Understanding Inequalities | 不等式全解析

📚 Understanding Inequalities | 不等式全解析

Inequalities are a fundamental topic in IGCSE Mathematics. They describe the relative size or order of two expressions, and solving them is a key skill for algebra, geometry, and real-world problem solving. This article will guide you through every essential concept, from basic notation to graphing and word problems.

不等式是 IGCSE 数学中的基础内容。它用于描述两个表达式之间的大小关系或顺序，而解不等式是代数、几何以及实际应用问题中的关键技能。本文将带你系统地掌握从基本符号到图像表示、再到应用题的全部核心考点。

1. Inequality Symbols | 不等式符号

Before solving inequalities, you must be fluent with the four main symbols. Each one compares two values or expressions.

在解不等式之前，你首先必须熟练掌握四个主要符号。每个符号都用来比较两个数值或表达式。

< means “less than”. Example: 3 < 5.

< 表示“小于”。例如：3 < 5。
> means “greater than”. Example: 7 > 2.

> 表示“大于”。例如：7 > 2。
≤ means “less than or equal to”. Example: x ≤ 4 allows x = 4.

≤ 表示“小于或等于”。例如：x ≤ 4 允许 x = 4。
≥ means “greater than or equal to”. Example: y ≥ -3 allows y = -3.

≥ 表示“大于或等于”。例如：y ≥ -3 允许 y = -3。

A helpful way to remember the direction is to think of the symbol as an arrow pointing to the smaller value. Also remember that ≤ and ≥ include the boundary value, while < and > do not.

记忆方向的一个好方法是：把符号看作一个指向较小值的箭头。另外要记住，≤ 和 ≥ 包含边界值，而 < 和 > 不包含边界值。

2. Solving Linear Inequalities | 解线性不等式

Solving a linear inequality is very similar to solving a linear equation, with one critical difference: when you multiply or divide both sides by a negative number, the inequality sign reverses.

解线性不等式与解线性方程非常相似，但有一个关键区别：当两边同时乘以或除以一个负数时，不等号的方向必须反转。

Example 1: Solve 2x + 5 < 13.

例 1：解 2x + 5 < 13。

2x + 5 < 13
2x < 8
x < 4

Subtract 5 from both sides, then divide by 2. Since 2 is positive, the sign stays the same. The solution is x < 4.

两边先减去 5，再除以 2。因为 2 是正数，所以不等号方向不变。解为 x < 4。

Example 2: Solve 3 – 2x ≥ 7.

例 2：解 3 – 2x ≥ 7。

3 – 2x ≥ 7
-2x ≥ 4
x ≤ -2

Subtract 3 from both sides to get -2x ≥ 4. Then divide by -2. Because we divided by a negative number, ≥ becomes ≤. The solution is x ≤ -2.

两边减去 3 得到 -2x ≥ 4。然后除以 -2。因为除以了负数，所以 ≥ 变为 ≤。解为 x ≤ -2。

3. Double Inequalities | 双重不等式

A double inequality combines two inequalities into one statement, such as -3 < x ≤ 5. It means x is greater than -3 and also less than or equal to 5.

双重不等式把两个不等式合并成一个表述，如 -3 < x ≤ 5。它表示 x 大于 -3 且小于或等于 5。

Example: Solve -4 < 2x + 6 ≤ 10.

例：解 -4 < 2x + 6 ≤ 10。

Remember that whatever you do to the middle, you must do to all three parts.

记住：你对中间部分进行的任何运算，都必须同时对三个部分进行。

-4 < 2x + 6 ≤ 10
-10 < 2x ≤ 4
-5 < x ≤ 2

First subtract 6 from all three parts: -10 < 2x ≤ 4. Then divide all parts by 2: -5 < x ≤ 2. This means x can be any number greater than -5 and up to and including 2.

首先在三部分中都减去 6：-10 < 2x ≤ 4。然后三部分都除以 2：-5 < x ≤ 2。这意味着 x 可以是大于 -5 且不超过 2（含 2）的任何数。

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

Number lines are a visual way to show the set of values that satisfy an inequality. You must know how to draw both open and closed circles correctly.

数轴是以可视化方式表示满足不等式的值集合的方法。你必须正确掌握空心圆和实心圆的使用。

Open circle (∘) at a number means that number is NOT included. Use this for < or >.

空心圆 (∘) 表示该数值不被包含。用于 < 或 >。
Closed circle (●) at a number means that number IS included. Use this for ≤ or ≥.

实心圆 (●) 表示该数值被包含。用于 ≤ 或 ≥。
Draw an arrow or a solid line to show all values that satisfy the inequality.

画箭头或实线来表示所有满足不等式的值。

For example, x > 2 is shown with an open circle at 2 and an arrow pointing to the right. For x ≤ -1, use a closed circle at -1 and an arrow pointing to the left.

例如，x > 2 在 2 处画空心圆并向右画箭头。对于 x ≤ -1，在 -1 处画实心圆并向左画箭头。

5. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities involve x². The solution is often a range or two separate ranges. A standard method is to sketch the parabola or use a sign table.

二次不等式涉及 x²。其解通常是一个区间或两个分离的区间。标准方法是画出抛物线草图或使用符号表。

Example: Solve x² – x – 6 > 0.

例：解 x² – x – 6 > 0。

Step 1: Factorise the quadratic expression.

第一步：对二次表达式进行因式分解。

x² – x – 6 = (x – 3)(x + 2)

The roots are x = 3 and x = -2. Since the coefficient of x² is positive, the parabola opens upwards. The expression is greater than 0 when the graph is above the x-axis, which occurs outside the interval between the roots.

根为 x = 3 和 x = -2。由于 x² 的系数为正，抛物线开口向上。当图像在 x 轴上方时，表达式大于 0，这发生在两根之间的区间之外。

x < -2 or x > 3

If the inequality had been x² – x – 6 < 0, the solution would be -2 < x < 3.

如果原不等式是 x² – x – 6 < 0，则解为 -2 < x < 3。

6. Representing Inequalities on a Graph | 在图像上表示不等式

For linear inequalities in two variables, such as y > 2x – 1, we draw the boundary line and shade the correct region. This is a common IGCSE question.

对于含两个变量的线性不等式，如 y > 2x – 1，我们需要画出边界线并正确的区域进行阴影标记。这是 IGCSE 的常见题型。

First draw the line as if it were an equation (y = 2x – 1). Use a solid line for ≥ or ≤, and a dashed line for > or <.

首先把不等式当作方程画出直线（y = 2x – 1）。对于 ≥ 或 ≤ 用实线，对于 > 或 < 用虚线。
Choose a test point not on the line, such as (0,0). Substitute it into the inequality. If the statement is true, shade the side containing the test point; otherwise shade the other side.

选择一个不在直线上的测试点，例如 (0,0)。代入不等式。如果成立，则对包含测试点的一侧进行阴影标记；否则对另一侧进行阴影标记。

For y > 2x – 1, substituting (0,0) gives 0 > -1, which is true, so we shade above the line. In exam questions, you may be asked to shade the region that satisfies several inequalities simultaneously.

对于 y > 2x – 1，代入 (0,0) 得 0 > -1，成立，所以对直线上方进行阴影标记。在考试中，你可能需要标出同时满足多个不等式的区域。

7. Systems of Linear Inequalities | 线性不等式组

When two or more inequalities are given together, the solution is the intersection (overlap) of all their solution regions on a graph.

当两个或更多不等式同时给定时，解集是它们在图像上所有解区域的交集（重叠部分）。

Example: Shade the region that satisfies y < 3, x ≥ 1, and y ≥ x.

例：标出满足 y < 3、x ≥ 1 和 y ≥ x 的区域。

Steps: Draw each boundary line, decide solid/dashed, and then find the area where all three conditions hold. Practice with different combinations until you are comfortable.

步骤：画出每条边界线，确定实线/虚线，然后找出同时满足三个条件的区域。多练习不同组合，直到熟练为止。

8. Inequalities with Fractions | 含分数的不等式

When an inequality contains fractions, clear the denominators first by multiplying every term by the common denominator. However, if that denominator could be negative, you must be careful about reversing the sign. In most IGCSE questions, the denominator is positive or you handle cases separately.

当不等式中含有分数时，首先要乘以公分母消去分母。但如果分母可能为负数，则必须小心符号反转。在大多数 IGCSE 题目中，分母为正，或需要分情况处理。

Example: Solve (x/3) + 1 < (x/2).

例：解 (x/3) + 1 < (x/2)。

Multiply both sides by 6 (the LCM of 3 and 2):

两边同时乘以 6（3 和 2 的最小公倍数）：

6 × (x/3 + 1) < 6 × (x/2)
2x + 6 < 3x
6 < x

The solution is x > 6.

解为 x > 6。

9. Word Problems Involving Inequalities | 不等式应用题

Real-world situations often translate into inequalities. Key phrases: “at least” means ≥, “at most” means ≤, “more than” means >, “no more than” means ≤.

现实情境常常可以转化为不等式。关键词：“至少”对应 ≥，“至多”对应 ≤，“多于”对应 >，“不超过”对应 ≤。

Example: A student needs at least 40 marks to pass. If he already has 25 marks, how many more marks m does he need?

例：一名学生至少需要 40 分才能及格。他已经得了 25 分，他至少还需要多少分 m？

25 + m ≥ 40
m ≥ 15

He needs at least 15 more marks.

他至少还需要 15 分。

10. Special Cases and Common Mistakes | 特殊情况与常见错误

Inequalities have a few tricky cases. Watch out for these to avoid losing marks.

不等式有一些容易出错的情况。注意以下陷阱，避免失分。

Situation \| 情况	Example \| 例子	Correct Approach \| 正确做法
Multiplying/dividing by a negative \| 乘以或除以负数	-3x < 6	Divide by -3 and reverse sign: x > -2
Multiplying by an unknown variable \| 乘以未知变量	1/x > 2	Consider cases x > 0 and x < 0 separately
Swapping sides \| 交换左右两边	5 > x	Rewrite as x < 5 (sign points to the same value)

Also remember to check whether the boundary value is included. A common mistake is writing x < 4 when the solution is x ≤ 4. Always read the original inequality carefully.

同时要检查边界值是否被包含。常见错误是把 x ≤ 4 写成 x < 4。做题时务必仔细审题。

11. Practice Questions | 练习题目

Test yourself with these typical exam-style questions.

用以下典型考试题目来测试自己。

Solve 5x – 7 ≥ 3x + 9.

解 5x – 7 ≥ 3x + 9。
Solve -2 < 3x + 1 ≤ 10 and show the answer on a number line.

解 -2 < 3x + 1 ≤ 10 并在数轴上表示答案。
Find the integer values of x that satisfy 2 < x² < 15.

找出满足 2 < x² < 15 的整数 x。
Shade the region defined by y ≥ x – 2 and y < 2x + 1.

标出满足 y ≥ x – 2 与 y < 2x + 1 的区域。

Answers: 1) x ≥ 4. 2) -1 < x ≤ 3, open circle at -1, closed circle at 3, line between. 3) x = -3, -2, 2, 3. 4) Check graph in your textbook or with a teacher.

答案：1) x ≥ 4。2) -1 < x ≤ 3，-1 处空心圆，3 处实心圆，连线。3) x = -3, -2, 2, 3。4) 请对照课本或请教老师检查图像。

12. Summary | 总结

Inequalities appear in almost every IGCSE Mathematics paper. Master these key points:

不等式几乎出现在每份 IGCSE 数学试卷中。掌握以下关键点：

Know the four symbols and what they mean.

熟悉四种符号及其含义。
When multiplying or dividing by a negative number, reverse the inequality sign.

乘以或除以负数时，反转不等号方向。
For double inequalities, perform the same operation on all three parts.

对于双重不等式，对三部分进行同样的运算。
Use open and closed circles correctly on number lines.

在数轴上正确使用空心圆和实心圆。
For quadratic inequalities, factorise first and consider the parabola shape.

对于二次不等式，先因式分解，再考虑抛物线形状。
For graph representations, use solid/dashed lines and test points.

对于图像表示，使用实线/虚线和测试点。

With regular practice, you will solve inequalities quickly and accurately in your exam.

通过定期练习，你将在考试中快速准确地解决不等式问题。

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Understanding Inequalities | 不等式全解析