IGCSE Mathematics: Mastering Linear Equations & Inequalities | 攻克线性方程与不等式

📚 IGCSE Mathematics: Mastering Linear Equations & Inequalities | 攻克线性方程与不等式

Linear equations and inequalities form the backbone of IGCSE Mathematics. From simple one-step equations to word problems involving unknown quantities, mastering these skills is essential for success in both Paper 1 (non-calculator) and Paper 2 (calculator) examinations. This revision guide breaks down the key techniques, common pitfalls, and exam-style strategies you need.

线性方程与不等式是 IGCSE 数学的基石。从简单的一步方程到涉及未知量的应用题，掌握这些技能对 Paper 1（非计算器）和 Paper 2（计算器）考试都至关重要。本复习指南将为你梳理关键技巧、常见错误以及应试策略。

1. The Basics: What Is a Linear Equation? | 基础：什么是线性方程？

A linear equation is an equation where the unknown variable has an exponent of 1. For example, 3x + 5 = 20 is linear because x appears only to the first power. The goal is always to isolate the variable on one side of the equation.

线性方程是指未知变量的指数为 1 的方程。例如，3x + 5 = 20 是线性的，因为 x 只以一次方的形式出现。解方程的目标始终是将变量单独放在等式的一侧。

Key idea: Whatever you do to one side, do to the other. | 核心思想：对等式一边所做的操作，必须对另一边同样进行。
Check your answer: Substitute your solution back into the original equation. | 验证答案：将解代回原方程进行检验。

3x + 5 = 20 → 3x = 15 → x = 5

Always show your working. In IGCSE, method marks are often awarded even if the final answer is wrong.

始终展示你的解题过程。在 IGCSE 考试中，即使最终答案错误，过程分也常常会被给予。

2. Solving Two-Step Equations | 解两步方程

Two-step equations require two inverse operations to isolate the variable. The order of operations in reverse is your guide: undo addition/subtraction first, then multiplication/division.

两步方程需要两次逆运算来分离变量。运算顺序的逆序是你的指南：先消去加减，再消去乘除。

Consider the equation 2x + 7 = 19. First, subtract 7 from both sides to get 2x = 12. Then divide both sides by 2 to obtain x = 6.

以方程 2x + 7 = 19 为例。首先，两边同时减去 7，得到 2x = 12。然后两边同时除以 2，得到 x = 6。

Example A: 5x − 9 = 31 → 5x = 40 → x = 8 | 例 A：5x − 9 = 31 → 5x = 40 → x = 8
Example B: 3(x + 4) = 30 → x + 4 = 10 → x = 6 | 例 B：3(x + 4) = 30 → x + 4 = 10 → x = 6

When brackets appear, you may either expand them first or divide both sides by the coefficient outside the bracket. Both methods are valid, but choose the one that makes the numbers simpler.

当出现括号时，你可以先展开括号，也可以两边同时除以括号外的系数。两种方法都正确，但应选择使数字更简单的那种。

3. Equations with Unknowns on Both Sides | 未知数在等式两边的方程

When the variable appears on both sides of the equation, your first step is to collect all variable terms on one side and all constant terms on the other.

当变量出现在等式两边时，第一步是将所有含变量的项移到一边，所有常数项移到另一边。

Solve 4x − 3 = 2x + 9. Subtract 2x from both sides: 2x − 3 = 9. Then add 3: 2x = 12. Finally divide by 2: x = 6.

解 4x − 3 = 2x + 9。两边同时减去 2x：2x − 3 = 9。然后加 3：2x = 12。最后除以 2：x = 6。

Tip: Move the smaller coefficient of x to avoid negative coefficients when possible.

提示：尽量移动较小的 x 系数，以避免出现负系数。

However, if you do get a negative coefficient, do not panic — just add the term to both sides. For example, 2x = 6 − x becomes 3x = 6, giving x = 2.

然而，如果确实得到了负系数，不要慌张——只需将该项加到两边。例如，2x = 6 − x 变为 3x = 6，解得 x = 2。

4. Linear Equations with Fractions | 含分数的线性方程

Fractions in equations can look intimidating, but the strategy is simple: multiply every term by the lowest common multiple (LCM) of all denominators to eliminate fractions.

方程中的分数看起来可能令人畏惧，但策略很简单：将每一项乘以所有分母的最小公倍数（LCM），从而消去分数。

Solve x/2 + x/3 = 5. The LCM of 2 and 3 is 6. Multiply each term by 6: 3x + 2x = 30. This simplifies to 5x = 30, so x = 6.

解 x/2 + x/3 = 5。2 和 3 的最小公倍数是 6。每项乘以 6：3x + 2x = 30。化简为 5x = 30，所以 x = 6。

Remember: Multiply every term by the LCM, not just the fraction terms. | 记住：每一项都要乘以 LCM，而不仅仅是分数项。
Watch out for: (x + 1)/3 = 2. Multiply both sides by 3 first: x + 1 = 6, so x = 5. | 注意：(x + 1)/3 = 2。先两边乘以 3：x + 1 = 6，所以 x = 5。

For equations with a fraction on one side equal to a fraction on the other, cross-multiplication is often the fastest method. If a/b = c/d, then ad = bc.

对于一边是分数、另一边也是分数的方程，交叉相乘通常是最快的方法。如果 a/b = c/d，则 ad = bc。

5. Forming Equations from Word Problems | 从应用题中建立方程

Word problems test your ability to translate English sentences into mathematical language. Look for keywords: “sum” means addition, “difference” means subtraction, “product” means multiplication, “is” means equals.

应用题考查你将英文句子转化为数学语言的能力。注意关键词：”sum” 表示加法，”difference” 表示减法，”product” 表示乘法，”is” 表示等于。

Example: “Three more than twice a number is 17. Find the number.” Let the number be n. Then 2n + 3 = 17, so 2n = 14, and n = 7.

例如：”某个数的两倍再加 3 等于 17，求这个数。” 设这个数为 n。则 2n + 3 = 17，所以 2n = 14，n = 7。

Define your variable first: Let x represent the unknown quantity. | 先定义变量：设 x 表示未知量。
Break the problem into parts: Write down expressions for each piece of information. | 分解问题：为每条信息写下对应的表达式。
Interpret your answer: Check if the answer makes sense in context. | 解释答案：检查答案在上下文中是否合理。

Age problems, money problems, and geometry problems are the most common types in IGCSE. Practice reading carefully — one wrong word can change the equation entirely.

年龄问题、金钱问题和几何问题是 IGCSE 中最常见的类型。务必仔细阅读——一个词理解错误可能完全改变方程。

6. Introduction to Inequalities | 不等式入门

An inequality compares two expressions using signs: > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to). Solving inequalities is similar to solving equations, with one critical exception.

不等式使用符号比较两个表达式：>（大于）、<（小于）、≥（大于或等于）、≤（小于或等于）。解不等式与解方程类似，但有一个关键区别。

Golden rule: When you multiply or divide both sides by a negative number, reverse the inequality sign.

黄金法则：当两边同时乘以或除以一个负数时，必须反转不等号。

Solve −2x < 6. Divide both sides by −2 and reverse the sign: x > −3. To check, try x = 0: −2(0) = 0 < 6, which is true. Try x = −4: −2(−4) = 8 < 6 is false, so x > −3 is correct.

解 −2x < 6。两边除以 −2 并反转不等号：x > −3。验证一下，取 x = 0：−2(0) = 0 < 6，成立。取 x = −4：−2(−4) = 8 < 6 不成立，所以 x > −3 是正确的。

7. Solving Linear Inequalities | 解线性不等式

The steps for solving linear inequalities mirror those for equations: simplify both sides, collect variable terms, and isolate the variable. The only difference is the inequality sign and the reversal rule for negatives.

解线性不等式的步骤与解方程相似：两边化简、合并变量项、分离变量。唯一的区别在于不等号以及负数时的反转规则。

Solve 3x + 2 ≥ 11. Subtract 2: 3x ≥ 9. Divide by 3: x ≥ 3. The solution set is all real numbers greater than or equal to 3.

解 3x + 2 ≥ 11。减去 2：3x ≥ 9。除以 3：x ≥ 3。解集是所有大于或等于 3 的实数。

Example: 2(4 − x) > 6 → 8 − 2x > 6 → −2x > −2 → x < 1. | 例：2(4 − x) > 6 → 8 − 2x > 6 → −2x > −2 → x < 1。
Number line representation: Use an open circle for < or >, a closed circle for ≤ or ≥. | 数轴表示：< 或 > 用空心圆，≤ 或 ≥ 用实心圆。

In IGCSE, you may be asked to represent the solution on a number line or write it in set notation. For example, x ≥ 3 can be written as {x : x ≥ 3}.

在 IGCSE 中，你可能被要求在数轴上表示解集，或用集合符号写出。例如，x ≥ 3 可以写作 {x : x ≥ 3}。

8. Double Inequalities | 双重不等式

A double inequality shows a variable between two values, such as −2 < x ≤ 5. This means x is greater than −2 and less than or equal to 5. You can solve compound inequalities by working on all three parts simultaneously.

双重不等式表示变量介于两个值之间，例如 −2 < x ≤ 5。这意味着 x 大于 −2 且小于或等于 5。你可以同时处理三个部分来解复合不等式。

Solve −3 < 2x + 1 < 7. Subtract 1 from all parts: −4 < 2x < 6. Divide all parts by 2: −2 < x < 3.

解 −3 < 2x + 1 < 7。所有部分同时减去 1：−4 < 2x < 6。所有部分同时除以 2：−2 < x < 3。

Always apply the same operation to all three parts of a double inequality.

对双重不等式的三个部分始终施以相同的运算。

If you multiply or divide by a negative number, you must reverse both inequality signs. For instance, −2 < −x < 4 becomes 2 > x > −4, which is better written as −4 < x < 2.

如果乘以或除以负数，必须反转两个不等号。例如，−2 < −x < 4 变为 2 > x > −4，更规范的写法是 −4 < x < 2。

9. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Even high-achieving students lose marks on linear equations and inequalities due to small but avoidable errors. Here is a list of the most frequent pitfalls.

即使是成绩优异的学生，也会在线性方程与不等式上因细小但可避免的错误而丢分。以下是最高频的陷阱清单。

Mistake \| 错误	Correction \| 纠正
Forgetting to reverse the inequality sign when dividing by a negative \| 除以负数时忘记反转不等号	Always reverse the sign when multiplying/dividing by a negative \| 乘以/除以负数时必须反转不等号
Only multiplying fraction terms by the LCM \| 只将分数项乘以 LCM	Multiply every term on both sides by the LCM \| 两边每一项都要乘以 LCM
Dropping the negative sign when moving a term \| 移项时丢掉负号	Write down the sign carefully before moving each term \| 移项前仔细写下每一项的符号
Not checking the answer \| 不检验答案	Substitute the solution back into the original equation \| 将解代回原方程验证

Additionally, always write your final answer clearly. In inequalities, the final answer should have the variable on the left side, as in x > 2 rather than 2 < x, unless the question asks for a number line.

此外，始终清晰写出最终答案。在不等式中，最终答案应将变量放在左侧，如 x > 2 而不是 2 < x，除非题目要求数轴表示。

10. Exam-Style Practice Questions | 考试风格练习题

Now let’s apply these skills to real exam-style questions. Try each one before reading the solution.

现在让我们将这些技能应用到真实的考试风格题目中。先尝试解答每一题，再看解答。

Question 1: Solve 5(x − 2) = 3x + 4. | 题目 1：解 5(x − 2) = 3x + 4。

Expand: 5x − 10 = 3x + 4. Subtract 3x: 2x − 10 = 4. Add 10: 2x = 14. Divide: x = 7.

展开：5x − 10 = 3x + 4。减去 3x：2x − 10 = 4。加 10：2x = 14。除以 2：x = 7。

Question 2: Solve (x + 3)/4 = 2. | 题目 2：解 (x + 3)/4 = 2。

Multiply both sides by 4: x + 3 = 8. Subtract 3: x = 5.

两边乘以 4：x + 3 = 8。减去 3：x = 5。

Question 3: Solve −3 ≤ 2x − 1 < 5. | 题目 3：解 −3 ≤ 2x − 1 < 5。

Add 1 to all parts: −2 ≤ 2x < 6. Divide by 2: −1 ≤ x < 3. The solution is all real numbers from −1 (included) up to but not including 3.

所有部分加 1：−2 ≤ 2x < 6。除以 2：−1 ≤ x < 3。解集是大于等于 −1 且小于 3 的所有实数。

11. Word Problem Worked Example | 应用题详解

Let’s work through a classic IGCSE word problem step by step.

让我们逐步解析一道经典的 IGCSE 应用题。

“A rectangle has a length of 3x cm and a width of x cm. The perimeter is 48 cm. Find the area of the rectangle.”

“一个长方形的长为 3x cm，宽为 x cm。周长为 48 cm。求长方形的面积。”

Perimeter formula: 2(length + width) = 48. So 2(3x + x) = 48, which gives 2(4x) = 48, so 8x = 48, and x = 6.

周长公式：2(长 + 宽) = 48。所以 2(3x + x) = 48，即 2(4x) = 48，因此 8x = 48，x = 6。

Length = 3 × 6 = 18 cm. Width = 6 cm. Area = length × width = 18 × 6 = 108 cm².

长 = 3 × 6 = 18 cm。宽 = 6 cm。面积 = 长 × 宽 = 18 × 6 = 108 cm²。

Step 1: Write down what you know. | 第一步：写下已知条件。
Step 2: Form an equation using the given relationship. | 第二步：根据已知关系建立方程。
Step 3: Solve the equation. | 第三步：解方程。
Step 4: Use the solution to answer the actual question asked. | 第四步：利用解来回答题目真正问的问题。

Notice that the question asked for the area, not for x. Many students solve for x and stop — they lose marks. Always read the final line of the question twice.

注意题目要求的是面积，而不是 x。许多学生求出 x 就停笔了——因此丢分。务必把题目的最后一句话读两遍。

12. Final Revision Tips | 最终复习建议

To wrap up, here are concise revision strategies to solidify your mastery of linear equations and inequalities before the exam.

最后，这里是简明扼要的复习策略，帮助你在考试前巩固对线性方程与不等式的掌握。

Practice daily: Solve at least 5 equations and 3 inequalities every day for two weeks before the exam. | 每日练习：考前两周每天至少解 5 个方程和 3 个不等式。
Memorise key phrases: “Reverse the sign” for negative multiplication/division in inequalities. | 记住关键口诀：不等式乘除负数要”反转不等号”。
Use estimation: Before solving, estimate what the answer should roughly be. This helps catch silly errors. | 先估算：解题前先大致估计答案范围，有助于发现离谱的错误。
Review past papers: IGCSE questions repeat similar patterns. Identify your weak area and target it. | 刷历年真题：IGCSE 题目有相似的出题模式。找出你的薄弱环节并针对性突破。

Mastering linear equations and inequalities is not just about passing an exam — it builds the algebraic foundation needed for quadratic equations, functions, and calculus at A-Level. Invest the time now, and you will benefit for years to come.

掌握线性方程与不等式不仅仅是为了通过考试——它为你学习二次方程、函数以及 A-Level 阶段的微积分奠定代数基础。现在投入时间，未来多年你都将受益。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mistake \| 错误	Correction \| 纠正
Forgetting to reverse the inequality sign when dividing by a negative \| 除以负数时忘记反转不等号	Always reverse the sign when multiplying/dividing by a negative \| 乘以/除以负数时必须反转不等号
Only multiplying fraction terms by the LCM \| 只将分数项乘以 LCM	Multiply every term on both sides by the LCM \| 两边每一项都要乘以 LCM
Dropping the negative sign when moving a term \| 移项时丢掉负号	Write down the sign carefully before moving each term \| 移项前仔细写下每一项的符号
Not checking the answer \| 不检验答案	Substitute the solution back into the original equation \| 将解代回原方程验证

IGCSE Mathematics: Mastering Linear Equations & Inequalities | 攻克线性方程与不等式