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

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

Linear equations and inequalities form the foundation of algebra. In the IGCSE syllabus, mastering them is essential for success in extended mathematics and beyond. This guide will walk you through every key skill, from algebraic notation to solving complex equations and inequalities, with worked examples and common pitfalls explained.

线性方程与不等式是代数的基础。在IGCSE课程中，掌握它们对于取得扩展数学高分以及后续学习至关重要。本指南将带你熟悉每一项核心技能，从代数符号到求解复杂的方程与不等式，并通过例题与常见易错点进行详细讲解。

1. Variables, Constants, and Expressions | 变量、常量与表达式

A variable is a symbol, usually a letter, that represents an unknown quantity. A constant is a fixed number. For example, in the expression 3x + 5, x is the variable, 3 is the coefficient, and 5 is the constant.

变量是表示未知数量的符号，通常用字母表示。常量是固定不变的数。例如，在表达式 3x + 5 中，x 是变量，3 是系数，5 是常量。

An algebraic expression contains variables, constants, and operations. An equation has an equals sign, while an inequality uses symbols such as >, <, ≥, or ≤. Distinguishing these is the first step toward solving them.

代数表达式包含变量、常量和运算。方程含有等号，而不等式使用 >、<、≥ 或 ≤ 等符号。分清这三者是求解的第一步。

Term: a number, a variable, or a product of both (e.g., 2x, -3, ½y).

项：一个数、一个变量或二者的乘积（例如 2x、-3、½y）。
Expression: a sum of terms (e.g., 4x – 7).

表达式：若干项的和（例如 4x – 7）。
Equation: two expressions connected by ‘=’ (e.g., 4x – 7 = 9).

方程：两个表达式用等号相连（例如 4x – 7 = 9）。

2. The Structure of Linear Equations | 线性方程的结构

A linear equation has no powers higher than 1. Its general form is ax + b = c, where a, b, and c are constants and a ≠ 0. The solution is the value of x that makes the equation true.

线性方程中未知数的最高次数为1。其一般形式为 ax + b = c，其中 a、b、c 是常数，且 a ≠ 0。方程的解是使等式成立的 x 的值。

ax + b = c (a ≠ 0)

For example, 2x + 3 = 11 is linear. Solving it gives x = 4. In IGCSE exams, you may encounter equations with fractions, brackets, and variables on both sides, but the underlying structure remains linear.

例如，2x + 3 = 11 是线性方程，其解为 x = 4。在IGCSE考试中，你可能会遇到含分数、括号和未知数在等号两侧的方程，但其基本结构依然是线性的。

3. Solving Linear Equations Using Inverse Operations | 用逆运算解线性方程

To isolate x, we use inverse operations. Addition and subtraction are inverses; multiplication and division are inverses. Whatever operation is applied to one side must be applied to the other side to keep the equation balanced.

为了分离出 x，我们要使用逆运算。加法与减法互为逆运算，乘法与除法互为逆运算。对一边进行的运算也必须对另一边进行，以保持等式的平衡。

Example: Solve 4x – 5 = 15.

例：解方程 4x – 5 = 15。

Add 5 to both sides: 4x = 20.

两边同时加 5：4x = 20。
Divide both sides by 4: x = 5.

两边同时除以 4：x = 5。

4x – 5 = 15 → 4x = 20 → x = 5

Always check your answer by substituting it back into the original equation: 4(5) – 5 = 15 ✓.

始终将答案代回原方程进行检验：4(5) – 5 = 15，成立。

4. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them using the distributive law: a(b + c) = ab + ac. Then solve the resulting equation using inverse operations.

当方程中含有括号时，先用乘法分配律展开：a(b + c) = ab + ac，然后用逆运算求解展开后的方程。

Example: Solve 3(2x – 1) = 15.

例：解方程 3(2x – 1) = 15。

Expand: 6x – 3 = 15.

展开：6x – 3 = 15。

Add 3: 6x = 18. Divide by 6: x = 3.

加 3：6x = 18。除以 6：x = 3。

3(2x – 1) = 15 → 6x – 3 = 15 → x = 3

5. Variables on Both Sides | 未知数在等式两边

If x appears on both sides of the equation, collect all x-terms on one side and constants on the other. This is done by adding or subtracting terms from both sides.

如果等式两边都含有 x，就将所有含 x 的项移到一边，常数项移到另一边。方法是在两边同时加上或减去相应的项。

Example: Solve 5x – 3 = 2x + 9.

例：解方程 5x – 3 = 2x + 9。

Subtract 2x from both sides: 3x – 3 = 9.

两边同时减去 2x：3x – 3 = 9。
Add 3 to both sides: 3x = 12.

两边同时加 3：3x = 12。
Divide by 3: x = 4.

两边同时除以 3：x = 4。

5x – 3 = 2x + 9 → 3x = 12 → x = 4

6. Equations with Fractions | 含分数的方程

A common IGCSE question involves fractions. Multiply every term by the common denominator to eliminate fractions, then solve as usual.

IGCSE常见题型涉及分数。将等式两边每一项都乘以公分母以去掉分数，然后照常求解。

Example: Solve x/2 + x/3 = 5.

例：解方程 x/2 + x/3 = 5。

The common denominator is 6. Multiply every term by 6:

公分母是 6。每一项都乘以 6：

6 × (x/2) + 6 × (x/3) = 6 × 5 → 3x + 2x = 30 → 5x = 30 → x = 6

Be careful to multiply every term, including the number on the right side of the equals sign.

注意要乘以每一项，包括等号右边的常数项。

7. Introduction to Linear Inequalities | 线性不等式入门

Inequalities show the relative size of two expressions. The symbols are < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Solving inequalities is similar to solving equations, but there is one crucial difference when multiplying or dividing by a negative number.

不等式表示两个表达式之间的大小关系。符号包括 <（小于）、>（大于）、≤（小于或等于）、≥（大于或等于）。求解不等式的方法与方程类似，但乘以或除以负数时有一个关键区别。

Example: Solve 2x + 5 > 11.

例：解不等式 2x + 5 > 11。

Subtract 5 from both sides: 2x > 6. Divide by 2: x > 3.

两边同时减 5：2x > 6。两边同时除以 2：x > 3。

2x + 5 > 11 → 2x > 6 → x > 3

8. The Critical Rule: Negative Multiplication and Division | 关键规则：乘以或除以负数

When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This is a common source of errors.

当不等式两边同时乘以或除以一个负数时，必须反转不等号方向。这是常见的错误来源。

Example: Solve -3x < 12.

例：解不等式 -3x < 12。

Divide both sides by -3 and reverse the sign:

两边同时除以 -3，并反转不等号：

-3x < 12 → x > -4

Check with a number greater than -4, say 0: -3(0) = 0 < 12 ✓. A number less than -4, say -5: -3(-5) = 15, which is not < 12, so the direction is correct.

取一个大于 -4 的数检验，比如 0：-3(0)=0 < 12，成立。取一个小于 -4 的数，比如 -5：-3(-5)=15，不小于 12，因此方向正确。

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

Solutions to linear inequalities can be shown on a number line. An open circle (○) means the value is not included (< or >). A closed circle (●) means the value is included (≤ or ≥).

线性不等式的解集可以在数轴上表示。空心圆（○）表示该点不包含在解集内（< 或 >），实心圆（●）表示该点包含在解集内（≤ 或 ≥）。

For example, x < 2 is shown as an open circle at 2 with an arrow pointing left. x ≥ -1 is shown as a closed circle at -1 with an arrow pointing right.

例如，x < 2 在数轴上表现为在 2 处画空心圆，并向左画箭头。x ≥ -1 则在 -1 处画实心圆，并向右画箭头。

Inequality	Circle	Arrow Direction
x > a	Open	Right
x ≥ a	Closed	Right
x < a	Open	Left
x ≤ a	Closed	Left

10. Writing and Solving Word Problems | 建立并求解应用题

IGCSE questions often require translating a real-world situation into a linear equation or inequality. Identify the unknown, assign a variable, and write the relationship in symbols.

IGCSE题目常常需要将实际情境转化为线性方程或不等式。先找出未知量，设变量，然后用符号写出数量关系。

Example: A number increased by 7 is greater than 20. Find the smallest integer that satisfies this condition.

例：一个数增加7后大于20。求满足该条件的最小整数。

Let the number be x. Then x + 7 > 20, so x > 13. The smallest integer is 14.

设这个数为 x，则 x + 7 > 20，所以 x > 13。最小整数是 14。

x + 7 > 20 → x > 13 → smallest integer = 14

11. Common Mistakes and How to Avoid Them | 常见错误与避坑指南

Many students make predictable mistakes. Here are three key ones to avoid in your exam.

许多学生会在考试中犯一些常见的错误。以下是三个需要避免的关键错误。

Forgetting to apply operations to both sides: ‘3x = 4x + 5; add 4x’ instead of subtracting 4x.

忘记对方程两边同时进行相同操作：例如“3x = 4x + 5，加 4x”而不是“减去 4x”。
Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.

乘以或除以负数时忘记反转不等号方向。
Not checking the answer. Always substitute your solution back into the original equation or inequality.

不检验答案。要始终将解代回原方程或不等式检验。

12. Exam Strategies and Summary | 考试策略与总结

In the exam, read the question carefully. Show all steps clearly — you gain method marks even if the final answer is wrong. Use inverse operations systematically, and write the answer as required (integer, fraction, decimal, or inequality).

在考试中，仔细审题，清晰展示每一个步骤——即使最终答案错误，也能获得方法分。系统地使用逆运算，并按题目要求写出答案（整数、分数、小数或不等式）。

Remember these key ideas:

请牢记以下核心要点：

Linear equations: isolate x by using inverse operations.

线性方程：使用逆运算分离出 x。
Linear inequalities: same as equations, but reverse the sign when multiplying/dividing by a negative number.

线性不等式：求解方法与方程相同，但乘以或除以负数时要反转不等号。
Always verify your solution by substitution.

始终通过代入法验证解的正确性。

Master these skills, and you will confidently tackle linear equations and inequalities in your IGCSE mathematics exams.

掌握这些技能，你就能在IGCSE数学考试中自信地解决线性方程与不等式问题。

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