Understanding Inequalities | 不等式精讲

📚 Understanding Inequalities | 不等式精讲

Inequalities are one of the most important topics in IGCSE Mathematics. They appear not only in algebra but also in coordinate geometry, problem solving, and even in real-life applications such as budgeting and engineering limits. In this article, we will break down everything you need to know about inequalities, from basic notation to solving linear and quadratic examples, and finally representing solutions on number lines and graphs.

不等式是 IGCSE 数学中最重要的话题之一。它不仅出现在代数中，还出现在坐标几何、应用题，甚至现实生活中的预算与工程限制中。在这篇文章中，我们将系统地讲解不等式的所有核心内容，从基本符号到线性与二次不等式的求解，再到如何在数轴和坐标系中表示解集。

1. Inequality Symbols | 不等式符号

Before solving inequalities, you must be completely comfortable with the four basic symbols. The symbol ‘>’ means ‘greater than’, ‘

在解不等式之前，你必须完全熟悉四个基本符号。’>’ 表示“大于”，’

‘x > 5’ means x is any number greater than 5, for example 6, 7.5, or 100.
‘x < 5' means x is any number less than 5, for example 4, 0, or -3.
‘x ≥ 5’ includes 5 itself and all numbers greater than 5.
‘x ≤ 5’ includes 5 itself and all numbers less than 5.

In the English curriculum, you will often see the notation ‘x ∈ ℝ’ which means x is a real number. If the solution set is restricted to integers, the notation ‘x ∈ ℤ’ is used, and you would only list whole number values.

在英国课程中，你常会看到记号 ‘x ∈ ℝ’，意思是 x 是实数。如果解集限制为整数，则使用记号 ‘x ∈ ℤ’，此时你只需列出整数值。

2. 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. The only major difference is: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

解线性不等式与解线性方程非常相似。你可以在两边同时加减相同的数，或同时乘以、除以同一个正数。唯一重要的区别是：如果两边同时乘以或除以一个负数，你必须把不等号的方向反转。

Let us look at a worked example:

我们来看一个完整的例子：

Solve 3x – 7 ≤ 2x + 5

First, subtract 2x from both sides:

首先，两边减去 2x：

x – 7 ≤ 5

Then add 7 to both sides:

然后两边加上 7：

x ≤ 12

The solution is all real numbers less than or equal to 12.

解是所有小于或等于 12 的实数。

Now consider an example with a negative multiplier:

现在看一个涉及负数乘法的例子：

Solve 4 – 2x > 10

Subtract 4 from both sides:

两边减去 4：

-2x > 6

Now divide both sides by -2, and remember to reverse the inequality sign:

现在两边除以 -2，并记得反转不等号：

x < -3

This is a classic trap in the exam. Many students forget to reverse the sign when dividing by a negative number, so always double-check this step.

这是考试中的经典陷阱。许多学生在除以负数时忘记反转符号，所以一定要反复检查这一步。

3. Double Inequalities | 双重不等式

A double inequality is a compact way to write two inequalities at the same time. For example, ‘3 < x ≤ 7' means x is greater than 3 and at the same time x is less than or equal to 7. Every value in the solution must satisfy both conditions.

双重不等式是一种同时书写两个不等式的简洁方式。例如，’3 < x ≤ 7' 表示 x 大于 3，同时 x 小于或等于 7。解中的每一个值都必须同时满足这两个条件。

To solve a double inequality, perform the same operation on all three parts: the left side, the middle, and the right side.

要解双重不等式，需要对三部分同时进行相同的运算：左边、中间和右边。

Example: Solve -5 ≤ 2x + 1 < 9

Subtract 1 from all three parts:

三部分同时减去 1：

-6 ≤ 2x < 8

Divide all three parts by 2:

三部分同时除以 2：

-3 ≤ x < 4

So x can be any real number from -3 up to but not including 4.

因此 x 可以是任意实数，从 -3 到小于 4（不含 4）。

Be careful when the middle part has a negative coefficient. For example:

当中间部分带有负系数时要特别小心。例如：

Solve 2 < 4 - x ≤ 6

Subtract 4 from all parts:

三部分同时减去 4：

-2 < -x ≤ 2

Multiply all parts by -1 and reverse both inequality signs:

三部分同时乘以 -1，并反转两个不等号：

2 > x ≥ -2

It is conventional to write this as ‘-2 ≤ x < 2' by reordering the parts.

按惯例，我们通常将其重排为 ‘-2 ≤ x < 2'。

4. Number Line Representation | 数轴表示法

In IGCSE exams, you are often asked to represent an inequality on a number line. The rules are simple: use a solid circle ‘•’ for ‘≥’ or ‘≤’ because the boundary value is included, and use an open circle ‘○’ for ‘>’ or ‘

在 IGCSE 考试中，你经常被要求在数轴上表示不等式。规则很简单：使用实心圆 ‘•’ 表示 ‘≥’ 或 ‘≤’，因为边界值包含在解集中；使用空心圆 ‘○’ 表示 ‘>’ 或 ‘

For example, to represent x ≥ 2, place a solid dot at 2 and draw an arrow pointing to the right. To represent x < -1, place an open dot at -1 and draw an arrow pointing to the left.

例如，要表示 x ≥ 2，在 2 处画一个实心点，然后向右画箭头。要表示 x < -1，在 -1 处画一个空心点，然后向左画箭头。

For a double inequality such as -2 < x ≤ 3, you draw an open circle at -2 and a solid circle at 3, then connect the two circles with a thick line.

对于双重不等式，例如 -2 < x ≤ 3，在 -2 处画空心圆，在 3 处画实心圆，然后用粗线连接两个圆。

Remember to always label the number line with a scale and include arrowheads at both ends unless the scale clearly continues beyond the region shown.

记住，画数轴时一定要标出刻度，并且在两端画出箭头，除非画面明显继续延伸。

5. Solving Quadratic Inequalities | 解二次不等式

A quadratic inequality is one that can be written in the form ax² + bx + c > 0, ax² + bx + c ≥ 0, ax² + bx + c < 0, or ax² + bx + c ≤ 0. The standard method is to first solve the corresponding quadratic equation, then use a sketch of the parabola to decide the sign of the expression in each region.

二次不等式是指可以写成 ax² + bx + c > 0、ax² + bx + c ≥ 0、ax² + bx + c < 0 或 ax² + bx + c ≤ 0 形式的不等式。标准方法是先解对应的二次方程，然后用抛物线草图判断每一区域中表达式的符号。

Let us work through a full example:

让我们完整地解一个例子：

Solve x² – 5x + 6 > 0

Step 1: Solve x² – 5x + 6 = 0. Factorise to get (x – 2)(x – 3) = 0, so x = 2 or x = 3.

第一步：解 x² – 5x + 6 = 0。因式分解得到 (x – 2)(x – 3) = 0，所以 x = 2 或 x = 3。

Step 2: Sketch the parabola. Since the coefficient of x² is positive, the parabola opens upwards. It crosses the x-axis at 2 and 3. The expression is positive when the curve is above the x-axis, which occurs to the left of 2 and to the right of 3.

第二步：画出抛物线草图。由于 x² 的系数为正，抛物线开口向上。它与 x 轴相交于 2 和 3。当曲线位于 x 轴上方时，表达式的值为正，这发生在 2 的左侧和 3 的右侧。

Step 3: Write the solution. The solution is x < 2 or x > 3.

第三步：写出解集。解为 x < 2 或 x > 3。

Notice that we use the word ‘or’, not ‘and’, because a number cannot be both less than 2 and greater than 3 at the same time.

注意，我们使用“或”而不是“且”，因为一个数不可能同时小于 2 又大于 3。

Now let us look at the opposite case:

现在我们来看相反的情况：

Solve x² – 5x + 6 ≤ 0

The same roots are 2 and 3. The parabola opens upwards, so the curve is below or on the x-axis between the two roots. Therefore the solution is 2 ≤ x ≤ 3.

相同的根是 2 和 3。抛物线开口向上，因此曲线在两个根之间位于 x 轴下方或正好在轴上。所以解为 2 ≤ x ≤ 3。

If the quadratic does not factorise easily, use the quadratic formula to find the roots. The formula is:

如果二次多项式不易因式分解，可以使用求根公式来找到根。公式如下：

x = (-b ± √(b² – 4ac)) / 2a

Once the roots are found, use the same sketching method to determine the sign regions.

一旦求出根，就可以使用同样的草图方法来确定符号区域。

6. Graphical Representation of Quadratic Inequalities | 二次不等式的图形表示

In some exam questions, you may be given a graph of a quadratic curve and asked to solve an inequality. The graph makes the solution much easier to visualise. For instance, if you see a parabola and a horizontal line, the region where the parabola is above the line corresponds to the solution of the quadratic inequality where the expression is greater than the constant.

在某些考试题中，你可能会看到一个二次曲线的图像，并被要求解不等式。图像将使解集更容易直观理解。例如，如果你看到一条抛物线和一条水平线，抛物线位于水平线上方的区域就对应着表达式大于该常数的二次不等式的解。

Consider the inequality x² – 4x + 3 < 0. Factorising gives (x - 1)(x - 3) < 0, so the roots are 1 and 3. The graph of y = x² - 4x + 3 is a U-shaped parabola. The part of the curve below the x-axis lies between x = 1 and x = 3. Therefore the solution is 1 < x < 3.

考虑不等式 x² – 4x + 3 < 0。因式分解得到 (x - 1)(x - 3) < 0，所以根为 1 和 3。y = x² - 4x + 3 的图像是 U 形抛物线。曲线位于 x 轴下方的部分位于 x = 1 和 x = 3 之间。因此解为 1 < x < 3。

When drawing the graph to solve an inequality, always mark the x-intercepts clearly and label the regions with ‘+’ or ‘-‘ signs. This will help you avoid confusion when writing the final answer.

画图求解不等式时，一定要清晰标出 x 轴截距，并在各个区域标注 ‘+’ 或 ‘-‘ 号。这能帮助你避免在写出最终答案时产生混淆。

7. Solving Inequalities with Fractions | 含分数的不等式

Linear inequalities may also contain fractions. The best strategy is to eliminate the denominators first by multiplying every term by the lowest common multiple (LCM) of all denominators. However, you must be careful: if the denominator is negative, the inequality direction may be affected.

线性不等式也可能包含分数。最佳策略是先通过将所有项乘以所有分母的最小公倍数（LCM）来消除分母。但是，你必须小心：如果分母为负，不等号方向可能会受到影响。

Consider the example:

考虑这个例子：

Solve x/3 + 1 ≥ x/2 – 2

Multiply every term by 6 (the LCM of 3 and 2):

将所有项乘以 6（3 和 2 的最小公倍数）：

2x + 6 ≥ 3x – 12

Subtract 2x from both sides:

两边减去 2x：

6 ≥ x – 12

Add 12 to both sides:

两边加上 12：

18 ≥ x

This can be written as x ≤ 18.

这可以写成 x ≤ 18。

When dealing with an inequality involving a fractional expression like (2x – 1)/(x + 3) > 0, note that x cannot be -3 because division by zero is undefined. Also the sign of the expression depends on the signs of both numerator and denominator in each interval defined by the critical values. This type of question is more common in A-Level, but IGCSE students should be aware of the domain restriction.

当处理像 (2x – 1)/(x + 3) > 0 这样的分式不等式时，注意 x 不能等于 -3，因为除以零没有定义。此外，表达式的符号取决于分子和分母在每个由临界值划分的区间内的符号。这类问题在 A-Level 中更常见，但 IGCSE 学生应该注意定义域限制。

8. Inequality Word Problems | 不等式应用题

Many IGCSE questions present inequalities in the form of word problems. Translating a real-life situation into an inequality is a skill that requires practice. Look for key phrases: ‘at least’ means ≥, ‘no more than’ means ≤, ‘more than’ means >, and ‘less than’ means

许多 IGCSE 题目以应用题的形式呈现不等式。将现实情境转化为不等式是一项需要练习的技能。注意关键词：’at least’ 表示 ≥，’no more than’ 表示 ≤，’more than’ 表示 >，’less than’ 表示

Example: A taxi company charges a base fare of $3 and $0.40 per mile. If the total fare must be less than or equal to $15, what distance can you travel?

例子：一家出租车公司收取基础费用 3 美元，每英里 0.40 美元。如果总费用必须小于或等于 15 美元，你可以行驶多少英里？

Let m be the number of miles. The total cost is 3 + 0.4m. The inequality is 3 + 0.4m ≤ 15. Subtracting 3 gives 0.4m ≤ 12, so m ≤ 30. Therefore you can travel at most 30 miles.

设 m 为英里数。总费用为 3 + 0.4m。不等式为 3 + 0.4m ≤ 15。两边减 3 得 0.4m ≤ 12，所以 m ≤ 30。因此你最多可以行驶 30 英里。

Always check whether the answer should be a whole number in the context of the problem. For example, if m represents the number of people, it must be an integer.

始终检查在题目情境中答案是否应为整数。例如，如果 m 代表人数的数量，那么它必须是整数。

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

Here are the most frequent mistakes students make in IGCSE inequality questions, along with tips to avoid them.

以下是学生在 IGCSE 不等式题目中最常犯的错误，以及避免这些错误的建议。

Mistake \| 错误	Correct Approach \| 正确做法
Forgetting to reverse the sign when dividing by a negative number	Always check if the number being divided or multiplied is negative; if so, flip the inequality sign.
Using an open circle for ‘≥’ or ‘≤’ on the number line	Use a solid dot for ‘included’ boundaries and an open circle for ‘not included’ boundaries.
Writing ‘and’ instead of ‘or’ for a quadratic inequality solution that is split into two regions	When the solution has two separate regions, use ‘or’, not ‘and’.
Forgetting to exclude values that make the denominator zero in fractional inequalities	Always state that the denominator cannot be zero before solving.

Another tip: when you finish solving an inequality, always test a value from your solution set in the original inequality to verify that it works.

另一个技巧：当你解完一个不等式后，务必从解集中选一个值代回原不等式检验，确保它成立。

10. Practice Questions | 练习题

To consolidate your understanding, try these five questions. Solve them on paper before checking the answers below.

为了巩固理解，请尝试以下五道题。先在纸上写出解答，再核对答案。

Solve 7 – 3x ≥ 2x + 12.
Solve -4 < 3x + 2 ≤ 11 and represent the solution on a number line.
Solve x² + x – 12 < 0.
Solve 2x² + 3x – 2 ≥ 0.
A rectangle has length (2x + 1) cm and width (x – 3) cm. The area must be at least 20 cm². Find the set of possible values of x.

Answers: 1. x ≤ -1. 2. -2 < x ≤ 3. 3. -4 < x < 3. 4. x ≤ -2 or x ≥ 0.5. 5. x ≥ 4 (since x > 3 from the width condition, and solving (2x + 1)(x – 3) ≥ 20 gives x ≥ 4 or x ≤ -1.5, so the valid range is x ≥ 4).

答案：1. x ≤ -1。2. -2 < x ≤ 3。3. -4 < x < 3。4. x ≤ -2 或 x ≥ 0.5。5. x ≥ 4（因为宽度条件要求 x > 3，并且解 (2x + 1)(x – 3) ≥ 20 得到 x ≥ 4 或 x ≤ -1.5，所以有效范围为 x ≥ 4）。

11. Summary | 总结

In this article, we covered the essential knowledge of inequalities for IGCSE Mathematics: the four symbols, solving linear and double inequalities, representing solutions on number lines, solving quadratic and fractional inequalities, and handling word problems. The most critical rule to remember is the reversal of the inequality sign when multiplying or dividing by a negative number.

在这篇文章中，我们涵盖了 IGCSE 数学中不等式的核心知识：四个基本符号、线性与双重不等式的求解、在数轴上的表示、二次与分式不等式的解法，以及应用题的处理。最需要记住的关键规则是：当两边乘以或除以一个负数时，不等号方向必须反转。

With regular practice, these methods will become natural. Always draw a sketch when solving quadratic inequalities and always test your solutions. This will significantly reduce avoidable mistakes in the exam.

通过定期练习，这些方法会变得非常自然。解二次不等式时务必画草图，并且始终检验解集。这将大大减少本可以避免的考试错误。

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Understanding Inequalities | 不等式精讲