Linear Equations and Inequalities | 线性方程与不等式

📚 Linear Equations and Inequalities | 线性方程与不等式

This revision guide covers linear equations and inequalities for Cambridge Lower Secondary Stage 8 Mathematics. You will learn how to solve equations by balancing, handle brackets and variables on both sides, form equations from real-life problems, and solve inequalities with correct number line diagrams.

本复习指南涵盖剑桥初中数学第八阶段的线性方程与不等式。你将学习如何通过平衡法解方程、处理括号和两边含未知数的方程、从实际问题建立方程，并正确求解不等式和绘制数轴图示。

These skills are tested regularly in Checkpoint and school assessments, so practising the steps shown here will help you avoid losing marks.

这些技能在 Checkpoint 和校内评估中经常出现，因此练习本文展示的步骤将帮助你避免失分。

1. Understanding Linear Equations | 认识线性方程

A linear equation is an equation in which the unknown, usually written as x, is only raised to the power of 1. This means there are no x², x³, or 1/x terms.

线性方程是指未知数（通常用 x 表示）的指数仅为 1 的方程。这意味着方程中没有 x²、x³ 或 1/x 这样的项。

For example, 2x + 3 = 11 and x − 5 = 2x + 4 are linear equations.

例如，2x + 3 = 11 和 x − 5 = 2x + 4 都是线性方程。

The solution of an equation is the value of x that makes the statement true. An equation is like a language: it tells you that two expressions are equal.

方程的解就是使等式成立的 x 的值。方程就像一种语言：它告诉你两个表达式相等。

In Stage 8, most linear equations have one unknown and one solution, but some special equations may have no solution or infinitely many solutions.

在第八阶段，大多数一元一次方程只有一个未知数和一个解，但有些特殊方程可能无解或有无穷多个解。

2. Solving Equations by Balancing | 用平衡法解方程

Think of an equation as a balance scale. Whatever you do to one side, you must also do to the other side to keep the balance.

把方程想象成一个天平。无论你对一边做什么运算，另一边也必须做同样的运算，才能保持平衡。

To solve 2x + 3 = 11, subtract 3 from both sides first, then divide both sides by 2.

要解 2x + 3 = 11，先在两边同时减去 3，再将两边同时除以 2。

2x + 3 = 11 → 2x = 8 → x = 4

Always write each step on a new line. This is important in exams because method marks are often awarded for correct balancing, even if the final answer is wrong.

每一步都要另起一行书写。这在考试中很重要，因为即使最终答案错误，正确的平衡步骤通常也能获得方法分。

Use inverse operations to undo operations: addition is undone by subtraction, multiplication is undone by division, and vice versa.

使用逆运算来撤销原来的运算：加法用减法撤销，乘法用除法撤销，反之亦然。

3. Equations with Variables on Both Sides | 两边都含未知数的方程

When both sides contain x, collect the x terms on one side and the number terms on the other side.

当方程两边都含有 x 时，把含有 x 的项集中到一边，把常数项集中到另一边。

Example: solve 5x − 2 = 2x + 7.

例如：解 5x − 2 = 2x + 7。

5x − 2 = 2x + 7 → 3x = 9 → x = 3

In the first step, subtract 2x from both sides to remove the smaller x term. Then add 2 to both sides to isolate the x term.

第一步，先在两边同时减去 2x，以消去较小的 x 项。然后在两边同时加 2，使含 x 的项单独出现。

Remember: a term changes sign when it moves to the other side via inverse operations. Writing ‘change side, change sign’ can help you avoid mistakes.

记住：通过逆运算移项时，项的符号会改变。记住“移项变号”可以帮助你避免错误。

4. Expanding Brackets in Equations | 方程中的去括号

If an equation contains a bracket, expand it first before collecting like terms.

如果方程中含有括号，先展开括号，再合并同类项。

For example, solve 4(x − 3) = 2x + 6.

例如，解 4(x − 3) = 2x + 6。

4(x − 3) = 2x + 6 → 4x − 12 = 2x + 6 → 2x = 18 → x = 9

Expanding means multiplying the term outside the bracket by each term inside the bracket. Be careful when the outside term is negative: for example, −3(x − 2) = −3x + 6.

展开就是用括号外的项乘以括号内的每一项。当括号外的项是负数时要特别小心：例如，−3(x − 2) = −3x + 6。

After expanding, use the balancing method to solve the equation as usual. Always simplify both sides before moving terms.

展开之后，再像往常一样使用平衡法解方程。在移项之前，一定要先化简两边。

5. Forming Equations from Word Problems | 由实际问题建立方程

Word problems can often be translated into linear equations. Identify the unknown, give it a letter, and write an equation from the given relationship.

文字题通常可以转化为线性方程。先确定未知数并用字母表示，然后根据已知关系写出方程。

Example: A rectangle has length 3 cm more than its width. Its perimeter is 26 cm. Find the width.

例如：一个长方形的长比宽多 3 cm，周长为 26 cm。求宽。

Let the width be x cm. The length is then (x + 3) cm. Perimeter = 2 × width + 2 × length.

设宽为 x cm，则长为 (x + 3) cm。周长 = 2 × 宽 + 2 × 长。

2x + 2(x + 3) = 26 → 4x + 6 = 26 → x = 5

The width is 5 cm and the length is 8 cm. Always answer the question asked, not just the value of x.

宽为 5 cm，长为 8 cm。一定要回答题目所问，而不仅仅是给出 x 的值。

6. Introduction to Inequalities | 不等式入门

An inequality compares two expressions using one of these symbols: <, >, ≤, ≥.

不等式使用以下符号比较两个表达式：<、>、≤、≥。

x < 4 means x is less than 4; x ≥ −3 means x is greater than or equal to −3.

x < 4 表示 x 小于 4；x ≥ −3 表示 x 大于或等于 −3。

Symbol	Meaning	Example	中文含义
<	less than	x < 4	小于
>	greater than	x > −3	大于
≤	less than or equal to	x ≤ 7	小于或等于
≥	greater than or equal to	x ≥ −1	大于或等于

Inequalities often have infinitely many solutions, unlike simple equations that usually have one solution.

与通常只有一个解的简单方程不同，不等式往往有无穷多个解。

7. Solving Linear Inequalities | 解一元一次不等式

Solving linear inequalities uses the same balancing method as equations, with one important difference.

解一元一次不等式使用的平衡法与方程相同，但有一个重要区别。

When you multiply or divide both sides by a negative number, reverse the inequality symbol.

当两边同时乘以或除以一个负数时，必须反转不等号的方向。

Example: solve −2x ≤ 6.

例如：解 −2x ≤ 6。

−2x ≤ 6 → x ≥ −3

The inequality reverses because dividing by −2 is dividing by a negative number. If no negative multiplication or division occurs, do not reverse the sign.

因为除以 −2 是除以负数，所以不等号反转。如果没有乘以或除以负数，就不要反转不等号。

Always check your solution by testing a value from the solution set in the original inequality.

始终通过从解集中选取一个值代入原不等式来进行检验。

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

Use an open circle for strict inequalities (< or >) and a closed circle for inclusive inequalities (≤ or ≥).

严格不等式（< 或 >）使用空心圆圈，包含不等式（≤ 或 ≥）使用实心圆点。

Shade the number line in the direction of all possible values.

在数轴上沿所有可能值的方向涂色。

For example, x ≥ −3 is shown with a closed circle at −3 and shading to the right.

例如，x ≥ −3 在数轴上表示为 −3 处画实心圆点，并向右涂色。

For x < 2, draw an open circle at 2 and shade to the left. These diagrams are often worth marks, so draw them carefully.

对于 x < 2，在 2 处画空心圆圈并向左涂色。这些图经常有分，所以要仔细绘制。

9. Common Mistakes and Checking Answers | 常见错误与答案检验

Many marks are lost through avoidable errors. The most common ones include the following.

许多分数因可避免的错误而丢失。最常见的错误包括以下几点。

Forgetting to reverse the inequality when dividing by a negative number. / 除以负数时忘记反转不等号。
Combining unlike terms, such as writing 2x + 3 = 5x. / 合并不同类项，例如把 2x + 3 写成 5x。
Only checking one side of an equation when substituting. / 代入检验时只检查方程的一边。
Losing the sign of a term when moving it across the equals sign. / 移项时丢失项的符号。

To check an equation solution, substitute your answer back into the original equation and confirm both sides are equal.

要检验方程的解，请将答案代回原方程，并确认两边相等。

For inequalities, choose a number inside your solution range and test it. If the original inequality is true, your answer is likely correct.

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Linear Equations and Inequalities | 线性方程与不等式