GCSE Edexcel Maths: Inequalities – Complete Revision Guide | GCSE Edexcel 数学：不等式 考点精讲

📚 GCSE Edexcel Maths: Inequalities – Complete Revision Guide | GCSE Edexcel 数学：不等式 考点精讲

Inequalities are a fundamental part of GCSE Edexcel maths, bridging the gap between equations and real-world constraints. In this revision guide, you will learn how to manipulate inequality symbols, solve linear and quadratic inequalities, represent solutions on number lines and graphs, and apply these skills to exam-style questions. Whether you are studying Foundation or Higher tier, mastering inequalities is essential for building confidence and achieving top marks.

不等式是 GCSE Edexcel 数学的基础内容，它连接了方程与现实世界中的限制条件。在本考点精讲中，你将学习如何正确处理不等号、解一元一次不等式和二次不等式、在数轴和坐标系中表示解集，并将这些技能应用到考试题型中。无论你是在备考基础层级还是高级层级，掌握不等式对于建立信心和取得高分都至关重要。

1. What are Inequalities? | 什么是不等式？

An inequality compares two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Unlike equations, inequalities describe a range of possible values rather than a single fixed number.

不等式使用 >（大于）、<（小于）、≥（大于等于）和 ≤（小于等于）等符号来比较两个表达式。与方程不同，不等式描述的是一系列可能的数值，而不是一个固定的数。

For example, x > 3 means x can be any number larger than 3, such as 3.1, 4, 5.2, or even 100. The value 3 itself is not included. If we have x ≥ 3, then 3 is included in the solution set.

例如，x > 3 表示 x 可以是任何一个比 3 大的数，比如 3.1、4、5.2，甚至是 100。数值 3 本身并不包含在内。如果我们有 x ≥ 3，那么 3 就包含在解集中。

---

2. Inequality Symbols and Their Meanings | 不等式符号及其含义

You must be confident with the four main inequality symbols. Edexcel exam questions often test whether students understand the strictness of each symbol, especially when reading word problems or shading graphs.

你必须熟练掌握四个主要的不等号。Edexcel 考题经常考查学生是否理解每个符号的严格程度，特别是在阅读文字题或为图形涂阴影时。

Symbol | 符号	Meaning | 含义	Example | 例子
<	less than | 小于	x < 5 → x is any number below 5
>	greater than | 大于	x > −2 → x is any number above −2
≤	less than or equal to | 小于等于	y ≤ 4 → y can be 4 or any number below
≥	greater than or equal to | 大于等于	n ≥ 10 → n can be 10 or larger

Remember: an open circle on a number line means the endpoint is not included (< or >); a closed circle means it is included (≤ or ≥).

记住：数轴上的空心圆表示端点不包含在内（用于 < 或 >）；实心圆表示端点包含在内（用于 ≤ 或 ≥）。

---

3. Solving Simple Linear Inequalities | 解简单的一元一次不等式

Solving a linear inequality follows the same basic rules as solving an equation: you can add, subtract, multiply or divide both sides, but with one crucial exception. If you multiply or divide by a negative number, you must reverse the inequality symbol.

解一元一次不等式的基本规则和解方程相同：你可以对两边进行加、减、乘、除，但有一个关键的例外。如果你乘以或除以一个负数，必须将不等号的方向反转。

Example: Solve 2x + 3 < 11.

Subtract 3 from both sides: 2x < 8.

Divide both sides by 2: x < 4.

例子：解 2x + 3 < 11。

两边同时减去 3：2x < 8。

两边同时除以 2：x < 4。

Now consider −4x ≤ 20. Dividing by −4 gives x ≥ −5. Notice the symbol changed from ≤ to ≥ because we divided by a negative.

现在考虑 −4x ≤ 20。两边除以 −4 得到 x ≥ −5。注意到不等号从 ≤ 变成了 ≥，因为我们除以了负数。

---

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

Edexcel frequently asks you to draw or interpret solutions on a number line. For a single inequality, you mark the boundary with an open or closed circle and draw an arrow in the direction of the inequality.

Edexcel 经常要求你在数轴上画出或读出不等于式的解。对于单个不等式，你用空心或实心圆标记边界，然后朝着不等号的方向画箭头。

If x > −1: draw an open circle at −1 and an arrow pointing to the right.

If y ≤ 3.5: draw a closed circle at 3.5 and an arrow pointing to the left.

如果 x > −1：在 −1 处画空心圆，并画一个指向右边的箭头。

如果 y ≤ 3.5：在 3.5 处画实心圆，并画一个指向左边的箭头。

When representing a compound inequality like −2 ≤ x < 3, you draw a closed circle at −2, an open circle at 3, and a line connecting them.

当表示像 −2 ≤ x < 3 这样的复合不等式时，你在 −2 处画实心圆，在 3 处画空心圆，然后用一条线段将它们连接起来。

---

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

If an inequality contains brackets, expand them first, just as you would with equations. Then collect like terms and isolate the variable.

如果不等式中含有括号，先展开括号，就像解方程一样。然后合并同类项，求出未知数。

Example: 3(2x − 1) < 5x + 4

Expand: 6x − 3 < 5x + 4

Subtract 5x: x − 3 < 4

Add 3: x < 7

例子：3(2x − 1) < 5x + 4

展开：6x − 3 < 5x + 4

减去 5x：x − 3 < 4

加上 3：x < 7

Always check by substituting a value from your solution set back into the original inequality to ensure it holds true.

总是要通过从你的解集中代入一个数值到原不等式中来检验，以确保不等式成立。

---

6. Inequalities with Unknowns on Both Sides | 两边都有未知数的不等式

When the unknown appears on both sides of the inequality, use the same strategy as with equations: collect the variable terms on one side and the constants on the other.

当未知数出现在不等式的两边时，采用与方程相同的策略：将含未知数的项移到一边，常数项移到另一边。

Example: 2x + 5 ≥ 5x − 7

Subtract 2x from both sides: 5 ≥ 3x − 7

Add 7: 12 ≥ 3x

Divide by 3: 4 ≥ x, which is the same as x ≤ 4.

例子：2x + 5 ≥ 5x − 7

两边同时减去 2x：5 ≥ 3x − 7

加上 7：12 ≥ 3x

除以 3：4 ≥ x，这与 x ≤ 4 相同。

Note how the inequality remains equivalent when we rewrite it with x on the left. This is a useful step to avoid confusion when presenting the final answer.

注意，当我们把 x 写在左边重写不等式时，它仍然是等价的。这是避免最终答案混淆的有用步骤。

---

7. Compound Inequalities | 复合不等式

A compound inequality combines two constraints on the same variable. For instance, −3 < 2x + 1 ≤ 5 means that 2x + 1 must be greater than −3 and at the same time less than or equal to 5. You solve the two parts together.

复合不等式将两个对同一变量的限制条件结合在一起。例如，−3 < 2x + 1 ≤ 5 表示 2x + 1 必须大于 −3，同时还要小于等于 5。你需要同时解这两个部分。

Solve −3 < 2x + 1 ≤ 5:

Subtract 1 from all three parts: −4 < 2x ≤ 4

Divide all parts by 2: −2 < x ≤ 2

This solution means x is greater than −2 and up to and including 2.

解 −3 < 2x + 1 ≤ 5：

将三部分同时减去 1：−4 < 2x ≤ 4

将三部分同时除以 2：−2 < x ≤ 2

这个解表示 x 大于 −2 并且最大到包含 2 在内。

On a number line you would mark an open circle at −2, a closed circle at 2, and shade the region in between.

在数轴上，你会在 −2 处标空心圆，在 2 处标实心圆，并给两者之间的区域涂上阴影。

---

8. Graphical Inequalities and Shading Regions | 图形不等式与区域涂色

Edexcel Higher candidates must be able to represent inequalities on a coordinate grid. You first draw the boundary line (or curve) and then shade the region that satisfies the inequality. Use a dashed line for strict inequalities (< or >) and a solid line for non-strict ones (≤ or ≥).

Edexcel 高级层级的考生必须能够在坐标网格上表示不等式。首先画出边界线（或曲线），然后给满足不等式的区域涂上阴影。严格不等式（< 或 >）使用虚线，非严格不等式（≤ 或 ≥）使用实线。

For example, to graph y < 2x + 1:

Draw the line y = 2x + 1 with a dashed line.

Choose a test point, often (0,0) if it is not on the line.

Substitute (0,0): 0 < 2(0) + 1 → 0 < 1, which is true.

Therefore shade the side containing (0,0).

例如，要画出 y < 2x + 1 的图形：

用虚线画出直线 y = 2x + 1。

选择一个测试点，如果 (0,0) 不在直线上，通常使用它。

代入 (0,0)：0 < 2(0) + 1 → 0 < 1，成立。

因此对包含 (0,0) 的那一侧涂阴影。

If a question asks for a region defined by multiple inequalities, shade each inequality and identify the overlapping region – this is the solution set.

如果题目要求多个不等式共同定义的区域，则分别对每个不等式涂阴影，并找出重叠的区域——这就是解集。

---

9. Quadratic Inequalities (Higher Tier) | 二次不等式（高级内容）

Quadratic inequalities such as x² − 4x − 5 < 0 require a different approach. First, solve the corresponding quadratic equation x² − 4x − 5 = 0 to find the critical values. Factorising gives (x − 5)(x + 1) = 0, so x = 5 or x = −1.

像 x² − 4x − 5 < 0 这样的二次不等式需要不同的解法。首先，解对应的二次方程 x² − 4x − 5 = 0 以找到临界值。因式分解得到 (x − 5)(x + 1) = 0，所以 x = 5 或 x = −1。

These critical values divide the number line into three intervals: x < −1, −1 < x < 5, and x > 5. Test a value from each interval in the original inequality to see where it is negative.

这些临界值将数轴分成三个区间：x < −1，−1 < x < 5，和 x > 5。从每个区间中取一个值代入原不等式，检查在哪里取值为负。

Since the quadratic opens upwards (coefficient of x² is positive), the graph is below the x-axis between the roots. So the solution is −1 < x < 5.

由于这个二次函数开口向上（x² 的系数为正），图形在两根之间位于 x 轴下方。因此解为 −1 < x < 5。

For x² − 4x − 5 ≥ 0, the solution would be x ≤ −1 or x ≥ 5. Remember to sketch the parabola to avoid sign errors.

对于 x² − 4x − 5 ≥ 0，解将是 x ≤ −1 或 x ≥ 5。记得画一个抛物线的草图以避免符号错误。

---

10. Inequalities with Fractions | 含有分数的不等式

When an inequality contains algebraic fractions, you must multiply both sides by the denominator. However, you need to consider the sign of the denominator, because multiplying by a negative flips the inequality. It is often safer to multiply by the square of the denominator, which is always non-negative.

当不等式中含有代数分数时，你必须将两边同乘以分母。但是，你需要考虑分母的符号，因为乘以负数会反转不等号。通常更安全的做法是乘以分母的平方，因为它总是非负的。

Example: (x + 1)/3 < 2

Since 3 is positive, multiply by 3: x + 1 < 6 → x < 5.

例子：(x + 1)/3 < 2

因为 3 是正数，两边乘以 3：x + 1 < 6 → x < 5。

For something like 1/(x − 2) > 3, Edexcel typically expects you to bring all terms to one side and consider critical values, but this is mostly Higher-tier content.

对于像 1/(x − 2) > 3 这样的不等式，Edexcel 通常要求你将所有项移到一边，并考虑临界值，但这主要是高级层级的内容。

---

11. Word Problems with Inequalities | 不等式应用题

Many GCSE problems embed inequalities in real-life contexts: budgets, speeds, weights, temperatures, etc. The key is to translate the written constraints into mathematical inequalities and then solve.

许多 GCSE 问题将不等式嵌入到现实生活情境中：预算、速度、重量、温度等。关键是要将文字中的限制条件翻译成数学不等式，然后求解。

Example: A student has £50 to spend on pens costing £1.50 each and notebooks costing £3 each. Write an inequality for the maximum number of pens (p) and notebooks (n) he can buy. The inequality is 1.5p + 3n ≤ 50.

例子：一个学生有 50 英镑，可以买每支 1.50 英镑的笔和每本 3 英镑的笔记本。写出他能购买的笔 (p) 和笔记本 (n) 的最大数量的不等式。不等式为 1.5p + 3n ≤ 50。

Often the question will then ask you to find possible combinations, requiring integer solutions. Always check that solutions make sense in context (e.g., you cannot purchase a negative number of items).

题目通常接着会让你找出可能的组合，这就要求得出整数解。始终检查解在上下文中是否有意义（例如，你不能购买负数的物品）。

---

12. Common Mistakes and Exam Tips | 常见错误与应试技巧

One of the most frequent errors is forgetting to flip the inequality sign when multiplying or dividing by a negative number. Always double-check this step, especially in multi-step problems. Another common pitfall is misreading the direction of compound inequalities or forgetting to use open/closed circles correctly when drawing.

最常见的错误之一是在乘以或除以负数时忘记反转不等号方向。特别是在多步骤问题中，一定要反复检查这一步。另一个常见的陷阱是读错复合不等式的方向，或者在画图时忘记正确使用空心/实心圆。

Exam tips:

– When solving a quadratic inequality, always sketch the parabola to visualise where the expression is positive or negative.

– Show all your working clearly; marks are awarded for correct method even if the final answer is wrong.

– In graph shading questions, label the shaded region if requested, and use the correct line style (dashed or solid).

– Practice with past Edexcel papers to familiarise yourself with the wording and mark schemes.

应试技巧：

– 解二次不等式时，一定要画抛物线草图，以直观看出表达式何时为正、何时为负。

– 清晰地展示所有解题步骤；即使最终答案错误，方法正确也能得分。

– 在图形涂色问题中，如果要求的话给涂色区域做标记，并使用正确的线性（虚线或实线）。

– 通过练习 Edexcel 往年的试卷来熟悉措辞和评分方案。

Published by TutorHao | GCSE Edexcel Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）