Linear Inequalities | 线性不等式

📚 Linear Inequalities | 线性不等式

Linear inequalities are algebraic expressions that use inequality symbols (>, <, ≥, ≤) to compare two quantities. They are a fundamental topic in the IGCSE Mathematics syllabus, requiring students to solve, represent, and interpret solutions on number lines and in real-world contexts.

线性不等式是使用不等号（>、<、≥、≤）来比较两个量的代数表达式。它们是 IGCSE 数学考纲中的基础专题，要求学生求解不等式、在数轴上表示解集，并在实际情境中进行解读。

1. Inequality Symbols and Their Meanings | 不等号及其含义

There are four inequality symbols that appear throughout IGCSE Mathematics. The open symbols > and < are called "strict inequalities," meaning the boundary value is not included. The symbols ≥ and ≤ are "non-strict inequalities," meaning the boundary value is included.

IGCSE 数学中会出现四种不等号。开口符号 > 和 < 被称为"严格不等式"，表示边界值不包含在内。符号 ≥ 和 ≤ 是"非严格不等式"，表示边界值包含在内。

> greater than — e.g. x > 3 means x can be 3.1, 4, 100, but not 3 itself.
< less than — e.g. y < 2 means y can be 1, 0, −5, but not 2.
≥ greater than or equal to — e.g. x ≥ 3 means x can be 3, 3.5, 10, etc.
≤ less than or equal to — e.g. y ≤ 2 means y can be 2, 1.9, −3, etc.

> 大于——例如 x > 3 表示 x 可以是 3.1、4、100，但不能是 3 本身。
< 小于——例如 y < 2 表示 y 可以是 1、0、−5，但不能是 2。
≥ 大于或等于——例如 x ≥ 3 表示 x 可以是 3、3.5、10 等。
≤ 小于或等于——例如 y ≤ 2 表示 y 可以是 2、1.9、−3 等。

A useful memory aid: the pointed end of the symbol always points toward the smaller quantity.

一个有用的记忆技巧：符号的尖端始终指向较小的量。

2. Interval Notation and Number Lines | 区间表示法与数轴

In IGCSE Mathematics, solutions to inequalities are typically shown on a number line. An open circle (○) indicates that a boundary value is not included (strict inequality), while a closed circle (●) indicates that the boundary value is included (non-strict inequality). The solution region is shaded along the number line.

在 IGCSE 数学中，不等式的解通常表示在数轴上。空心圆圈（○）表示边界值不包含在内（严格不等式），实心圆圈（●）表示边界值包含在内（非严格不等式）。解的区域沿数轴用阴影标出。

Inequality	Number Line Representation	Interval Notation
x > 2	Open circle at 2, shade right	(2, ∞)
x ≥ 2	Closed circle at 2, shade right	[2, ∞)
x < −1	Open circle at −1, shade left	(−∞, −1)
−2 ≤ x < 3	Closed circle at −2, open at 3, shade between	[−2, 3)

不等式	数轴表示	区间记法
x > 2	2 处空心圆，向右画阴影	(2, ∞)
x ≥ 2	2 处实心圆，向右画阴影	[2, ∞)
x < −1	−1 处空心圆，向左画阴影	(−∞, −1)
−2 ≤ x < 3	−2 处实心圆，3 处空心圆，之间画阴影	[−2, 3)

In interval notation, square brackets [ ] indicate that the endpoint is included, while round parentheses ( ) indicate that the endpoint is excluded. The symbol ∞ (infinity) is always used with a parenthesis, never a bracket.

在区间记法中，方括号 [ ] 表示端点包含在内，圆括号 ( ) 表示端点不包含。符号 ∞（无穷大）总是配合圆括号使用，绝不用方括号。

3. Core Rules for Solving Inequalities | 解不等式的基本规则

Solving inequalities follows the same fundamental operations as solving equations, with one crucial exception: when you multiply or divide both sides by a negative number, the direction of the inequality sign must be reversed.

解不等式所遵循的基本运算与解方程相同，但有一个关键例外：当两边同时乘以或除以一个负数时，必须改变不等号的方向。

Rule 1: Adding or subtracting the same quantity from both sides — the inequality direction is unchanged.

规则一：两边加减同一个数——不等号方向不变。

Rule 2: Multiplying or dividing both sides by a positive number — the inequality direction is unchanged.

规则二：两边乘除以同一个正数——不等号方向不变。

Rule 3: Multiplying or dividing both sides by a negative number — the inequality direction is reversed.

规则三：两边乘除以同一个负数——不等号方向必须反转。

Example: −2x < 6. Divide both sides by −2 and reverse the sign:

例如：−2x < 6。两边除以 −2 并反转符号：

x > −3

This rule applies to both strict and non-strict inequalities. Reversing the inequality sign is equivalent to swapping the left and right sides of the expression.

这条规则同时适用于严格不等式和非严格不等式。反转不等号相当于交换表达式左右两边。

4. Solving Linear Inequalities Step by Step | 分步求解线性不等式

Linear inequalities involve algebraic expressions of the first degree. The goal is to isolate the variable on one side of the inequality, treating the process systematically.

线性不等式涉及一次代数表达式。目标是将变量单独隔离到不等式的一侧，系统化地处理整个过程。

Worked example: Solve 3x − 7 ≥ 2x + 5.

示例：求解 3x − 7 ≥ 2x + 5。

Step 1: Subtract 2x from both sides to group x terms on the left.

步骤一：两边同时减去 2x，将含 x 的项集中在左边。

3x − 2x − 7 ≥ 5

Step 2: Simplify.

步骤二：化简。

x − 7 ≥ 5

Step 3: Add 7 to both sides.

步骤三：两边同时加 7。

x ≥ 12

The solution is x ≥ 12, which can be represented on a number line with a closed circle at 12 and shading to the right, or in interval notation as [12, ∞).

解为 x ≥ 12，在数轴上表示为 12 处实心圆并向右画阴影，或用区间记法表示为 [12, ∞)。

5. Solving Inequalities with Negative Coefficients | 含负系数的不等式求解

When the variable has a negative coefficient, special care is required. The strategy is to first rearrange the inequality so that the x-term is positive, or to divide by the negative coefficient and remember to flip the inequality sign.

当变量的系数为负数时，需要格外小心。策略是先将不等式整理为 x 项为正，或者除以负数系数并记得翻转不等号。

Worked example: Solve 4 − 5x < 19.

示例：求解 4 − 5x < 19。

Method 1: Subtract 4 from both sides first.

方法一：先两边同时减去 4。

−5x < 15

Now divide both sides by −5, which reverses the inequality.

现在两边同时除以 −5，不等号方向反转。

x > −3

Method 2: Add 5x to both sides so the coefficient becomes positive.

方法二：两边同时加 5x，使系数变为正数。

4 < 19 + 5x

Then subtract 19 from both sides.

然后两边同时减去 19。

−15 < 5x

Divide by 5.

两边除以 5。

x > −3

Both methods yield the same solution. Method 2 is often preferred by examiners because it avoids the sign-reversal step entirely.

两种方法得到相同的结果。方法二通常更受考官青睐，因为它完全避免了符号反转的步骤。

6. Double Inequalities (Compound Inequalities) | 双重不等式（复合不等式）

A double inequality such as −2 ≤ 3x + 1 ≤ 7 combines two inequalities into one statement. Solving it requires performing all operations on all three parts simultaneously.

双重不等式如 −2 ≤ 3x + 1 ≤ 7 将两个不等式合并为一个语句。求解时需要同时对三个部分执行相同的运算。

Worked example: Solve −2 ≤ 3x + 1 ≤ 7.

示例：求解 −2 ≤ 3x + 1 ≤ 7。

Step 1: Subtract 1 from all three parts.

步骤一：三部分同时减去 1。

−3 ≤ 3x ≤ 6

Step 2: Divide all three parts by 3.

步骤二：三部分同时除以 3。

−1 ≤ x ≤ 2

The solution is the interval [−1, 2], represented by a closed circle at −1 and another at 2, with shading between them.

解为区间 [−1, 2]，在数轴 −1 和 2 处画实心圆，并在两者之间画阴影。

7. Solving Simultaneous Linear Inequalities | 联立线性不等式的求解

When two separate inequalities involving the same variable are given, the solution is the intersection of the two individual solution sets. Each inequality is solved independently, then the common region on the number line is identified.

当给出两个涉及同一变量的独立不等式时，解是两个单独解集的交集。先分别求解每个不等式，然后在数轴上找出公共区域。

Worked example: Solve the following inequalities simultaneously: 2x + 1 > 5 and 3x − 2 ≤ 10.

示例：联立求解：2x + 1 > 5 和 3x − 2 ≤ 10。

First inequality: 2x + 1 > 5.

第一个不等式：2x + 1 > 5。

2x > 4, so x > 2

Second inequality: 3x − 2 ≤ 10.

第二个不等式：3x − 2 ≤ 10。

3x ≤ 12, so x ≤ 4

The solution is the intersection of {x : x > 2} and {x : x ≤ 4}, giving:

解是 {x : x > 2} 和 {x : x ≤ 4} 的交集，即：

2 < x ≤ 4

On the number line, this is shown with an open circle at 2, a closed circle at 4, and shading between them.

在数轴上，2 处画空心圆，4 处画实心圆，并在两者之间画阴影。

8. Simplifying Inequalities That Contain Fractions | 含分数不等式的化简

Fractional expressions appear frequently in IGCSE inequalities. A reliable strategy is to multiply both sides by the common denominator to clear fractions before rearranging terms.

分数表达式在 IGCSE 不等式中经常出现。一个可靠的策略是先将两边乘以公分母以消除分数，然后再整理各项。

Worked example: Solve (x/3) − 2 > (x/4) + 1.

示例：求解 (x/3) − 2 > (x/4) + 1。

The common denominator of 3 and 4 is 12. Multiply both sides by 12.

3 和 4 的最小公分母是 12。两边同时乘以 12。

4x − 24 > 3x + 12

Subtract 3x from both sides.

两边同时减去 3x。

x − 24 > 12

Add 24 to both sides.

两边同时加 24。

x > 36

When multiplying by a positive common denominator, the inequality direction is preserved. However, if a negative common denominator were used, the direction would flip.

当乘以正的公分母时，不等号方向保持不变。然而，如果使用负的公分母，方向就会翻转。

9. Representing Inequalities with Integer Solutions | 整数解的表示

In the IGCSE exam, questions often ask for the “integer solutions” of an inequality. For example, if the solution is −3 < x ≤ 5, the integer solutions are −2, −1, 0, 1, 2, 3, 4, 5.

在 IGCSE 考试中，题目常常要求给出不等式的”整数解”。例如，若解为 −3 < x ≤ 5，那么整数解为 −2、−1、0、1、2、3、4、5。

Key considerations for integer solutions:

整数解的关键要点：

List all integers that satisfy the inequality in ascending order.
Pay attention to which endpoints are included. x > −3 excludes −3, but x ≤ 5 includes 5.
If the inequality has infinitely many integer solutions in a given range, state that “x is any integer greater than …” or use the listing notation.

按升序列出所有满足不等式的整数。
注意哪些端点是包含的。x > −3 不包含 −3，但 x ≤ 5 包含 5。
如果不等式在给定范围内有无限多个整数解，用”x 为任意大于……的整数”或列举记号来表示。

Worked example: The inequality 2 ≤ x < 6 has integer solutions 2, 3, 4, 5. Note that 6 is not included.

示例：不等式 2 ≤ x < 6 的整数解为 2、3、4、5。注意 6 不包含在内。

10. Applying Inequalities to Real-World Problems | 不等式在实际问题中的应用

Linear inequalities are powerful tools for modelling real-life constraints: minimum spending limits, maximum capacity, profit thresholds, and scoring requirements all translate naturally into inequality statements.

线性不等式是建模现实约束的有力工具：最低消费限额、最大容量、利润阈值和得分要求都可以自然地转化为不等式语句。

Worked example: A student scores 72 and 85 on two tests. Let the score on the third test be x. To achieve an average score of at least 80, what score must the student obtain on the third test?

示例：一名学生在两次测试中分别得了 72 分和 85 分。设第三次测试的成绩为 x。要使得平均分至少达到 80 分，第三次测试的得分应满足什么条件？

The average score condition is:

平均分条件为：

(72 + 85 + x) ÷ 3 ≥ 80

Multiply both sides by 3.

两边同时乘以 3。

157 + x ≥ 240

Subtract 157 from both sides.

两边同时减去 157。

x ≥ 83

The student must score at least 83 on the third test.

该学生第三次测试至少需要得 83 分。

11. Common Mistakes and How to Avoid Them | 常见错误与规避方法

Several recurring errors appear in exam scripts. Recognising these pitfalls is critical for achieving full marks on inequality questions.

考试答卷中有几个反复出现的错误。识别这些陷阱对于在不等式题目中获得满分至关重要。

Common Mistake	Correct Approach
Forgetting to flip the sign when dividing by a negative number	Always check the sign of the divisor; if negative, reverse the inequality
Writing the answer in the wrong direction (e.g., x < 5 instead of x > 5)	Substitute a value back into the original inequality to verify
Misreading open versus closed circles on number lines	Associate open ○ with > or <, closed ● with ≥ or ≤
Treating inequalities exactly like equations when the variable term is subtracted	Use the “make the x-term positive” method to avoid sign confusion

常见错误	正确做法
除以负数时忘记反转符号	始终检查除数的符号；若为负，反转不等号
答案方向写反（如写成 x < 5 而非 x > 5）	将某个值代回原不等式进行验证
在数轴上混淆空心圆与实心圆	将空心 ○ 与 > 或 < 对应，实心 ● 与 ≥ 或 ≤ 对应
当变量项被减去时仍按方程方式处理	使用”使 x 项为正”的方法来避免符号混淆

Always verify your solution by selecting a value from the solution set and substituting it into the original inequality. This takes only a few seconds and prevents careless errors.

务必通过从解集中选取一个值代入原不等式来验证你的解。这只需要几秒钟，却能防止粗心错误。

12. Summary and Exam Tips | 总结与考试技巧

Linear inequalities are a reliable source of marks on the IGCSE paper. Mastery requires three skills: correct algebraic manipulation, accurate number line representation, and careful interpretation of word problems.

线性不等式是 IGCSE 试卷中稳定拿分的考点。掌握它需要三项技能：正确的代数运算、准确的数轴表示，以及仔细解读文字应用题。

Key exam strategies to remember:

需要记住的关键考试策略：

Always prioritise making the x-term positive when solving.
Reverse the inequality sign only when multiplying or dividing by a negative quantity.
In double inequalities, perform every operation on all three parts simultaneously.
When listing integer solutions, write them in ascending order and check the endpoints carefully.
Draw the number line clearly, distinguishing open and closed circles precisely.
Substitute your final answer back into the original inequality to confirm correctness.

求解时优先让 x 项变为正数。
仅在乘以或除以负数时才反转不等号。
对于双重不等式，所有运算必须同时作用于三个部分。
列举整数解时按升序书写，并仔细检查端点。
清晰绘制数轴，准确区分空心圆和实心圆。
将最终答案代回原不等式以确认正确性。

With consistent practice, solving linear inequalities becomes an automatic and reliable process, contributing to a strong overall performance in the IGCSE Mathematics examination.

通过持续练习，求解线性不等式会成为一个自动而可靠的过程，为你在 IGCSE 数学考试中的整体优异表现奠定坚实基础。

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Linear Inequalities | 线性不等式