📚 Mastering Inequalities for GCSE CIE Mathematics | GCSE CIE 数学：不等式 考点精讲
Inequalities are a fundamental topic in GCSE CIE Mathematics, appearing in both core and extended papers. They extend the idea of equations by describing ranges of values rather than single solutions. From solving simple linear statements to sketching regions on graphs, inequalities demand logical thinking and clear notation. This guide covers all essential skills you need, with step-by-step explanations, worked examples, and tips for avoiding common pitfalls. Master these techniques and you will be ready for any inequality-related question in your examination.
不等式是 GCSE CIE 数学的核心主题，既出现在核心试卷也出现在扩展试卷中。它们扩展了方程的思想，描述数值范围而非单一解。从解简单的线性不等式到在图上绘制区域，不等式需要逻辑思维和清晰的符号表达。本指南涵盖你所需的所有关键技能，配有逐步解析、例题和避免常见错误的技巧。掌握这些方法，你将能应对试卷中任何与不等式相关的问题。
1. Understanding Inequality Symbols | 理解不等式符号
Inequalities use five main symbols: ‘>’ (greater than), ‘<' (less than), '≥' (greater than or equal to), '≤' (less than or equal to), and '≠' (not equal to). Each one defines a relationship between two expressions. For example, x > 3 means that x can be any number strictly larger than 3, but not 3 itself. When the symbol includes an equal sign, the boundary value is part of the solution set.
不等式使用五个主要符号：’>’（大于）、’<'（小于）、'≥'（大于等于）、'≤'（小于等于）以及 '≠'（不等于）。每个符号定义了两个表达式之间的关系。例如，x > 3 表示 x 可以是任意严格大于 3 的数，但不包括 3 本身。当符号包含等号时，边界值属于解集的一部分。
A common misunderstanding is reversing the direction of the symbol. Remember: the small end points to the smaller quantity. So ‘5 < x' and 'x > 5′ mean exactly the same thing. As a learner, it is safer to keep the variable on the left side, but be prepared to interpret both forms in exam questions.
一个常见的误解是把符号方向弄反。记住：小的一端指向更小的量。因此 ‘5 < x' 和 'x > 5′ 意思完全相同。作为学生，把变量放在左边更安全，但要能在试题中解读两种形式。
2. Solving Linear Inequalities | 解线性不等式
Solving a linear inequality is very similar to solving a linear equation, with one crucial exception: when you multiply or divide both sides by a negative number, you must reverse the inequality sign. For instance, to solve 8 – 3x ≤ 2, first subtract 8 from both sides: –3x ≤ –6. Now divide by –3, reversing the sign: x ≥ 2. Always check your solution by substituting a value back into the original inequality.
解线性不等式与解线性方程非常相似，但有一个关键例外：当你在不等式两边乘以或除以一个负数时，必须反转不等号的方向。例如，解 8 – 3x ≤ 2，首先两边减 8：–3x ≤ –6。现在除以 –3，反转符号：x ≥ 2。始终通过将数值代回原不等式来检验你的解。
You may also encounter inequalities that need expanding brackets first. Simplify step by step, collecting like terms, and only reverse the sign if the step involves multiplying or dividing by a negative. Keep the variable on the side that gives a positive coefficient if possible, to avoid forgetting to flip the symbol.
你也会遇到需要先展开括号的不等式。逐步化简，合并同类项，只有当步骤涉及乘以或除以负数时才改变符号。尽可能让变量系数为正，以避免忘记反转符号。
3. Representing Inequalities on a Number Line | 在数轴上表示不等式
Number lines provide a visual way to show solution sets. Use an open circle (o) for strict inequalities (<, >) and a closed circle (•) for inclusive inequalities (≤, ≥). Draw an arrow in the direction of all possible values. For example, y ≤ –2 is shown with a closed circle at –2 and a line extending to the left, marked with an arrowhead. If the variable can be any real number, the line continues indefinitely.
数轴提供了一种直观展示解集的方式。严不等式（<, >）使用空心圆圈（o），包含等号的不等式（≤, ≥）使用实心圆点（•）。朝所有可能取值的那个方向画一条带箭头的线。例如，y ≤ –2 表示为在 –2 处画实心圆点，并向左延伸一条带箭头的线。若变量可以是任意实数，线条无限延伸。
Sometimes you must represent a compound inequality like 0 < x < 5. This is shown with an open circle at 0, an open circle at 5, and the segment between them shaded. Exam questions often ask you to write down the inequality represented by a given diagram, so practice reading number lines in both directions.
有时你需要表示复合不等式，如 0 < x < 5。这时在 0 和 5 处画空心圆，并将两点之间的部分涂上阴影。试题常会要求你根据给定的图形写出不等式，因此要练习从两个方向阅读数轴。
4. Solving Double Inequalities | 解双重不等式
A double inequality combines two inequalities, such as –3 ≤ 2x + 1 < 7. The goal is to isolate x in the middle. Perform the same operation on all three parts simultaneously. Subtract 1 from all parts: –4 ≤ 2x < 6. Then divide all parts by 2: –2 ≤ x < 3. No sign reversal is needed here because we divide by a positive number. Write the final answer in its simplest form, ensuring the smaller number is on the left.
双重不等式组合了两个不等式，如 –3 ≤ 2x + 1 < 7。目标是把中间的 x 分离出来。同时对三部分进行相同运算：全部减去 1，得到 –4 ≤ 2x < 6。然后全部除以 2，得到 –2 ≤ x < 3。因为除以正数，无需反转不等号。将最终答案写成最简形式，确保较小的数在左边。
If the coefficient of x is negative, you may need to split the double inequality into two separate ones to avoid confusion. For example, –5 < 3 – 2x ≤ 9 can be split into –5 < 3 – 2x and 3 – 2x ≤ 9. Solve each part, keeping track of sign changes, then combine the results into the correct intersection. Double inequalities appear frequently in extended papers, so this skill is essential.
若 x 的系数为负，你可能需要把双重不等式拆分成两个独立不等式以避免混淆。例如，–5 < 3 – 2x ≤ 9 可以拆成 –5 < 3 – 2x 和 3 – 2x ≤ 9。分别解每一部分，注意符号变化，再把结果合并成正确的交集。双重不等式在扩展试卷中经常出现，因此这项技能至关重要。
5. Solving Quadratic Inequalities | 解二次不等式
Quadratic inequalities like x² – 4x + 3 ≤ 0 require a different approach. First, treat the inequality as an equation to find critical values. Solve x² – 4x + 3 = 0 by factorising: (x – 1)(x – 3) = 0, so x = 1 or x = 3. These values divide the number line into intervals: x < 1, 1 < x < 3, and x > 3. Test a point from each interval in the original inequality to determine which regions satisfy it.
像 x² – 4x + 3 ≤ 0 这样的二次不等式需要不同的方法。首先，把不等式当作方程来求临界值。通过因式分解解 x² – 4x + 3 = 0：(x – 1)(x – 3) = 0，所以 x = 1 或 x = 3。这些值把数轴分为几个区间：x < 1，1 < x < 3 和 x > 3。从每个区间选一个测试点代入原不等式，以确定哪些区域满足条件。
For x² – 4x + 3 ≤ 0, test x = 0: 0 – 0 + 3 = 3, which is not ≤ 0, so x < 1 fails. Test x = 2: 4 – 8 + 3 = –1, which is ≤ 0, so 1 ≤ x ≤ 3 works. Test x = 4: 16 – 16 + 3 = 3, not ≤ 0. The solution is 1 ≤ x ≤ 3. Because the inequality includes equality, the critical values themselves are part of the solution set. Always write the final answer in interval notation or using inequality signs as required.
对于 x² – 4x + 3 ≤ 0，测试 x = 0：0 – 0 + 3 = 3，不 ≤ 0，因此 x < 1 不成立。测试 x = 2：4 – 8 + 3 = –1，≤ 0，所以 1 ≤ x ≤ 3 成立。测试 x = 4：16 – 16 + 3 = 3，不 ≤ 0。解为 1 ≤ x ≤ 3。因为不等式包含等号，临界值本身属于解集。始终按要求用区间表示法或不等号形式写出最终答案。
6. Graphical Approach to Quadratic Inequalities | 二次不等式的图解法
You can also solve quadratic inequalities by sketching the graph of y = ax² + bx + c. A positive leading coefficient (a > 0) gives a ∪-shaped parabola, while a negative one gives ∩-shaped. The x-intercepts are the critical values where the expression equals zero. To solve ax² + bx + c > 0, look for the x-values where the graph lies above the x-axis. For < 0, look for where it lies below.
你还可以通过画出 y = ax² + bx + c 的图像来解二次不等式。正的二次项系数（a > 0）给出 ∪ 形抛物线，负的给出 ∩ 形。与 x 轴的交点就是表达式等于零的临界值。要解 ax² + bx + c > 0，找出图像位于 x 轴上方的 x 值；对于 < 0，则找出图像位于 x 轴下方的 x 值。
For example, to solve –x² + 4x – 3 ≥ 0, note that a = –1. The parabola opens downward. The roots from –x² + 4x – 3 = 0 are x = 1 and x = 3 (multiply by –1: x² – 4x + 3 = 0). The graph is above or on the x-axis between the roots, so the solution is 1 ≤ x ≤ 3. This method is quick once you can sketch parabolas confidently. CIE exams may provide the graph or expect you to sketch it roughly.
例如，解 –x² + 4x – 3 ≥ 0，注意 a = –1。抛物线开口向下。由 –x² + 4x – 3 = 0 解得根为 x = 1 和 x = 3（乘以 –1：x² – 4x + 3 = 0）。在两根之间图像位于 x 轴上方或轴上，因此解为 1 ≤ x ≤ 3。一旦你能自信地画出抛物线，这种方法就非常快捷。CIE 考试可能会提供图像，或要求你大致画出。
7. Inequalities in Word Problems | 应用题中的不等式
Real-world problems often require you to form and solve an inequality. Phrases like ‘at least’, ‘not more than’, ‘a minimum of’, and ‘exceeds’ give clues. For instance, ‘the sum of three times a number and 7 is at least 19’ translates to 3x + 7 ≥ 19. Solving gives x ≥ 4. Always define your variable clearly and check if the solution makes sense in context (e.g., integer constraints, positive lengths).
现实世界的问题常常需要你建立并解一个不等式。诸如 ‘至少’、’不超过’、’最小值’ 和 ‘超过’ 等短语提供线索。例如，’一个数的三倍加 7 至少为 19′ 可转化为 3x + 7 ≥ 19。解得 x ≥ 4。始终清晰地定义变量，并检查解在上下文中是否有意义（例如，整数约束、正的长度等）。
In CIE questions, you might need to combine inequalities. For example, a rectangle has length 2x cm and width x cm, and its perimeter is less than 50 cm, while its area is at least 24 cm². The perimeter inequality is 2(2x + x) < 50, simplified to 6x < 50, so x < 25/3. The area inequality is 2x² ≥ 24, giving x² ≥ 12, so x ≥ √12 or x ≤ –√12 (ignore negative). Thus x satisfies √12 ≤ x < 25/3. Such problems test your ability to link algebra with geometry.
在 CIE 题目中，你可能需要组合不等式。例如，一个矩形长为 2x cm、宽为 x cm，其周长小于 50 cm，面积至少为 24 cm²。周长不等式为 2(2x + x) < 50，化简得 6x < 50，所以 x < 25/3。面积不等式为 2x² ≥ 24，得 x² ≥ 12，故 x ≥ √12 或 x ≤ –√12（舍去负值）。因此 x 满足 √12 ≤ x < 25/3。这类题目考查你将代数与几何联系起来的能力。
8. Representing Inequalities on a Coordinate Plane | 平面直角坐标系中的不等式区域
An inequality in two variables defines a region on the xy-plane. Start by drawing the boundary line. For strict inequalities (< or >), use a dashed line. For ≤ or ≥, use a solid line. For example, y > 2x + 1 has a dashed line y = 2x + 1. To decide which side to shade, pick a test point not on the line, such as (0,0). Substitute: 0 > 2(0) + 1? That gives 0 > 1, which is false. So shade the other side, the region above the line.
二元不等式定义 xy 平面上的一个区域。首先画出边界线。对于严不等式（< 或 >），使用虚线。对于 ≤ 或 ≥，使用实线。例如，y > 2x + 1 的边界线 y = 2x + 1 为虚线。要确定涂哪一侧，选取一个不在线上的测试点，如 (0,0)。代入得：0 > 2(0) + 1？即 0 > 1，不成立。因此涂另一侧，即直线上方的区域。
When an inequality is given in the form x ≤ a or y ≥ b, the boundary is vertical or horizontal. For x ≤ 3, draw a solid vertical line at x = 3 and shade the left side. For y < –2, draw a dashed horizontal line at y = –2 and shade below. Shading the required region is a common exam task, and sometimes you must label the region clearly with a letter, such as R.
当不等式形如 x ≤ a 或 y ≥ b 时，边界是垂直或水平的。对于 x ≤ 3，在 x = 3 处画实线，涂左侧；对于 y < –2，在 y = –2 处画虚线，涂下方。给要求区域涂阴影是常见的考试任务，有时你需要用字母（如 R）清晰地标记区域。
9. Systems of Linear Inequalities in Two Variables | 二元线性不等式组
Exam questions often ask you to shade the region that satisfies a set of inequalities, such as x ≥ 0, y ≤ x + 2, and y > –x + 1. Draw each boundary line using the appropriate style (solid or dashed) and then shade each individual inequality lightly, or use a combined approach. The feasible region is the intersection where all shadings overlap. This region should be clearly identified, usually by a bolder shading or by labelling with R.
考试题目经常要求你涂出满足一组不等式的区域，例如 x ≥ 0，y ≤ x + 2，以及 y > –x + 1。用适当的线型（实线或虚线）画出每条边界线，然后分别涂出每个不等式的区域，或者采用综合方法。可行区域是所有涂色重叠的交集。这个区域应被清晰标识，通常用更深的涂色或标记 R 表示。
Sometimes the question provides the graph and asks you to write down the three inequalities that define a shaded region. Work systematically: identify each boundary line’s equation, note whether it is solid or dashed, and determine the inequality direction by testing a point inside the shaded region. For vertical and horizontal lines, inequalities are straightforward; for slanted lines, use y = mx + c form and test a point to get the sign. Practice with past papers to become efficient at this skill.
有时题目给出图形，要求你写出定义某个阴影区域的三条不等式。有条理地操作：找出每条边界线的方程，注意线是实线还是虚线，并通过在阴影区域内选点测试来确定不等式方向。对于水平或垂直线，不等式很简单；对于斜线，用 y = mx + c 形式并选点测试得到符号。通过真题练习来熟练这项技能。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing by a negative. Always pause before that step and ask: ‘Am I using a negative number?’ Another mistake is misreading the symbol on a number line diagram, confusing open and closed circles. Double-check whether the boundary is included. Also, when solving quadratic inequalities, students sometimes write the answer as two disjoint intervals when it should be a single continuous interval (or vice versa). Sketching the graph helps avoid this.
最常见的错误之一是在乘以或除以负数时忘记反转不等号。始终在执行这一步之前暂停，问自己：“我是在用负数吗？”另一个错误是误读数轴图上的符号，混淆空心和实心圆圈。反复确认边界是否包含在内。另外，解二次不等式时，学生有时在应当写成单个连续区间时却写成两个分离区间（反之亦然）。画草图有助于避免这个错误。
In coordinate geometry shading, ensure your dashed and solid lines are clearly distinguishable. Use a ruler and pencil in the exam. If you are asked to find integer points in a region, systematically check each coordinate pair near the boundaries. Finally, always present your final answer in the format requested: as an inequality, interval, set notation, or a list of integers. Read the question carefully to know what is expected.
在坐标系涂色时，确保虚线和实线明显可区分。考试中使用尺子和铅笔。如果要求找出区域内的整数点，系统地检查边界附近的每个坐标对。最后，始终按照要求的格式给出最终答案：不等式、区间、集合符号或整数列表。仔细阅读题目，明确期望的答案形式。
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