Unit 2 Algebra Essentials: Expressions, Equations and Inequalities | 第二单元代数核心：表达式、方程与不等式

📚 Unit 2 Algebra Essentials: Expressions, Equations and Inequalities | 第二单元代数核心：表达式、方程与不等式

Welcome to this Unit 2 algebra revision guide for Grade 9 Mathematics. This article focuses on the core skills you need for algebraic expressions, linear equations, inequalities and formulae. Each section gives a clear explanation followed by a worked example so you can check your understanding as you progress.

欢迎使用本九年级数学第二单元代数复习指南。本文重点讲解代数式、线性方程、不等式和公式所需的核心技能。每个部分都先给出清晰讲解，再配以例题，便于你随时检测理解程度。

1. Reading and Writing Algebraic Expressions | 读与写代数式

An algebraic expression uses letters such as x, y or n to stand for unknown numbers. For example, “5 more than a number” can be written as x + 5, while “3 times a number” is written as 3x. The letter is called a variable, and the number in front of it is the coefficient.

代数式使用 x、y 或 n 等字母表示未知数。例如，“比一个数大 5”可以写成 x + 5，而“一个数的 3 倍”写成 3x。这个字母称为变量，它前面的数字称为系数。

When writing expressions, multiplication signs are usually omitted between a number and a variable, so 4 × m is written as 4m. Division is often shown as a fraction, so a divided by 3 can be written as a / 3 or a ÷ 3.

书写代数式时，数字与变量之间的乘号通常省略，因此 4 × m 写作 4m。除法常用分数形式表示，因此 a 除以 3 可以写作 a / 3 或 a ÷ 3。

Let n = a number. Then “twice the number increased by 7” is 2n + 7.

设 n 为一个数，则“这个数的两倍再加 7”为 2n + 7。

2. Collecting Like Terms | 合并同类项

Like terms contain exactly the same variable or combination of variables raised to the same power. For example, 3a and -5a are like terms, but 3a and 3a² are not. To simplify an expression, add or subtract the coefficients of like terms.

同类项包含完全相同的变量或变量组合，且指数相同。例如，3a 和 -5a 是同类项，但 3a 和 3a² 不是。化简代数式时，只需要将同类项的系数相加或相减。

Always keep the variable part unchanged when collecting like terms. For instance, 7x – 2x = 5x, not 5x². You can also collect constant terms such as 4 and -9 to get -5.

合并同类项时，变量部分始终保持不变。例如，7x – 2x = 5x，而不是 5x²。常数项也可以合并，例如 4 和 -9 合并得到 -5。

5a + 3b – 2a + 7b = 3a + 10b

5a – 2a = 3a
3b + 7b = 10b
Final answer: 3a + 10b

3. Expanding Single Brackets | 单项式乘括号展开

To expand a bracket, multiply each term inside the bracket by the term outside. This uses the distributive property: a(b + c) = ab + ac. Always be careful with negative signs and remember that the sign travels with the term.

展开括号时，要用括号外的项乘以括号内的每一项。这运用了分配律：a(b + c) = ab + ac。务必注意负号，符号要跟随相应的项一起移动。

For example, -2(3x – 4) means -2 × 3x plus -2 × -4. The result is -6x + 8, because a negative times a negative gives a positive.

例如，-2(3x – 4) 表示 -2 × 3x 加上 -2 × -4。结果是 -6x + 8，因为负数乘负数得正数。

4(2x + 3) = 8x + 12 and 3(x – 5) = 3x – 15

4(2x + 3) = 8x + 12，3(x – 5) = 3x – 15

4. Factorising Linear Expressions | 线性表达式因式分解

Factorising is the reverse of expanding. It means writing an expression as a product of brackets. To factorise a linear expression, find the highest common factor of all the terms and place it outside the bracket.

因式分解是展开的逆运算，也就是把一个代数式写成括号乘积的形式。对线性表达式进行因式分解时，先找出各项的最大公因数，并把它放在括号外。

For example, 8x + 12 has a highest common factor of 4, so it can be written as 4(2x + 3). You can always check your answer by expanding the bracket back.

例如，8x + 12 的最大公因数是 4，所以可以写成 4(2x + 3)。你可以随时通过重新展开括号来检验答案是否正确。

15x – 10 = 5(3x – 2)

5. One-Step and Two-Step Equations | 一步与两步方程

An equation states that two expressions are equal. To solve it, perform inverse operations to isolate the variable on one side. Whatever you do to one side of the equation, you must also do to the other side.

方程表示两个代数式相等。解方程时，需要运用逆运算把变量单独留在方程的一边。对方程一边所做的任何运算，另一边也必须同步进行。

A one-step equation requires only one inverse operation, such as x + 6 = 11, so x = 5. A two-step equation requires two operations, such as 3x – 4 = 11; first add 4 to both sides, then divide both sides by 3.

一步方程只需要一次逆运算，例如 x + 6 = 11，所以 x = 5。两步方程需要两次运算，例如 3x – 4 = 11；先两边加 4，再两边同时除以 3。

3x – 4 = 11 ⇒ 3x = 15 ⇒ x = 5

6. Equations with Brackets and Unknowns on Both Sides | 带括号及未知数在两侧的方程

If an equation contains brackets, expand them first. Then collect like terms on each side before isolating the variable. This often turns a complicated-looking equation into a simple two-step or multi-step equation.

如果方程含有括号，应先展开括号。然后在方程两边分别合并同类项，再分离变量。这样做通常能把看似复杂的方程转化为简单的两步或多步方程。

When the unknown appears on both sides, remove the smaller variable term first. For example, in 5x + 2 = 3x + 10, subtract 3x from both sides to get 2x + 2 = 10, then solve as usual.

当未知数出现在方程两边时，先消去较小的变量项。例如，在 5x + 2 = 3x + 10 中，两边同时减去 3x，得到 2x + 2 = 10，然后按正常方法求解。

2(x + 3) = x + 8 ⇒ 2x + 6 = x + 8 ⇒ x = 2

7. Forming Equations from Word Problems | 根据文字题列方程

Many exam questions ask you to form an equation from a real-life situation. Start by defining a variable for the unknown quantity, then translate the words into algebraic language. Look for key phrases such as “total”, “difference”, “product” and “is equal to”.

许多考试题目要求根据实际情境列出方程。首先为未知量设一个变量，然后把文字转化为代数语言。注意“总和”“差”“乘积”“等于”等关键词语。

Always state clearly what your variable represents. For example, let x be the cost of one notebook. If three notebooks and a £2 pen cost £14, the equation is 3x + 2 = 14.

务必清晰说明变量代表什么。例如，设 x 为一本笔记本的价格。如果三本笔记本和一支 2 英镑的笔共花费 14 英镑，那么方程就是 3x + 2 = 14。

3x + 2 = 14 ⇒ 3x = 12 ⇒ x = 4, so one notebook costs £4.

3x + 2 = 14 ⇒ 3x = 12 ⇒ x = 4，所以一本笔记本价格为 4 英镑。

8. Introduction to Inequalities | 不等式入门

An inequality compares two expressions using signs such as <, >, ≤ or ≥. The symbol < means “less than”, > means “greater than”, ≤ means “less than or equal to”, and ≥ means “greater than or equal to”.

不等式使用 <、>、≤ 或 ≥ 等符号比较两个代数式。符号 < 表示“小于”，> 表示“大于”，≤ 表示“小于或等于”，≥ 表示“大于或等于”。

Inequalities can be shown on a number line. A hollow circle is used for strict inequalities such as x < 3 or x > 1, while a filled circle is used for x ≤ 3 or x ≥ 1.

不等式可以在数轴上表示。严格不等式如 x < 3 或 x > 1 使用空心圆，而 x ≤ 3 或 x ≥ 1 使用实心圆。

Symbol \| 符号	Meaning \| 含义
<	less than \| 小于
>	greater than \| 大于
≤	less than or equal to \| 小于或等于
≥	greater than or equal to \| 大于或等于

9. 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 by the same positive number without changing the inequality sign.

解线性不等式与解线性方程非常相似。你可以对两边同时加、减、乘或除以同一个正数，而不改变不等号方向。

However, if you multiply or divide both sides by a negative number, you must reverse the inequality sign. For example, -2x < 6 becomes x > -3 after dividing both sides by -2 and flipping the sign.

但是，如果两边同时乘或除以一个负数，则必须反转不等号方向。例如，-2x < 6 两边同时除以 -2 后变为 x > -3，并且不等号方向翻转。

2x – 3 < 7 ⇒ 2x < 10 ⇒ x < 5

Always present your final answer as a clear range, such as x > -3 or -2 ≤ x < 4, and remember to draw the number line if the question requires it.

最终答案要写成明确的范围，例如 x > -3 或 -2 ≤ x < 4。如果题目要求画数轴，记得正确标出空心或实心圆。

10. Rearranging Formulae and Exam Tips | 变换公式与考试技巧

A formula is an equation that shows the relationship between different variables. To make a different variable the subject, use the same inverse operations as you would when solving an equation. For example, v = u + at can be rearranged to a = (v – u) / t.

公式是表示不同变量之间关系的方程。要把另一个变量变为主项，使用与解方程相同的逆运算即可。例如，v = u + at 可以变换为 a = (v – u) / t。

In exams, always show each step clearly. Common mistakes include forgetting to reverse the inequality sign, losing a negative sign when expanding brackets, and failing to check answers by substitution. A quick substitution check can save many marks.

考试时务必清晰写出每一步。常见错误包括忘记反转不等号、展开括号时丢失负号，以及没有通过代入检验答案。快速代入检验往往能避免大量失分。

Given E = mc², make m the subject: m = E / c²

已知 E = mc²，将 m 作为主项：m = E / c²

Practice forming equations from word problems and rearranging formulae under timed conditions. The more you practise, the more confident you will be in identifying the correct operation and avoiding sign errors.

请在限时条件下多练习根据文字题列方程和变换公式。练习越多，你就越能自信地判断正确运算并避免符号错误。

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Symbol \| 符号	Meaning \| 含义
<	less than \| 小于
>	greater than \| 大于
≤	less than or equal to \| 小于或等于
≥	greater than or equal to \| 大于或等于

Unit 2 Algebra Essentials: Expressions, Equations and Inequalities | 第二单元代数核心：表达式、方程与不等式