGCSE introduces simple quadratic inequalities, often of the form x² > a or x² < a, or ones that can be factorised like x² − 5x + 6 < 0. The method relies on sketching the related quadratic graph or using a sign table. For x² > 9, think of the roots x = −3 and x = 3. The parabola y = x² − 9 opens upward; the region where the graph is above the x‑axis is outside the roots. Therefore, the solution is x < −3 or x > 3.
CCEA IGCSE 引入简单的二次不等式，常见形式有 x² > a 或 x² < a，或可因式分解的如 x² − 5x + 6 < 0。方法依赖画出相关二次图形或使用符号表。对于 x² > 9，考虑根 x = −3 和 x = 3。抛物线 y = x² − 9 开口向上；图形位于 x 轴上方的区域在两根之外。因此，解集为 x < −3 或 x > 3。
For x² < 9, the parabola is below the x‑axis between the roots, so the solution is −3 < x < 3. When the quadratic can be factorised, such as x² − 5x + 6 < 0, factorise to (x − 2)(x − 3) < 0. The critical values are 2 and 3. Test intervals: for x < 2, both factors are negative, product positive; between 2 and 3, one factor negative, one positive, product negative; for x > 3, both positive. The negative region gives the solution 2 < x < 3.
对于 x² < 9，抛物线在两根之间位于 x 轴下方，所以解集为 −3 < x < 3。当二次式可因式分解时，例如 x² − 5x + 6 < 0，分解为 (x − 2)(x − 3) < 0。关键值为 2 和 3。检验区间：x < 2 时两因式皆负，乘积正；介于 2 和 3 之间时一负一正，乘积负；x > 3 时皆正。负值区域给出解集 2 < x < 3。
8. Graphical Inequalities: Drawing Linear Regions | 图形不等式：绘制线性区域
When an inequality involves two variables, such as y > 2x + 1, the solution is a region of the coordinate plane. First, graph the boundary line y = 2x + 1. If the inequality is strict (> or <), draw the boundary as a dashed line to show it is not included. If it uses ≤ or ≥, draw a solid line. Then choose a test point not on the line – the origin (0,0) is often convenient – and substitute it into the inequality. If the test point satisfies the inequality, shade that side of the line; otherwise, shade the opposite side.
当不等式含有两个变量时，如 y > 2x + 1，解集是坐标平面上的一个区域。首先，画出边界线 y = 2x + 1。若不等式为严格不等（> 或 <），边界画为虚线，表示不包含线上点；若使用 ≤ 或 ≥，则画实线。然后选取一个不在直线上的测试点——原点 (0,0) 通常很方便——将其代入不等式。如果测试点满足不等式，则给直线该侧涂上阴影；否则涂另一侧。
For y > 2x + 1, testing (0,0) gives 0 > 1, which is false. Therefore, shade the side not containing (0,0). Label the region clearly. CCEA questions often combine two or more inequalities on the same grid, asking you to shade the region that satisfies all conditions simultaneously – the intersection.
对于 y > 2x + 1，测试 (0,0) 得 0 > 1，不成立。因此，在不含 (0,0) 的那侧涂阴影。清晰标注区域。CCEA 题目常在同一个方格图上结合两个或多个不等式，要求你涂出同时满足所有条件的区域——即交集。
9. Shading Techniques for y < mx + c and Vertical/Horizontal Lines | 为 y < mx + c 及竖直线水平线涂阴影的技巧
If the boundary line is rearranged into the form y = mx + c, it is easy to determine the side to shade: for y > … shade above the line; for y < … shade below. This rule works provided y has a positive coefficient. For vertical lines, such as x > −4, draw a dashed vertical line through x = −4 and shade the right-hand side. For horizontal lines, like y ≤ 3, draw a solid horizontal line at y = 3 and shade below.
如果边界线整理成 y = mx + c 的形式，判断涂哪一侧很容易：对于 y > … 涂在直线上方；对于 y < … 涂在下方。这个规则在 y 系数为正时成立。对于竖直线，如 x > −4，过 x = −4 画一条虚线，然后涂右侧区域。对于水平线，如 y ≤ 3，在 y = 3 处画一条实线，然后涂下方区域。
Sometimes you need to find the inequality that represents a given shaded region. Look at the boundary line equation first: find its gradient and y‑intercept. Identify whether the line is dashed or solid. Then pick a test point in the shaded region to decide which inequality sign is correct. For example, a shaded region above a dashed line through (0,1) with gradient 2 would be y > 2x + 1.
有时你需要找出表示给定阴影区域的不等式。先看边界线方程：找出其斜率和 y 轴截距。判断线是虚线还是实线。然后选阴影区域中的一个测试点，决定正确的不等号。例如，阴影区域在一条通过 (0,1) 且斜率为 2 的虚线上方，那么不等式为 y > 2x + 1。
10. Systems of Linear Inequalities | 线性不等式组
CCEA frequently asks you to shade the region defined by multiple inequalities, such as y ≥ x − 1, y < −½x + 4, and x > 0. The approach is to graph each boundary line using the correct line style (dashed or solid), lightly shade or indicate each half‑plane, and then identify the region where all shadings overlap. This region is the feasible region. Finally, make the overlap clearly visible and erase or cross out any redundant shading.
CCEA 经常要求你涂出由多个不等式定义的区域，例如 y ≥ x − 1, y < −½x + 4 和 x > 0。方法是：用正确的线型（虚线或实线）画出每条边界线，轻轻涂出或标出每个半平面，然后找出所有阴影重合的区域。这个区域就是可行区域。最后，让重合部分清晰可见，擦掉或划掉多余的阴影。
Questions may also ask you to list the integer coordinates that lie inside the region, or to find the maximum or minimum value of an expression like 3x + 2y over the region. This links to linear programming ideas, but at IGCSE level it usually involves testing vertices of the feasible region. Always read the question carefully to see if boundaries are included.
题目还可能要求你列出区域内的整数坐标，或者求表达式如 3x + 2y 在该区域上的最大值或最小值。这与线性规划的思想相关，但在 IGCSE 阶段通常涉及检验可行区域的顶点。务必仔细审题，看清边界是否包含在内。
11. Integer Solutions and Word Problems | 整数解与应用题
Word problems require you to form inequalities from a scenario, then solve them and find integer solutions that make practical sense. For example, “Arun has £20 to spend on pens costing £1.50 each and notebooks costing £2.50 each. He buys p pens and n notebooks, with the total cost not exceeding £20.” The inequality is 1.5p + 2.5n ≤ 20. If he also buys more pens than notebooks, we add p > n. List possible integer pairs (p, n) that satisfy both.
应用题要求你根据情景列出不等式，然后求解并找出符合实际的整数解。例如，“Arun 有 20 英镑，每支笔 1.50 英镑，每本笔记本 2.50 英镑。他买了 p 支笔和 n 本笔记本，总花费不超过 20 英镑。” 不等式为 1.5p + 2.5n ≤ 20。如果他买的笔比笔记本多，还要加上 p > n。列出满足两者的可能整数对 (p, n)。
When finding integer solutions from a number line or region, ensure you respect strict vs. non‑strict signs. For 2 < x ≤ 6, the integer solutions are 3, 4, 5, 6. For inequalities on graphs, identify the integer lattice points within the shaded region, paying attention to whether boundary points are included. Always present integer solutions as a list, e.g. x = 3, 4, 5, 6.
从数轴或区域中寻找整数解时，要确保区分严格与不严格不等号。对于 2 < x ≤ 6，整数解为 3, 4, 5, 6。对于图形中的不等式，找出阴影区域内的整数格点，注意边界点是否包含。始终以列表形式呈现整数解，如 x = 3, 4, 5, 6。
12. Summary and Exam Tips | 总结与考试技巧
Mastering inequalities for CCEA IGCSE boils down to a few consistent habits: always reverse the sign when multiplying or dividing by a negative; check your solution by substituting a value; use open and closed circles accurately on number lines; for graphs, draw boundaries first and use a test point; and when you finish, reread the question to confirm whether you need to find integer solutions or shade a specific region. Write your final inequality with the variable on the left, and present number line diagrams with clear labels.
要掌握 CCEA IGCSE 的不等式内容，关键在于养成几个一致的习惯：做乘除负数运算时总是反转不等号；通过代入数值检验解集；在数轴上准确使用空心圆和实心圆；对于图形，先画边界再用测试点；做完后重读题目，确认是否需要找整数解或涂特定区域。最终不等式将变量写在左边，数轴图示要标清楚。
Common pitfalls include forgetting to flip the sign, misreading a double inequality, and shading the wrong side of a boundary line when the test point is inconvenient. Practising a wide variety of problems—especially those that blend algebra and coordinate geometry—will build your confidence. Keep your working neat and logical; CCEA examiners reward clarity. With these tools, inequalities will become one of your strongest topics on the paper.
常见陷阱包括忘记反转符号、误读双重不等式以及测试点不当时涂错边界线一侧。练习多样化的题目——尤其是那些结合代数与坐标几何的题——会增强你的信心。保持解题过程整洁、有逻辑；CCEA 考官青睐清晰的表达。掌握这些工具后，不等式将成为你考卷上最拿手的题目之一。
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