Solving Linear Equations & Inequalities: A Complete IGCSE Guide | 解线性方程与不等式：IGCSE 数学完整指南

📚 Solving Linear Equations & Inequalities | 解线性方程与不等式：IGCSE 数学完整指南

Linear equations and inequalities are the foundation of algebra. In this revision guide, we will break down the key skills you need for IGCSE Mathematics, with worked examples, common mistakes, and exam-style practice.

线性方程与不等式是代数的基石。在本次复习指南中，我们将系统梳理 IGCSE 数学所需的解题技巧，并提供例题、常见错误分析和考试风格练习。

---

1. What Are Linear Equations and Inequalities? | 什么是线性方程与不等式？

A linear equation is an equation where every variable has an exponent of 1. It can be written in the form ax + b = c, where a, b and c are constants and x is the unknown.

线性方程是指所有变量的指数均为 1 的方程。它可以写成 ax + b = c 的形式，其中 a、b、c 是常数，x 是未知数。

A linear inequality compares two expressions using <, >, ≤ or ≥. For example, 2x + 3 > 7 is a linear inequality because x only appears to the first power.

线性不等式使用 <、>、≤ 或 ≥ 来比较两个表达式。例如，2x + 3 > 7 是一个线性不等式，因为 x 只以一次幂出现。

The solution of a linear equation is usually a single number. The solution of a linear inequality is usually a range of numbers.

线性方程的解通常是一个具体的数值；而线性不等式的解通常是一个数值范围。

---

2. The Balance Method | 平衡法（逆运算）

To solve a linear equation, think of the equation as a set of scales. Whatever you do to one side, you must do to the other side to keep it balanced.

解线性方程时，可以把方程看作一架天平。对一边做的运算，必须对另一边做同样的运算，才能保持平衡。

Use inverse operations to isolate the variable: addition and subtraction are opposites; multiplication and division are opposites.

使用逆运算来隔离未知数：加法与减法互为逆运算，乘法与除法互为逆运算。

Example: Solve x + 5 = 12.

例：解方程 x + 5 = 12。

x + 5 − 5 = 12 − 5 → x = 7

Check: 7 + 5 = 12, so the solution is correct.

验算：7 + 5 = 12，所以解是正确的。

Example: Solve 3x = 21.

例：解方程 3x = 21。

3x ÷ 3 = 21 ÷ 3 → x = 7

Always show the inverse operation clearly in your working.

在解题过程中，请清晰地写出每一步逆运算。

---

3. Equations with Variables on Both Sides | 含有变量在等号两边的方程

When the variable appears on both sides, collect the variable terms on one side and the constant terms on the other side.

当未知数同时出现在等号两边时，应把含未知数的项移到一边，把常数项移到另一边。

Example: Solve 5x − 2 = 3x + 8.

例：解方程 5x − 2 = 3x + 8。

5x − 3x = 8 + 2 → 2x = 10 → x = 5

When moving a term across the equals sign, change its sign: positive becomes negative, and negative becomes positive.

把项移到等号另一边时，需要改变符号：正变负，负变正。

For inequalities, the same rule applies when collecting variables, but remember to keep the inequality symbol pointing in the correct direction.

对于不等式，移项规则相同，但要确保不等号的方向始终正确。

---

4. Equations with Brackets and Fractions | 含括号和分数的方程

Brackets are removed using the distributive law: a(b + c) = ab + ac.

使用分配律去除括号：a(b + c) = ab + ac。

Example: Solve 4(x + 2) = 3x + 12.

例：解方程 4(x + 2) = 3x + 12。

4x + 8 = 3x + 12

4x − 3x = 12 − 8 → x = 4

To solve an equation with fractions, multiply every term by the lowest common multiple (LCM) of the denominators.

解含分数的方程时，用分母的最小公倍数（LCM）乘以方程中的每一项。

Example: Solve x/2 + 1 = x/3 + 3.

例：解方程 x/2 + 1 = x/3 + 3。

The LCM of 2 and 3 is 6. Multiply every term by 6:

2 和 3 的最小公倍数是 6。用 6 乘以每一项：

6 × (x/2) + 6 × 1 = 6 × (x/3) + 6 × 3

3x + 6 = 2x + 18 → x = 12

Always check whether your solution satisfies the original equation.

完成计算后，务必检验解是否满足原方程。

---

5. Forming Equations from Word Problems | 根据应用题列方程

Word problems require you to translate English or Chinese statements into algebraic equations.

应用题要求你把文字叙述转化为代数方程。

Step 1: Define the unknown clearly, for example “let x be the number of books”.

第一步：明确设未知数，例如“设 x 为书本数量”。

Step 2: Write down the relationships as an equation.

第二步：把数量关系写成方程。

Step 3: Solve the equation and answer the question with units.

第三步：解方程，并带上单位回答问题。

Example: The sum of a number and 9 is three times the number. Find the number.

例：一个数与 9 的和等于这个数的 3 倍，求这个数。

x + 9 = 3x

9 = 2x → x = 4.5

Check: 4.5 + 9 = 13.5 and 3 × 4.5 = 13.5, so the answer is correct.

验算：4.5 + 9 = 13.5，且 3 × 4.5 = 13.5，所以答案正确。

---

6. Solving Simple Linear Inequalities | 解简单线性不等式

Solve inequalities in the same way as equations, but write an inequality symbol instead of an equals sign.

解不等式的方法与解方程类似，只是把等号换成不等号。

Example: Solve x + 3 > 7.

例：解不等式 x + 3 > 7。

x > 7 − 3 → x > 4

This means every number greater than 4 is a solution.

这意味着所有大于 4 的数都是解。

Example: Solve 5x ≤ 30.

例：解不等式 5x ≤ 30。

x ≤ 30 ÷ 5 → x ≤ 6

Notice that x = 6 is included because of the ≤ symbol.

注意：因为使用 ≤，所以 x = 6 是包含在解集内的。

---

7. Representing Solutions on a Number Line | 在数轴上表示解集

On a number line, an open circle (○) means the endpoint is not included, while a closed circle (●) means the endpoint is included.

在数轴上，空心圆圈（○）表示不包含该端点，实心圆圈（●）表示包含该端点。

For x > 4, draw an open circle at 4 and an arrow to the right.

对于 x > 4，在 4 处画空心圆，并向右画箭头。

For x ≤ 6, draw a closed circle at 6 and an arrow to the left.

对于 x ≤ 6，在 6 处画实心圆，并向左画箭头。

For combined inequalities such as −2 < x ≤ 3, draw an open circle at −2 and a closed circle at 3, then connect them with a line.

对于组合不等式 −2 < x ≤ 3，在 −2 处画空心圆，在 3 处画实心圆，然后用线段连接两点。

You may also be asked to write the solution set in the form of an interval, such as x ∈ (−2, 3].

题目也可能要求你用区间表示解集，例如 x ∈ (−2, 3]。

---

8. Solving Double Inequalities | 解双重不等式

A double inequality has the form a < b < c and can be solved by performing the same operation on all three parts.

双重不等式的形式为 a < b < c，求解时需要对三部分同时进行相同的运算。

Example: Solve −3 < 2x + 1 ≤ 7.

例：解不等式 −3 < 2x + 1 ≤ 7。

Step 1: Subtract 1 from all three parts.

第一步：三部分同时减去 1。

−3 − 1 < 2x ≤ 7 − 1

−4 < 2x ≤ 6

Step 2: Divide all three parts by 2.

第二步：三部分同时除以 2。

−2 < x ≤ 3

The solution is every number greater than −2 and less than or equal to 3.

解是所有大于 −2 且小于或等于 3 的数。

---

9. Multiplying or Dividing by a Negative Number | 乘以或除以负数时改变不等号方向

This is the most important rule in solving inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign.

这是解不等式最重要的规则：两边同时乘以或除以一个负数时，不等号方向必须反转。

Example: Solve −2x < 6.

例：解不等式 −2x < 6。

x > 6 ÷ (−2) → x > −3

Notice that the < symbol became > because we divided by −2.

注意：因为两边除以 −2，所以 < 变成了 >。

Why does this happen? Consider −2 < 1. If you multiply both sides by −1, you get 2 > −1, so the inequality sign must flip.

为什么如此？以 −2 < 1 为例，两边同时乘以 −1 后得到 2 > −1，因此不等号必须调转方向。

Do not forget this rule when solving inequalities in the exam.

考试中解不等式时，切勿忘记这一规则。

---

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

Mistake 1: Forgetting to flip the inequality sign when multiplying or dividing by a negative number.

错误一：在乘以或除以负数时忘记反转不等号。

Mistake 2: Moving a term across the equals sign without changing its sign.

错误二：把项移到等号另一边时没有改变符号。

Mistake 3: Not checking the solution by substituting it back into the original equation.

错误三：没有把解代回原方程进行检验。

Mistake 4: Drawing the wrong circle on a number line: open vs closed.

错误四：在数轴上画错圆点的类型：空心圆与实心圆混淆。

Tip 1: Always write “x = …” clearly in your final answer.

技巧一：在最终答案中清楚写出“x = …”。

Tip 2: For word problems, include the units in your final answer.

技巧二：在应用题中，最终答案要包括单位。

Tip 3: If you are unsure about an inequality, test a number from your solution set in the original inequality.

技巧三：如果不确定不等式的解集，可以在原不等式中代入解集内的一个数进行验证。

---

11. Practice Questions | 练习题

Try these questions on your own before looking at the answers.

请先独立完成以下题目，再查看答案解析。

Question 1: Solve 7x − 4 = 2x + 11.

第 1 题：解方程 7x − 4 = 2x + 11。

Question 2: Solve 3(x + 2) = 2(2x − 1).

第 2 题：解方程 3(x + 2) = 2(2x − 1)。

Question 3: Solve x/4 + 2 = x/2 − 1.

第 3 题：解方程 x/4 + 2 = x/2 − 1。

Question 4: Solve 5 − 3x ≥ 14.

第 4 题：解不等式 5 − 3x ≥ 14。

Question 5: Solve −4 ≤ 2x − 6 < 8 and represent the solution on a number line.

第 5 题：解不等式 −4 ≤ 2x − 6 < 8，并在数轴上表示解集。

---

12. Answers and Explanations | 答案与解析

Answer 1: 7x − 4 = 2x + 11 → 7x − 2x = 11 + 4 → 5x = 15 → x = 3.

答案 1：7x − 4 = 2x + 11 → 7x − 2x = 11 + 4 → 5x = 15 → x = 3。

Answer 2: 3(x + 2) = 2(2x − 1) → 3x + 6 = 4x − 2 → 6 + 2 = 4x − 3x → x = 8.

答案 2：3(x + 2) = 2(2x − 1) → 3x + 6 = 4x − 2 → 6 + 2 = 4x − 3x → x = 8。

Answer 3: x/4 + 2 = x/2 − 1. Multiply by 4: x + 8 = 2x − 4 → 12 = x → x = 12.

答案 3：x/4 + 2 = x/2 − 1。两边乘以 4：x + 8 = 2x − 4 → 12 = x → x = 12。

Answer 4: 5 − 3x ≥ 14 → −3x ≥ 9 → divide by −3 and flip sign: x ≤ −3.

答案 4：5 − 3x ≥ 14 → −3x ≥ 9 → 两边除以 −3 并反转不等号：x ≤ −3。

Answer 5: −4 ≤ 2x − 6 < 8 → add 6 to all parts: 2 ≤ 2x < 14 → divide by 2: 1 ≤ x < 7. On a number line, draw a closed circle at 1 and an open circle at 7, then connect them.

答案 5：−4 ≤ 2x − 6 < 8 → 三部分加 6：2 ≤ 2x < 14 → 三部分除以 2：1 ≤ x < 7。在数轴上，1 处画实心圆，7 处画空心圆，然后连接两点。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）