📚 Mastering Inequalities for IB and Edexcel Mathematics | IB Edexcel 数学：不等式 考点精讲
Inequalities form a cornerstone of both IB and Edexcel A Level Mathematics, bridging algebraic manipulation and real-world problem solving. Whether you are preparing for IB Analysis & Approaches, IB Applications & Interpretation, or Edexcel International A Level, a robust understanding of inequalities is essential for success. This article distils key concepts, techniques, and exam strategies to help you tackle inequality questions with confidence.
不等式是 IB 和 Edexcel A Level 数学的核心内容之一，连接着代数运算与现实问题求解。无论你正在准备 IB 分析与方法、IB 应用与解释，还是 Edexcel 国际 A Level 考试，扎实掌握不等式对于取得高分至关重要。本文将提炼关键概念、解题技巧与应试策略，帮助你有把握地解决不等式考题。
1. Understanding Inequality Notation and Properties | 理解不等式的符号与性质
eater than), ≤ (less than or equal to), and ≥ (greater than or equal to). When working with inequalities, you can add or subtract the same quantity from both sides without changing the direction. Multiplying or dividing both sides by a positive number maintains the inequality sign, but multiplying or dividing by a negative number reverses the sign. This reversal is a critical rule that many students overlook under exam pressure.
不等式表示一个表达式大于、小于或不等于另一个表达式。基本符号有 <（小于）、>（大于）、≤（小于等于）和 ≥（大于等于）。处理不等式时，可以在两边同时加减同一个量，不等号方向不变。两边同乘或同除一个正数，不等号保持不变，但同乘或同除一个负数，不等号必须反向。这个反向规则是很多学生在考试中容易忽略的关键点。
2. Solving Linear Inequalities | 求解线性不等式
ering to flip the inequality sign if you multiply or divide by a negative number. The solution is typically expressed as a simple inequality (x < 4), in interval notation (-∞, 4), or on a number line. In IB exams, you may need to express solutions in set builder notation, while Edexcel frequently uses interval notation.
线性不等式涉及一次表达式，例如 3x - 5 < 7。求解时，通过逆运算分离变量，若乘或除以负数，切记调转不等号方向。解通常以简单不等式（x < 4）、区间记号（-∞, 4）或在数轴上表示。IB 考试常要求用集合建构式表示，而 Edexcel 更常用区间记号。
o two parts or by performing the same operation on all three sides simultaneously. Both methods are acceptable, but simultaneous operations are more efficient and reduce arithmetic errors.
双向不等式如 -3 < 2x + 1 ≤ 5 可以拆成两部分求解，或同时对三边执行相同运算。两种方法均可，但同步运算更高效，能减少算术错误。
3. Quadratic Inequalities and the Sign Chart Method | 二次不等式与符号表方法
h. First, find the critical values by solving the corresponding quadratic equation. Then sketch the parabola or use a sign chart (also called a table of intervals) to determine where the quadratic is positive or negative. If the coefficient of x² is positive, the graph is U-shaped, so the quadratic is negative between the roots and positive outside them.
二次不等式，如 x² - 5x + 6 ≥ 0，需要不同的处理方式。首先，解对应的二次方程求出临界值。然后画出抛物线草图或使用符号表（也称区间表）来确定二次式何时为正、何时为负。若 x² 的系数为正，图像呈 U 形，则二次式在两根之间为负，在两根之外为正。
For IB and Edexcel, the ability to express solutions as x ≤ 2 or x ≥ 3 is as important as using interval notation (-∞, 2] ∪ [3, ∞).
常见错误是当抛物线开口向下时错误地设置不等号。务必核对首项系数的正负。对 IB 和 Edexcel 而言，能以 x ≤ 2 或 x ≥ 3 表达解，与使用区间记号 (-∞, 2] ∪ [3, ∞) 同等重要。
4. Rational Inequalities and Finding Common Denominators | 分式不等式与通分法
When tackling rational inequalities like (x - 1)/(x + 2) > 0, do not simply multiply both sides by the denominator, as its sign may be unknown. Instead, bring all terms to one side so you have a single fraction compared to zero. Then find the critical values by setting the numerator and denominator equal to zero separately. These values divide the number line into intervals; test a point from each interval in the original inequality to determine the solution set.
处理形如 (x - 1)/(x + 2) > 0 的分式不等式时，不要贸然两边同乘分母，因为其符号未定。应将所有项移到一边，使一边为单个分式与 0 比较。然后分别令分子、分母为零，求出临界值。这些值将数轴分成若干区间；从每个区间内选一点代入原不等式，确定解集。
Pay special attention to the denominator's critical values: they are never included in the solution (unless the inequality allows equality and the denominator is strictly positive/negative in that neighborhood). Always state any excluded values clearly to secure method marks in Edexcel and IB exams.
特别留意分母的临界值：它们永远不包含在解中（除非原不等式带等号且分母在邻域内恒正或恒负）。务必清晰写出所有排除值，以在 Edexcel 和 IB 考试中确保得到步骤分。
5. Absolute Value Inequalities and Geometric Interpretation | 绝对值不等式及其几何意义
Absolute value inequalities, such as |2x - 3| < 5, can be approached either algebraically or geometrically. The inequality |expresssion| < a (with a > 0) means the distance from the expression to zero is less than a, leading to the compound inequality -a < expression < a. Conversely, |expression| > a splits into expression < -a or expression > a. This 'less thAND, greatOR' mnemonic is immensely helpful.
绝对值不等式，如 |2x - 3| < 5，可用代数或几何方法处理。不等式 |表达式| < a（a > 0）表示表达式到零的距离小于 a，导致复合不等式 -a < 表达式 < a。反之，|表达式| > a 拆分成 表达式 < -a 或 表达式 > a。"小于即 AND，大于即 OR" 的记忆法非常实用。
Graphical interpretation on a number line reinforces understanding: |x - c| represents the distance between x and c. Edexcel and IB examiners often expect candidates to be able to move fluently between algebraic manipulation and graphical reasoning.
在数轴上的几何解释能加深理解：|x - c| 表示 x 与 c 的距离。Edexcel 和 IB 考官常期望考生能在代数操作与图形推理之间自如转换。
When absolute values appear on both sides, square both sides (after checking they are non-negative) or consider the different cases for the signs of the expressions inside the modulus. Both approaches are valid, but squaring is often quicker.
当两边都出现绝对值时，可在检验非负后两边平方，或根据模内表达式的符号分情况讨论。两种方法均可，但平方常常更快。
6. Systems of Linear Inequalities and Feasible Regions | 线性不等式组与可行域
In both IB AI and Edexcel Decision Mathematics, systems of linear inequalities define a feasible region on the coordinate plane. Each inequality corresponds to a half-plane; the intersection of these half-planes is the feasible region. This concept is central to linear programming, where an objective function is optimised over this region. Accurate graphing and shading are essential skills.
在 IB AI 和 Edexcel 决策数学中，线性不等式组在坐标平面上定义了一个可行域。每个不等式对应一个半平面；这些半平面的交集即为可行域。这个概念是线性规划的核心，即在该区域上优化目标函数。精确的绘图与阴影表示是必备技能。
ines clearly and use dashed lines for strict inequalities (< or >) and solid lines for inclusive inequalities (≤ or ≥).
画阴影时，遵循考纲规定的惯例：IB 常使用"否定区域"法，而 Edexcel 可能不同。务必清晰标记直线，严格不等式 (< 或 >) 用虚线，带等号的不等式 (≤ 或 ≥) 用实线。
Finding the vertices of the feasible region is often required. Solve simultaneous equations for the intersecting lines, but be careful to check that each vertex indeed satisfies all constraints before evaluating the objective function.
通常需要找出可行域的顶点。通过解直线交点的联立方程组来求得，但注意在代入目标函数前，要核实每个顶点确实满足所有约束。
7. Graphical Approaches to Quadratic and Higher-Degree Inequalities | 二次及高次不等式的图像解法
While a sign chart is efficient, sketching the graph of the function can provide a quick visual check. For a quadratic inequality, the intersection points with the x-axis are the roots. The solution set corresponds to the x-values for which the graph lies above or below the axis. This method extends naturally to cubic and quartic inequalities, where knowing the shape helps identify the sign in each interval.
虽然符号表很高效，但绘制函数图像可提供快速的视觉检验。对于二次不等式，图像与 x 轴的交点即根。解集对应于图像位于 x 轴上方或下方的 x 值。这种方法自然延伸到三次和四次不等式，了解图像形状有助于判断各区间的符号。
For higher-degree polynomial inequalities, factorisation is key. Once you factor the polynomial into linear and irreducible quadratic factors, determine the sign of each factor over the intervals separated by the real roots. The overall sign is the product of the individual signs. This technique is tested at both IB Higher Level and Edexcel Further Pure contexts.
对于高次多项式不等式，因式分解是关键。一旦将多项式分解为一次和不可约二次因式，便可判断每个因式在各实数根分隔的区间内的符号。总的符号即各因式符号的乘积。此技巧在 IB 高级水平及 Edexcel 进阶纯数中均有考查。
8. Exponential and Logarithmic Inequalities | 指数与对数不等式
Inequalities involving exponentials and logarithms require careful handling of monotonicity. For a base greater than 1, the exponential and logarithmic functions are strictly increasing, so the inequality direction is preserved when applying these functions. However, if 0 < base < 1, the functions are strictly decreasing, and the inequality sign flips when taking logs or exponentials. This nuance appears in challenging IB HL and Edexcel A2 questions.
涉及指数与对数的不等式需要小心处理单调性。当底数大于 1 时，指数函数和对数函数严格单调递增，因此在应用这些函数时不等号方向保持不变。然而，若 0 < 底数 < 1，则函数严格单调递减，取对数或指数时不等号会反向。这个细微之处在 IB HL 和 Edexcel A2 的挑战题中常有考查。
Always ensure the arguments of any logarithms remain positive. When solving log inequalities, start by writing the domain restrictions (e.g., x + 2 > 0) and then apply logarithmic properties. Combining logs into a single logarithm and comparing arguments can lead to a solution, but you must intersect that result with the domain restrictions to obtain the final answer.
务必确保任何对数的真数保持正数。求解对数不等式时，首先写出定义域限制（例如 x + 2 > 0），然后运用对数性质。将对数合并后比较真数可得出一组解，但必须将该结果与定义域限制求交集以得到最终答案。
9. Inequalities in Modulus with Quadratics and Non-linear Expressions | 含二次或非线性表达式的模不等式
Combining modulus with quadratics, such as |x² - 4| < 3, tests a deeper understanding. Here, the geometric meaning is that x² - 4 is between -3 and 3. This splits into two inequalities: x² - 4 > -3 and x² - 4 < 3. Solve each quadratic inequality and take the intersection of the solution sets. Such compound problems require organised work and careful handling of intersections and unions.
将模与二次式结合，如 |x² - 4| < 3，考查更深层的理解。其几何意义是 x² - 4 的取值在 -3 与 3 之间。这拆成两个不等式：x² - 4 > -3 和 x² - 4 < 3。分别解二次不等式，并取解集的交集。此类复合问题需要有组织的书写以及仔细处理交集与并集。
Alternatively, you can square both sides after isolating the modulus on one side, eliminating the absolute value. This yields a quartic inequality that can often be simplified, but always verify solutions in the original inequality to exclude extraneous results.
另一种方法是先将模隔离在一边，然后两边平方以消去绝对值。这样会得到一个四次不等式，通常可化简，但务必在原不等式中验证解以排除增根。
10. Common Errors and How to Avoid Them | 常见错误及避免方法
One pervasive error is forgetting to reverse the inequality sign when multiplying or dividing by a negative number, especially when the negative sign is 'hidden' inside a variable term like -x. Always make the coefficient of the variable positive first. Another pitfall is treating a rational inequality like an equation by cross-multiplying without concern for the sign of the denominator. Instead, use the method of bringing all terms to one side and constructing a sign chart.
一个普遍错误是乘或除以负数时忘记调转不等号，尤其是当负号'藏'在诸如 -x 这样的变量项中时。务必首先将变量的系数化为正。另一个陷阱是像解方程一样交叉相乘处理分式不等式，却忽略分母的符号。应使用移项通分、建立符号表的方法。
In quadratic inequalities, students often write the solution as 'between the roots' for all cases, disregarding whether the parabola opens upward or downward. Always sketch the graph or use a test point to confirm. For absolute value inequalities, mixing 'and' and 'or' logic is a frequent source of lost marks.
在二次不等式中，学生往往将所有情况都写成'两根之间'，忽略了抛物线的开口方向。务必画图或选取测试点确认。对于绝对值不等式，混淆'且'和'或'的逻辑也是丢分的常见原因。
Finally, failing to state the final answer in the required format – interval notation, set notation, or inequality form – can cost unnecessary marks in IB and Edexcel exams. Read the question instruction carefully.
最后，未能按照要求格式给出最终答案——区间记号、集合记号或不等式形式——会在 IB 和 Edexcel 考试中无故丢分。务必仔细阅读题目指令。
11. Exam-Style Strategy and Time Management | 考试策略与时间管理
When faced with an inequality problem in Section A or early Paper sections, allocate time based on mark weighting. For a 4-mark question, allow about 5 minutes. Start by identifying the type of inequality and recalling the standard algorithm. Write down each step clearly, including the critical values, sign chart, and the final solution. This structure makes it easier to spot arithmetic slips and enables the examiner to award partial credit.
面对试卷 A 部分或早期大题中的不等式问题时，根据分值分配时间。对于一道 4 分的题，预留约 5 分钟。首先判别不等式类型，回想标准解法步骤。清晰地写下每一步，包括临界值、符号表和最终解。这种结构便于发现算术失误，也让阅卷官更容易给步骤分。
In longer linear programming or applied problems, draw your graph accurately with a ruler, label axes and lines, and shade properly. Annotate the feasible region clearly. When evaluating the objective function at vertices, organise calculations in a small table to minimise errors. Underline the final decision variables and the optimal value.
在较长的线性规划或应用问题中，用尺规精确绘制图像，标记坐标轴和直线，正确画出阴影。清晰标注可行域。在顶点处评估目标函数时，用小表格整理计算以减少错误。在最终决策变量和最优值下划线突出显示。
ing tasks, inequalities often arise in constraints. Clearly link your algebraic inequalities to the context – negative quantities. Document your reasoning to show the moderator or examiner your logical flow.
在 IB 探究报告或 Edexcel 建模任务中，不等式常出现在约束条件中。明确将代数不等式与情境关联——例如，x ≥ 0, y ≥ 0 表示非负量。记录推理过程，向评审人或考官展示逻辑脉络。
12. Summary and Final Tips | 总结与最后提示
Inequalities are a unifying thread through algebra, functions, and optimisation. Master the core types: linear, quadratic, rational, and absolute value, and practise converting between graphical, numerical, and algebraic representations. Familiarise yourself with the notation required by your specific syllabus – IB and Edexcel have subtle differences that can matter. Regular practice with past paper questions builds speed and accuracy.
不等式是贯穿代数、函数和优化的一条统一线索。掌握核心类型：线性、二次、分式和绝对值不等式，并练习在图像、数值与代数表示之间转换。熟悉你所考纲要求的记号——IB 与 Edexcel 之间的细微差别可能很重要。定期练习历年真题有助于提高速度与准确性。
Remember the three golden rules: (1) flip the sign when multiplying/dividing by a negative; (2) always consider domain restrictions in rational and logarithmic inequalities; (3) sketch or test intervals rather than guessing. With these tools, you can handle any inequality question that appears in your IB or Edexcel mathematics exam.
牢记三条黄金法则：（1）乘或除以负数时调转不等号；（2）在分式及对数不等式中始终考虑定义域限制；（3）画图或检验区间，而非猜测。有了这些工具，你就能应对 IB 或 Edexcel 数学考试中任何不等式问题。
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