Unit 3 Higher Homework: Linear Equations and Inequalities | 第3单元提优作业：线性方程与不等式

📚 Unit 3 Higher Homework: Linear Equations and Inequalities | 第3单元提优作业：线性方程与不等式

This revision guide covers the core skills in Unit 3, pages 41-63 of the Higher Homework series. The focus is on solving linear equations and inequalities, forming equations from real-life contexts, and representing solutions on number lines. These topics appear frequently in GCSE Higher tier papers and in early A Level algebra review.

本复习指南涵盖 Higher Homework 系列第 3 单元第 41 至 63 页的核心技能。重点包括解一元一次方程和不等式、根据实际情境建立方程，以及在数轴上表示解集。这些主题在 GCSE 高阶段试卷和 A Level 代数复习中经常出现。

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1. Understanding Linear Equations | 理解线性方程

A linear equation is an algebraic statement in which the highest power of the variable is 1. Its graph is always a straight line. For example, 3x + 5 = 20 is linear because x appears only to the power 1.

一元一次方程是最高次数为 1 的代数等式。它的图像总是一条直线。例如，方程 3x + 5 = 20 就是一元一次方程，因为 x 只以 1 次方出现。

General form:

ax + b = c, where a ≠ 0

一般形式：

ax + b = c，其中 a ≠ 0

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2. The Balancing Method | 平衡法解方程

To solve a linear equation, keep the equation balanced by performing the same operation on both sides. Undo addition or subtraction first, then multiplication or division.

解一元一次方程时，要对等式两边同时进行相同运算以保持平衡。先消去加法或减法，再消去乘法或除法。

Example: Solve 2x + 3 = 11.

例如：解方程 2x + 3 = 11。

2x + 3 − 3 = 11 − 3

2x = 8

x = 8 ÷ 2 = 4

Always substitute your answer back into the original equation to check it.

始终将答案代回原方程进行检验。

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3. Equations with Fractions | 含分数的方程

When an equation contains fractions, multiply every term by the lowest common denominator (LCD) to clear the fractions. This makes the equation easier to solve.

当方程含有分数时，将每一项都乘以最小公分母（LCD）以去分母。这会使方程更容易求解。

Example: Solve x/3 + 2 = 5.

例如：解方程 x/3 + 2 = 5。

3 × (x/3) + 3 × 2 = 3 × 5

x + 6 = 15

x = 9

Be careful to multiply every term, including constants, by the LCD.

注意要将每一项，包括常数项，都乘以最小公分母。

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4. Forming Equations from Word Problems | 从应用题建立方程

Many Higher tier questions ask you to convert a written scenario into an equation. Identify the unknown, assign a variable, and translate keywords such as ‘total’, ‘difference’, and ‘product’ into operations.

高阶段试卷中的许多题目要求你将文字情境转化为方程。先确定未知量，设一个变量，再将“总共”“差”“乘积”等关键词转化为运算符号。

• ‘sum of x and 7’ becomes x + 7
• ‘three times a number’ becomes 3n
• ‘five less than y’ becomes y − 5

• “x 与 7 的和”可写成 x + 7
• “一个数的三倍”可写成 3n
• “比 y 少 5”可写成 y − 5

Always define the variable clearly before writing the equation.

在写出方程之前，始终要清楚地定义变量。

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5. Introduction to Inequalities | 不等式入门

An inequality compares two expressions using the symbols >, <, ≥, or ≤. Unlike an equation, an inequality usually has a range of solutions rather than one single value.

不等式使用 >、<、≥、≤ 等符号来比较两个表达式。与方程不同，不等式的解通常是一个取值范围，而不是一个单一的值。

Symbol	Meaning	Example
>	greater than	x > 4
<	less than	x < 4
≥	greater than or equal to	x ≥ 4
≤	less than or equal to	x ≤ 4

中文释义：> 大于；< 小于；≥ 大于或等于；≤ 小于或等于。

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

Solving a linear inequality is similar to solving a linear equation. However, if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

解一元一次不等式与解一元一次方程类似。但如果将两边同时乘以或除以一个负数，必须改变不等号的方向。

Example: Solve −2x < 10.

例如：解不等式 −2x < 10。

−2x ÷ (−2) > 10 ÷ (−2)

x > −5

This is the most common error in Higher homework, so always check the sign when

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