Inequalities for WJEC IGCSE Mathematics | IGCSE WJEC 数学：不等式考点精讲

📚 Inequalities for WJEC IGCSE Mathematics | IGCSE WJEC 数学：不等式考点精讲

Inequalities are a fundamental topic in the WJEC IGCSE Mathematics syllabus, appearing regularly in both calculator and non‑calculator papers. They extend the idea of equations by showing that one expression is greater than, less than, or not equal to another. This revision guide covers everything you need, from basic symbols and linear inequalities to quadratic inequalities and graphical regions, all tailored to the WJEC specification.

不等式是 WJEC IGCSE 数学大纲中的基础内容，经常在可使用与不可使用计算器的试卷中出现。它们将方程的概念加以推广，表明一个表达式大于、小于或不等于另一个表达式。本复习指南涵盖你所需的一切，从基本符号和线性不等式到二次不等式和图形区域，全部针对 WJEC 考试要求量身打造。

1. Inequality Symbols and Their Meanings | 理解不等式符号

The five key inequality symbols you must recognise are: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to) and ≠ (not equal to). Remember that the ‘open’ side of the symbol always faces the larger quantity. For example, 5 > 3 reads ‘5 is greater than 3’. In WJEC exams these symbols are used to form inequalities such as x > 2 or y ≤ -1.

你必须掌握五个关键的不等式符号：<（小于）、>（大于）、≤（小于或等于）、≥（大于或等于）和 ≠（不等于）。请记住，符号的“开口”一侧始终面向较大的量。例如，5 > 3 读作“5 大于 3”。在 WJEC 考试中，这些符号用于构建诸如 x > 2 或 y ≤ -1 的不等式。

2. Solving Linear Inequalities | 解线性不等式

Solving a linear inequality is very similar to solving a linear equation. You can add, subtract, multiply or divide both sides, with one crucial exception: if you multiply or divide by a negative number, you must reverse the direction of the inequality sign. For instance, to solve -2x < 8, divide by -2 and flip the sign to get x > -4. Always present your final answer with the variable on the left where possible.

解线性不等式与解线性方程非常相似。你可以加减、乘或除不等式的两边，但有一个至关重要的例外：如果乘以或除以一个负数，你必须反转不等号的方向。例如，解 -2x < 8 时，除以 -2 并反转符号，得到 x > -4。最终答案应尽可能将变量放在左边。

3. Representing Inequalities on a Number Line | 在数轴上表示不等式

WJEC often asks you to illustrate an inequality on a number line. Use an open circle (o) for strict inequalities < or >, and a closed circle (•) for ≤ or ≥. Then draw a solid line extending in the correct direction. For example, x ≥ 1 is shown with a closed circle at 1 and a line to the right. The same rules apply when you are asked to write down the inequality shown by a given diagram.

WJEC 经常要求你在数轴上表示一个不等式。对于严格的不等式 < 或 >，使用空心圆 (o)；对于 ≤ 或 ≥，使用实心圆 (•)。然后沿正确的方向画一条实线。例如，x ≥ 1 表示为在 1 处标实心圆，并向右侧画线。反过来，当你需要根据给出的示意图写出不等式时，也遵循同样的规则。

4. Reversing the Inequality Sign – The Negative Rule | 不等式符号反转——负数规则

This is the most common source of errors at IGCSE. If you multiply or divide both sides of an inequality by any negative value, you must reverse the inequality sign. For example, -3y ≤ 15 becomes y ≥ -5 after dividing by -3. Forgetting this step will cost you marks, even if your arithmetic is otherwise perfect. The same rule applies when working with reciprocals: if x > 4, then 1/x < 1/4 only when x is positive – take care with sign changes.

这是 IGCSE 中最常见的错误来源。如果你将不等式两边同时乘以或除以任何负数，则必须反转不等号的方向。例如，-3y ≤ 15 除以 -3 后变为 y ≥ -5。即使你的运算在其他方面都正确，忘记这一步也会导致失分。在处理倒数时同样适用此规则：当 x 为正数时，若 x > 4，则 1/x < 1/4 —— 注意符号变化。

5. Solving Inequalities with Brackets | 解带括号的不等式

Expand any brackets first, just as you would in an equation. Then collect like terms, and isolate the variable using inverse operations while respecting the negative rule. Example: 2(x – 3) < 8 → 2x – 6 < 8 → 2x < 14 → x < 7. Double-check by substituting a value just inside the solution set, such as x = 6.5, to ensure it satisfies the original inequality.

首先展开所有括号，就像解方程一样。然后合并同类项，利用逆运算隔离变量，同时谨记负数规则。例如：2(x – 3) < 8 → 2x – 6 < 8 → 2x < 14 → x < 7。验证时可以从解集中取一个边界内的值，如 x = 6.5，代入原不等式以确保成立。

6. Solving Double Inequalities | 解双重不等式

A double inequality like -3 < 2x + 1 ≤ 5 can be solved by performing the same operation on all three parts. Subtract 1: -4 < 2x ≤ 4. Divide by 2: -2 < x ≤ 2. The solution set is all numbers between -2 and 2, including 2 but not -2. On a number line, you would draw an open circle at -2 and a closed circle at 2, with a line connecting them.

像 -3 < 2x + 1 ≤ 5 这样的双重不等式可以对所有的三部分同时执行相同操作来求解。减 1：-4 < 2x ≤ 4。除以 2：-2 < x ≤ 2。解集为 -2 到 2 之间的所有数，包含 2 但不包含 -2。在数轴上，应在 -2 处画空心圆，在 2 处画实心圆，并用线段连接两者。

7. Inequality Regions on Graphs | 图形上的不等式区域

WJEC IGCSE includes problems where you must shade the region defined by one or more inequalities. A line is drawn dashed for strict inequalities (< or >) and solid for weak inequalities (≤ or ≥). After drawing the bounding line, test a point (usually (0,0) unless the line passes through the origin) to see which side to shade. In the exam, label the required region R if asked.

WJEC 的 IGCSE 考试包含需要给一个或多个不等式所定义的区域涂色的问题。严格不等式（< 或 >）使用虚线绘制边界，非严格不等式（≤ 或 ≥）使用实线。画出边界线后，选取一个检验点（通常为 (0,0)，除非直线经过原点）以确定应涂色的一侧。考试中如要求，请将所求区域标记为 R。

8. Solving Quadratic Inequalities | 解二次不等式

For higher tier, you may need to solve an inequality like x² – x – 6 > 0. First, solve the corresponding quadratic equation x² – x – 6 = 0 to get critical values x = -2 and x = 3. The parabola y = x² – x – 6 cuts the x-axis at these points. Since the coefficient of x² is positive, the graph is ∪-shaped. The inequality > 0 is satisfied outside the roots, so the solution is x < -2 or x > 3. Always sketch a quick graph to avoid sign errors. If the inequality had been x² – x – 6 < 0, the solution would be -2 < x < 3.

在高阶考试中，你可能需要解如 x² – x – 6 > 0 的不等式。首先，解对应的二次方程 x² – x – 6 = 0，得到关键值 x = -2 和 x = 3。抛物线 y = x² – x – 6 在这些点与 x 轴相交。由于 x² 系数为正，图像呈开口向上的 ∪ 形。不等式 > 0 在根的外侧成立，因此解为 x < -2 或 x > 3。务必快速画一个草图以避免符号错误。如果不等式是 x² – x – 6 < 0，则解为 -2 < x < 3。

9. Inequalities in Word Problems | 不等式在文字题中的应用

WJEC often embeds inequalities in real‑life contexts. You might be asked to find the maximum number of people a lift can carry, or to express a cost constraint. Set up an algebraic inequality from the information given. For example, a rectangular garden of width w metres and length 2w + 3 metres has a perimeter less than 50 m. This leads to 2w + 2(2w + 3) < 50, which simplifies to 6w + 6 < 50, so w < 7⅓. Since w is a width, state any practical restrictions, e.g. w > 0.

WJEC 经常将不等式嵌入实际生活情境。你可能会被要求计算一部电梯可承载的最多人数，或表达一个成本约束条件。根据所给信息建立一个代数不等式。例如，一个宽为 w 米、长为 2w + 3 米的矩形花园，其周长小于 50 米。可得 2w + 2(2w + 3) < 50，简化得到 6w + 6 < 50，因此 w < 7⅓。由于 w 是宽度，还需说明实际限制条件，例如 w > 0。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

The most frequent errors are: forgetting to flip the inequality sign when multiplying/dividing by a negative; misreading ≤ as < and vice versa on number lines; shading the wrong side of a boundary line; and failing to express the final answer clearly. In WJEC mark schemes, clarity matters – use ‘or’ for disjoint solutions and ‘and’ for combined ones. Always check your answer with a sample value, and re‑read the question to ensure you used the correct inequality symbol. Practise past paper questions to build speed and confidence.

最常见的错误有：在乘以或除以负数时忘记反转不等号；在数轴上将 ≤ 误读为 <，或反之；在边界线错误的一侧涂色；以及未能清晰地表达最终答案。在 WJEC 的评分方案中，清晰度很重要——不相交的解用“或”连接，联合的解用“且”连接。务必用样本值检查你的答案，并重新阅读题目以确保使用了正确的不等号。练习历年真题以提高速度和信心。

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Inequalities for WJEC IGCSE Mathematics | IGCSE WJEC 数学：不等式 考点精讲