📚 Edexcel Maths: Mastering Inequalities | Edexcel 数学：不等式 考点精讲
Inequalities are a core part of the Edexcel International A-Level and IGCSE Mathematics syllabuses, extending the idea of equations to ranges of values. In this revision guide, we break down every key concept, from solving simple linear inequalities to handling quadratics, absolute values, and graphing regions. Master these techniques to confidently tackle both pure and applied maths questions.
不等式是 Edexcel 国际 A-Level 和 IGCSE 数学课程的核心内容，它将方程的思想扩展到数值范围。本篇复习指南将逐一拆解所有关键考点，从求解简单线性不等式，到处理二次不等式、绝对值不等式以及区域作图。掌握这些技巧，你将从容应对纯数学和应用数学中的相关题目。
1. Inequality Notation and Number Lines | 不等式符号与数轴表示
Inequalities use symbols <, >, ≤, ≥ to compare expressions. On a number line, open circles represent strict inequalities (< or >) while closed circles represent inclusive inequalities (≤ or ≥). The direction of the arrow indicates all numbers greater than or less than the boundary.
不等式使用符号 <、>、≤、≥ 来比较表达式。在数轴上，空心圆圈表示严格不等式（< 或 >），而实心圆圈表示包含边界的不等式（≤ 或 ≥）。箭头的方向表示大于或小于边界的所有数值。
For –2 < x ≤ 3, draw an open circle at –2, a closed circle at 3, and shade the line between them. This compact representation helps visualise solution sets before expressing them in set or interval notation.
对于 –2 < x ≤ 3，在 –2 处画空心圆，在 3 处画实心圆，并涂阴影连接二者。这种紧凑的表示法有助于在写成集合或区间形式之前直观地呈现解集。
2. Solving Linear Inequalities | 求解线性不等式
Treat linear inequalities like linear equations, but remember: multiplying or dividing by a negative number reverses the inequality sign. For example, 3x – 4 > 5 simplifies to x > 3, while –2x ≤ 6 becomes x ≥ –3 after division by –2.
解线性不等式与解线性方程类似，但切记：乘或除以一个负数时，不等号方向要改变。例如，3x – 4 > 5 化简为 x > 3；而 –2x ≤ 6 除以 –2 后变为 x ≥ –3。
Always present your final answer in its simplest form, often using set notation like {x: x > 3} or interval notation (3, ∞). Double-check boundary conditions when the inequality is inclusive.
最终答案应以最简形式呈现，常用集合符号如 {x: x > 3}，或区间表示法 (3, ∞)。当不等式包含等号时，务必检查边界条件。
3. Compound Inequalities | 复合不等式
Compound inequalities combine two inequalities into one statement, such as –5 ≤ 2x + 1 < 7. Solve by breaking into two parts or by performing the same operation on all three sections simultaneously. Subtracting 1 gives –6 ≤ 2x < 6, then dividing by 2 yields –3 ≤ x < 3.
复合不等式将两个不等式合并成一个语句，例如 –5 ≤ 2x + 1 < 7。可以拆分为两部分求解，也可以同时对三个部分进行相同的运算。先减 1 得 –6 ≤ 2x < 6，再除以 2 得到 –3 ≤ x < 3。
Avoid common mistakes like forgetting to reverse an inequality when the whole compound inequality is multiplied by a negative. Write answers in the form a ≤ x < b, and represent them on a number line with appropriate circles.
避免常见错误，例如当整个复合不等式乘以负数时忘记调转不等号。答案应写成 a ≤ x < b 的形式，并在数轴上用合适的圆圈表示。
4. Quadratic Inequalities | 二次不等式
To solve ax² + bx + c > 0 (or < 0), first find the roots of the corresponding quadratic equation. Sketch the parabola: if a > 0, it opens upwards; if a < 0, it opens downwards. Determine where the graph lies above or below the x-axis to satisfy the inequality.
要求解 ax² + bx + c > 0（或 < 0），首先找到对应二次方程的根。绘制抛物线草图：若 a > 0，开口向上；若 a < 0，开口向下。确定图像在 x 轴上方或下方的部分，以满足不等式。
For x² – 5x + 6 < 0, roots are 2 and 3. The parabola opens upwards, so the graph is below the x-axis between the roots: 2 < x < 3. For x² – x – 6 ≥ 0, roots are –2 and 3; the solution is x ≤ –2 or x ≥ 3. Always express quadratics with a positive coefficient of x² when possible.
对于 x² – 5x + 6 < 0，根为 2 和 3。抛物线开口向上，因此图像在两根之间位于 x 轴下方：2 < x < 3。对于 x² – x – 6 ≥ 0，根为 –2 和 3；解为 x ≤ –2 或 x ≥ 3。尽可能将二次项系数化为正。
5. Using the Discriminant with Inequalities | 判别式在不等式中的应用
The discriminant Δ = b² – 4ac determines the nature of the roots. For a quadratic to be always positive and have no real roots (i.e. ax² + bx + c > 0 for all x), we need a > 0 and Δ < 0. Similarly, to be always negative, a < 0 and Δ < 0.
判别式 Δ = b² – 4ac 决定根的性质。要使二次函数恒正且无实根（即对所有 x 有 ax² + bx + c > 0），需满足 a > 0 且 Δ < 0。类似地，要恒负需满足 a < 0 且 Δ < 0。
This skill is often tested in Edexcel problems asking for a range of k such that kx² + 3x + k is positive for all real x. Set a = k > 0 and Δ = 9 – 4k² < 0, which leads to k > 1.5.
这一技巧在 Edexcel 考题中常出现，例如求 k 的取值范围使得 kx² + 3x + k 对所有实数 x 恒正。令 a = k > 0 且 Δ = 9 – 4k² < 0，解得 k > 1.5。
6. Inequalities Involving Rational Expressions | 含有理表达式的不等式
Rational inequalities like (x – 1)/(x + 2) > 0 require careful handling. Bring all terms to one side, combine into a single fraction, and find critical values where the numerator or denominator equals zero. Use a sign table or test intervals to determine where the fraction is positive or negative. Never multiply through by the denominator if its sign is unknown, as this could incorrectly reverse the inequality.
形如 (x – 1)/(x + 2) > 0 的有理不等式需谨慎处理。将所有项移到一端，合并为一个分式，找出使分子或分母为零的临界值。使用符号表或区间测试法判断分式在何处为正或负。切勿在分母符号未知的情况下直接乘以分母，否则可能错误地改变不等号方向。
For (x – 1)/(x + 2) ≥ 0, critical points are x = –2 (excluded) and x = 1 (included). Testing intervals gives x < –2 or x ≥ 1. Write the final answer using set notation, carefully excluding points that make the denominator zero.
对于 (x – 1)/(x + 2) ≥ 0，临界点为 x = –2（需排除）和 x = 1（包含）。经过区间测试得 x < –2 或 x ≥ 1。最终答案用集合符号表示，并仔细排除分母为零的点。
7. Absolute Value Inequalities | 绝对值不等式
Absolute value inequalities such as |x – 3| ≤ 4 translate to a compound inequality: –4 ≤ x – 3 ≤ 4, so –1 ≤ x ≤ 7. For |x + 2| > 5, split into x + 2 < –5 or x + 2 > 5, giving x < –7 or x > 3. Always isolate the absolute value expression first if there are other terms.
绝对值不等式如 |x – 3| ≤ 4 可转化为复合不等式：–4 ≤ x – 3 ≤ 4，因此 –1 ≤ x ≤ 7。对于 |x + 2| > 5，拆分为 x + 2 < –5 或 x + 2 > 5，得 x < –7 或 x > 3。若有其他项，务必先分离出绝对值表达式。
For more complex cases like |3x – 2| < x + 4, consider cases based on the expression inside the absolute value, or square both sides if both sides are non-negative. Squaring is a powerful technique but must be used cautiously, as it can sometimes introduce extraneous solutions—always verify your answers.
对于更复杂的情形如 |3x – 2| < x + 4，根据绝对值内部表达式的不同情况分情形讨论，或者当两边非负时两边平方。平方是一种强有力的方法，但必须谨慎使用，因为它有时会引入增根——务必验证所得解。
8. Graphical Inequalities and Shaded Regions | 图解法与阴影区域
To display y ≥ 2x + 1, draw the line y = 2x + 1 as a solid line (since ≥ includes equality) and shade the region above it. For strict inequalities like y < –x + 3, use a dashed line and shade below. When handling multiple inequalities, the feasible region is the overlap of all shaded areas.
要表示 y ≥ 2x + 1，画出直线 y = 2x + 1 并画成实线（因为 ≥ 包含等号），并将直线上方的区域涂上阴影。对于严格不等式如 y < –x + 3，使用虚线并涂下方区域。处理多个不等式时，可行域是所有阴影区域的重叠部分。
Edexcel often asks you to label the region that satisfies a set of inequalities, sometimes involving vertical or horizontal lines like x > 0. Always shade the intended region and clearly mark the boundaries; use a key or label the region R as requested.
Edexcel 常要求标出满足一组不等式的区域，有时会涉及 x > 0 这类垂直或水平线。务必在指定区域涂阴影并清晰标记边界；如有要求，使用图例或将区域标为 R。
9. Inequalities in Linear Programming | 线性规划中的不等式
Linear programming problems in Edexcel Decision Mathematics use inequalities to define a feasible region of production possibilities. Typical constraints include x ≥ 0, y ≥ 0, and linear combinations like 2x + 3y ≤ 24. The objective function is then maximised or minimised over this region.
Edexcel 决策数学中的线性规划问题使用不等式定义生产可行域。典型的约束条件包括 x ≥ 0、y ≥ 0 以及像 2x + 3y ≤ 24 这样的线性组合。然后在该区域上最大化或最小化目标函数。
Graph the boundary lines, determine whether they are solid or dashed based on the inequality, shade the unwanted region or the feasible region (depending on preference), and then slide the objective function line to find the optimal vertex. Always check the integer requirements if variables must be whole numbers.
绘制边界直线，根据不等式判断使用实线还是虚线，分别使用涂除非法区域或涂可行性区域（视个人偏好而定），然后平移目标函数直线以找到最优顶点。若变量必须是整数，需检查取整要求。
10. Algebraic Manipulations and Discriminant-Based Inequality Proofs | 代数变形与基于判别式的不等式证明
Many Edexcel proof questions require showing that a quadratic expression is always positive or that a certain inequality holds for all real numbers. Completing the square is a powerful technique here. For example, to prove x² – 4x + 5 > 0 for all real x, write it as (x – 2)² + 1 > 0, which is clearly positive.
许多 Edexcel 证明题要求证明一个二次表达式恒正，或某不等式对所有实数均成立。配方法在此非常有用。例如，要证明对所有实数 x 有 x² – 4x + 5 > 0，可写成 (x – 2)² + 1 > 0，显然恒正。
Another common proof involves showing that for distinct real numbers a and b, a² + b² > 2ab. Rearranging gives a² – 2ab + b² > 0, i.e. (a – b)² > 0, which is true for a ≠ b. Recognise these opportunities to simplify complex inequalities into perfect squares.
另一个常见证明是，对于任意不相等的实数 a 和 b，有 a² + b² > 2ab。移项得 a² – 2ab + b² > 0，即 (a – b)² > 0，在 a ≠ b 时显然成立。要善于识别这类将复杂不等式化为完全平方的机会。
11. Interval Notation and Set Builder Notation | 区间表示法与集合构造式
Edexcel expects you to express solutions using correct mathematical notation. Interval notation uses parentheses ( , ) for open boundaries and brackets [ , ] for closed boundaries. For example, –2 < x ≤ 5 is written (–2, 5]. The union symbol ∪ joins disjoint intervals: x < –1 or x > 2 becomes (–∞, –1) ∪ (2, ∞).
Edexcel 要求使用正确的数学符号表示解。区间表示法使用小括号 ( , ) 表示开边界，方括号 [ , ] 表示闭边界。例如 –2 < x ≤ 5 写作 (–2, 5]。并集符号 ∪ 连接不相交的区间：x < –1 或 x > 2 变为 (–∞, –1) ∪ (2, ∞)。
Set builder notation, {x: condition}, is also widely accepted. Be comfortable converting between these forms and number-line representations. Precision in notation counts—missing an equality can cost marks.
集合构造式 {x: 条件} 也被广泛接受。要能熟练在这些形式和数轴表示之间进行转换。符号的精确性很重要——遗漏一个等号就可能失分。
12. Common Pitfalls and Exam Tips | 常见陷阱与应试技巧
Never multiply or divide an inequality by an expression whose sign you do not know, unless you separate into cases. When squaring both sides of an absolute value inequality, confirm both sides are non-negative and check for false solutions. For quadratic inequalities, don’t blindly write the interval between roots—always sketch the graph to confirm the direction.
切勿在不知道表达式符号的情况下对不等式进行乘除运算，除非分情况讨论。对绝对值不等式两边平方时，要确认两边非负，并检查是否有增根。对于二次不等式，不要盲目地写出两根之间的区间——一定要画图确认开口方向。
Show all working clearly. Marks are awarded for method: rearranging to zero, finding critical values, and stating final intervals. In graphical questions, label lines and shade neatly. Finally, read the question carefully: does it ask for integer values of x, or all real values? Adjust your answer accordingly.
清晰展显所有步骤。方法的呈现将获得分数：化为零、找出临界值、并陈述最终区间。在作图题中，要标注直线并整齐涂阴影。最后，仔细审题：它问的是 x 的整数值，还是所有实数值？相应调整你的答案。
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