IGCSE Maths: Inequalities Exam Focus | IGCSE 数学：不等式考点精讲

📚 IGCSE Maths: Inequalities Exam Focus | IGCSE 数学：不等式考点精讲

Inequalities form a fundamental part of the IGCSE Mathematics syllabus, appearing both as standalone questions and within broader problem‑solving contexts. Mastering this topic means understanding the notation, solution methods on a number line, the effects of multiplying or dividing by negative values, and how to represent regions on the coordinate plane. This article breaks down every essential skill you will need, from simple linear inequalities to shading graphical regions, with clear bilingual explanations and exam‑style tips.

不等式是 IGCSE 数学大纲中的基础内容，既会单独出题，也常融入更广泛的解题场景中。要掌握这个主题，你需要理解不等式符号、在数轴上的表示方法、乘除以负数时不等号方向的变化，以及如何在坐标平面上表示区域。本文将逐一拆解你需要掌握的每一项核心技能，从简单的一元一次不等式到图形区域的阴影表示，并提供清晰的双语讲解和应试技巧。

1. Symbols and Meaning | 符号与含义

An inequality states that two expressions are not equal, using symbols to show the relationship. The four main symbols are: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). The open circle on a number line represents strict inequality (< or >), while a closed circle represents ≤ or ≥.

不等式表示两个表达式不相等，并用符号表明它们之间的关系。四个主要符号是：<（小于）、>（大于）、≤（小于或等于）和 ≥（大于或等于）。数轴上用空心圆圈表示严格不等式（< 或 >），用实心圆圈表示 ≤ 或 ≥。

For example, x < 3 means x can be any number strictly less than 3, such as 2.9, 0, –5, but not 3 itself. x ≥ –1 means x can be –1, 0, 4, or any number greater than or equal to –1.

例如，x < 3 表示 x 可以是任何严格小于 3 的数，如 2.9、0、–5，但不能是 3 本身。x ≥ –1 表示 x 可以是 –1、0、4 或任何大于或等于 –1 的数。

Always read the inequality from the variable side first: ‘x is greater than 2’ for x > 2. This avoids confusion when the variable is on the right, e.g. 2 < x is the same as x > 2.

一定要从变量那一边开始读不等式：对于 x > 2，读作“x 大于 2”。当变量在右边时，比如 2 < x 等同于 x > 2，这样读就不会混淆。

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

Drawing a number line is a quick way to visualise the solution set. Use a solid dot for ≤ or ≥, and a hollow dot for < or >. For a simple inequality like x ≥ 2, place a solid dot at 2 and draw an arrow to the right. For x < –1, place a hollow dot at –1 and draw an arrow to the left.

画数轴是一种快速直观展示解集的方法。对于 ≤ 或 ≥ 用实心圆点，对于 < 或 > 用空心圆点。对于像 x ≥ 2 这样的简单不等式，在 2 处画一个实心圆点，并向右画箭头。对于 x < –1，在 –1 处画空心圆点，并向左画箭头。

When a double inequality is given, such as –2 < x ≤ 3, you show it with a hollow dot at –2, a solid dot at 3, and a thick line segment connecting them. This compact representation is often required in exam answers.

当给出双向不等式，例如 –2 < x ≤ 3，你需要在 –2 处画空心圆点，在 3 处画实心圆点，并用一条粗线段将它们连接起来。这种紧凑的表示方法在考试答案中经常要求。

Exam tip: Always label the scale on your number line clearly, marking at least the boundary numbers. Even if you draw by hand, ensure the positions are reasonably proportional.

考试技巧：一定要在数轴上清楚地标出刻度，至少标出边界数字。即使是手绘，也要保证位置大致成比例。

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

Solving a linear inequality uses exactly the same steps as solving a linear equation: eliminate brackets, collect like terms, isolate the variable on one side. The only extra rule is that if you multiply or divide by a negative number, you must reverse the inequality sign.

解一元一次不等式所使用的步骤与解一元一次方程完全相同：去括号、合并同类项、将变量移到一边。唯一额外的一条规则是：如果乘以或除以一个负数，必须将不等号方向反转。

Example: Solve 3x + 5 > 14.

Subtract 5 from both sides: 3x > 9. Divide both sides by 3: x > 3. The sign stays the same because we divided by a positive number.

例如：解 3x + 5 > 14。

两边同时减去 5：3x > 9。两边除以 3：x > 3。因为除以的是正数，不等号方向不变。

Example with sign reversal: Solve 5 – 2x ≤ 11.

Subtract 5: –2x ≤ 6. Divide by –2 and reverse the sign: x ≥ –3. Always show the reversal step clearly to avoid losing marks.

需要反转符号的例子：解 5 – 2x ≤ 11。

减去 5：–2x ≤ 6。除以 –2 并反转符号：x ≥ –3。一定要清楚地展示反转符号的步骤，以免丢分。

4. The Golden Rule: Multiplying or Dividing by a Negative | 黄金法则：乘除以负数

This is the single most common pitfall in inequality questions. The rule applies not only when the coefficient of x is negative, but also when you multiply both sides by a negative number to eliminate a denominator or simplify. If you forget to flip the sign, your answer will be completely wrong.

这是不等式题目中最常见的陷阱。这条规则不仅适用于 x 的系数为负数时，也适用于当你为了消去分母或简化而两边同乘一个负数时。如果忘记翻转符号，答案就会完全错误。

Consider: –4x > 20. Divide by –4: x < –5. Check with a test value: x = –6 gives –4(–6)=24 > 20, works. x = –4 gives 16 > 20, false. The reversal is necessary.

考虑 –4x > 20。除以 –4：x < –5。用一个测试值检验：x = –6 时，–4(–6)=24 > 20，成立。x = –4 时，16 > 20，不成立。可见符号反转是必要的。

Another tricky case: If you multiply both sides of –x/2 < 3 by 2, you get –x < 6. Then multiply by –1 to get x > –6. Many students forget to reverse the sign in the second step.

另一个容易出错的例子：如果对 –x/2 < 3 两边同乘 2，得到 –x < 6。然后乘以 –1 得到 x > –6。很多学生在第二步忘记反转符号。

Remember: the sign does not change when adding or subtracting, only when multiplying or dividing by a negative value.

请记住：加减运算不会改变不等号方向，只有乘除一个负数时才会改变。

5. Solving Double Inequalities | 解双向不等式

A double inequality like –3 ≤ 2x + 1 < 5 can be solved in one go by performing the same operation on all three parts. The aim is to isolate x in the middle. Subtract 1 from each part: –4 ≤ 2x < 4. Then divide all parts by 2: –2 ≤ x < 2.

像 –3 ≤ 2x + 1 < 5 这样的双向不等式可以通过对三个部分同时进行相同运算来一次性求解。目标是将 x 孤立在中间。每个部分都减去 1：–4 ≤ 2x < 4。然后每个部分都除以 2：–2 ≤ x < 2。

If there is a negative coefficient for x, you still have to reverse the inequality signs after dividing by a negative number. For example, –1 < 5 – 3x ≤ 8. Subtract 5: –6 < –3x ≤ 3. Divide by –3: 2 > x ≥ –1. It is conventional to rewrite this as –1 ≤ x < 2, from smallest to largest.

如果 x 的系数是负数，除以负数后仍然需要反转两个不等号。例如 –1 < 5 – 3x ≤ 8。减去 5：–6 < –3x ≤ 3。除以 –3：2 > x ≥ –1。习惯上将其改写为 –1 ≤ x < 2，从小到大排列。

When the two inequality signs are of the same type (both ≤ or both <), you can also split the double inequality into two separate ones and solve them individually before combining. This split method is often easier for beginners.

当两个不等号类型相同时（都是 ≤ 或都是 <），你也可以把双向不等式拆成两个单独的不等式，分别求解后再合并。这种拆分法对初学者来说往往更简单。

6. Inequalities with Brackets and Fractions | 带括号与分数的不等式

Expand brackets carefully and treat fractions by multiplying both sides (or all three parts in a double inequality) by the lowest common denominator. Always remember: if that common denominator is negative, the inequality signs must be reversed.

要小心地展开括号，对于分数，可以在两边（或在双向不等式的三个部分）同乘以最小公分母。始终记住：如果这个公分母是负数，不等号必须反转。

Example: (3x – 1)/2 ≥ (x + 4)/3. Multiply everything by 6 (positive): 3(3x – 1) ≥ 2(x + 4) → 9x – 3 ≥ 2x + 8 → 7x ≥ 11 → x ≥ 11/7.

例子：(3x – 1)/2 ≥ (x + 4)/3。全式乘以 6（正数）：3(3x – 1) ≥ 2(x + 4) → 9x – 3 ≥ 2x + 8 → 7x ≥ 11 → x ≥ 11/7。

If the denominator contains a variable, the trick of multiplying through could be dangerous because you might be multiplying by a negative quantity without knowing. In IGCSE, denominators usually contain only numbers, so the method is safe.

如果分母中含有变量，直接去分母会比较危险，因为你可能在不知道正负的情况下乘以了一个负数。在 IGCSE 中，分母通常只包含数字，所以这个方法很安全。

7. Forming Inequalities from Word Problems | 从文字题建立不等式

Real‑world problems often require you to translate a written statement into an inequality. Keywords like ‘at least’, ‘not more than’, ‘maximum’, ‘minimum’, ‘fewer than’, ‘exceeds’ all give clues about which symbol to use.

现实世界中的问题常常要求你将文字叙述转化为不等式。像“至少”、“不多于”、“最大”、“最小”、“少于”、“超过”这些关键词都可以提示你该使用哪个符号。

Phrase / 短语	Inequality Symbol / 符号
at least / 至少	≥
not more than / 不多于	≤
maximum / 最大	≤
minimum / 最小	≥
fewer than / 少于	<
exceeds / 超过	>

Once the inequality is formed, solve it as usual. Always check that your answer makes sense in the context of the problem (for example, a negative number of people is impossible).

建立不等式后，像平常一样求解。一定要检查你的答案在题目情境下是否有意义（例如，人数不可能是负数）。

8. Graphs of Inequalities on the Cartesian Plane | 笛卡尔平面上的不等式图形

For two‑variable inequalities (usually x and y), the solution is a region on the coordinate plane. First, draw the boundary line: solid line if the inequality includes equality (≤ or ≥), dashed line if it is strict (< or >). Then shade the side that satisfies the inequality.

对于两个变量的不等式（通常是 x 和 y），解是坐标平面上的一个区域。首先画出边界线：如果不等式包含等号（≤ 或 ≥），用实线；如果是严格不等式（< 或 >），用虚线。然后给满足不等式的那一侧涂上阴影。

Example: Shade the region y > 2x + 1. Draw y = 2x + 1 with a dashed line. Pick a test point, e.g. (0,0). Substitute: 0 > 2(0)+1 → 0 > 1 is false, so shade the side not containing (0,0).

例子：为 y > 2x + 1 的区域涂阴影。用虚线画出 y = 2x + 1。选一个测试点，如 (0,0)。代入：0 > 2(0)+1 → 0 > 1 不成立，因此涂上没有包含 (0,0) 的那一侧。

A system of inequalities defines the region where all shaded areas overlap. Exam questions may ask you to find the set of inequalities that define a given shaded region. In such cases, start by writing the equations of the boundary lines, then determine the correct inequality signs using a test point inside the region.

一个不等式组定义的是所有阴影区域重叠的部分。考试题可能会要求你找出定义给定阴影区域的不等式组。这时，先从写出边界线的方程开始，然后利用区域内的一个测试点来确定正确的不等号方向。

9. Quadratic Inequalities (Higher Tier Only) | 二次不等式（仅限高阶）

Some IGCSE Higher Tier papers include quadratic inequalities such as x² – 5x + 6 < 0. To solve, factorise the quadratic first: (x – 2)(x – 3) < 0. The critical values are x = 2 and x = 3. Sketch a quick parabola (positive coefficient of x² means a ∪‑shape) to determine where the expression is negative.

某些 IGCSE 高阶试卷会包含二次不等式，例如 x² – 5x + 6 < 0。求解时，先对二次式进行因式分解：(x – 2)(x – 3) < 0。关键值为 x = 2 和 x = 3。快速画出抛物线的草图（x² 系数为正，开口向上），以确定表达式在何处为负值。

The product is negative between the two roots, so the solution is 2 < x < 3. For > 0, the solution would be x < 2 or x > 3. Always present the final answer clearly, often in set notation or on a number line.

乘积在两个根之间为负，因此解为 2 < x < 3。如果是 > 0，解则为 x < 2 或 x > 3。始终要清晰地给出最终答案，通常用集合符号或在数轴上表示。

10. Set Notation and Interval Notation | 集合符号与区间表示

IGCSE often expects answers in a clean format. For example, the solution x > 3 can be written as {x : x > 3} or using interval notation (3, ∞). For compound inequalities, –2 ≤ x < 5 can be expressed as [–2, 5) in interval notation – square bracket for inclusive, round bracket for exclusive.

IGCSE 往往要求以整洁的格式给出答案。例如，解 x > 3 可以写成 {x : x > 3}，或者使用区间表示 (3, ∞)。对于复合不等式，–2 ≤ x < 5 可以用区间表示 [–2, 5) ——方括号表示包含端点，圆括号表示不包含。

Familiarity with both forms is useful because certain questions may ask for the answer ‘using set notation’ or ‘in the form a < x < b’. Do not mix the two unless the question specifically requires a particular style.

熟悉这两种形式很有用，因为某些题目可能会要求“用集合符号”或“以 a < x < b 的形式”给出答案。除非题目特别要求某种格式，否则不要将两者混用。

11. Common Mistakes and How to Avoid Them | 常见错误与如何避坑

Mistake 1: Forgetting to flip the sign when dividing by a negative. Always highlight the step where the sign changes, and double‑check with a test value.

错误 1：除以负数时忘记翻转符号。始终在符号变化的步骤做标记，并用测试值进行双重检验。

Mistake 2: Misreading the inequality symbol when drawing a number line. A hollow dot for ≤ is unacceptable. Draw the dots carefully and label them.

错误 2：在数轴上画图时看错不等号。将 ≤ 画成空心圆点是不可接受的。仔细画出圆点并做好标记。

Mistake 3: When solving double inequalities, performing an operation on only two parts. Always apply the operation to all three sections simultaneously.

错误 3：解双向不等式时，只对其中两部分进行运算。一定要同时对三个部分都进行相同的运算。

Mistake 4: In diagram regions, using a solid line for a strict inequality. If the line is part of the region boundary and the inequality is strict, use a dashed line and erase any solid trace.

错误 4：在图形区域中，对严格不等式使用了实线。如果这条线是区域的边界，且不等式是严格的，必须用虚线，并擦掉任何实线的痕迹。

Mistake 5: Not simplifying the final answer. Always give the simplest form, and write double inequalities with the smaller number on the left (e.g. –1 < x < 5, not 5 > x > –1).

错误 5：最终答案没有化简。一定要给出最简形式，并以较小的数在左的方式书写双向不等式（如 –1 < x < 5，而不是 5 > x > –1）。

12. Exam Strategy and Quick Checklist | 考试策略与速查清单

Read the question: does it ask for the solution set? On a number line? Using set notation?
Solve step by step, showing all working clearly.
If you multiply/divide by a negative, draw a small ⚠ next to the step to remind yourself to flip the sign.
For graphical inequalities, label your axes, use a ruler for boundary lines, and clearly indicate which side is shaded. Use a test point to confirm.
If time allows, substitute a value from your solution back into the original inequality to verify.

审题：题目要求的是解集吗？在数轴上表示？还是用集合符号？
逐步求解，清晰展示所有步骤。
如果乘以或除以一个负数，在旁边画一个小 ⚠ 来提醒自己翻转符号。
对于图形不等式，要标注坐标轴，用直尺画出边界线，并清楚地标明阴影在哪个区域。用测试点进行确认。
如果时间允许，从你的解中选一个数值代回原不等式进行验证。

Mastering inequalities is about precision and consistency. Every step you practise brings you closer to a perfect score on this topic.

掌握不等式要做到精确和始终如一。你练习的每一步都会让你离这个主题的满分更近。

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IGCSE Maths: Inequalities Exam Focus | IGCSE 数学：不等式 考点精讲