📚 Inequalities: Key Concepts for IB and AQA Mathematics | 不等式：IB 与 AQA 数学核心考点精讲
Inequalities are a cornerstone of algebra, bridging numerical comparisons with graphical representations. In IB and AQA mathematics curricula, you will tackle a variety of inequality types: linear, quadratic, polynomial, rational and those involving absolute values. Understanding how to solve them and represent solutions accurately is crucial for higher-level topics such as calculus, linear programming, and statistical constraints.
不等式是代数的基石，架起了数值比较与图形表示之间的桥梁。在 IB 和 AQA 数学课程中，你将处理各种类型的不等式：线性、二次、多项式、有理以及含绝对值的不等式。准确求解并正确表示解集对于微积分、线性规划和统计约束等高阶主题至关重要。
This revision guide walks through every inequality family that appears on Paper 1 and Paper 2, gives you a reliable method for each, and flags the traps that cost marks most often in examinations.
本篇复习指南将逐一讲解 Paper 1 与 Paper 2 中出现的每一类不等式，为每一类给出可靠解法，并指出考试中最常丢分的陷阱。
1. What is an Inequality? | 不等式是什么？
An inequality expresses a relationship where two expressions are not necessarily equal. Instead of asking 'for which values are the two sides the same?', an inequality asks 'for which values is one side larger or smaller than the other?'. Because the answer is normally a whole range of values rather than a single number, the solution is usually an interval, or a union of intervals.
不等式表示两个表达式不一定相等的关系。它不问"两边在哪些取值下相等"，而是问"在哪些取值下，一边大于或小于另一边"。由于答案通常不是某一个数，而是一整段取值，所以解集一般是一个区间，或若干区间的并集。
The four comparison symbols you must know fluently are listed below.
你必须熟练掌握的四个比较符号如下。
- < means 'is less than' (strict, endpoint excluded).
- > means 'is greater than' (strict, endpoint excluded).
- ≤ means 'is less than or equal to' (inclusive, endpoint included).
- ≥ means 'is greater than or equal to' (inclusive, endpoint included).
- < 表示"小于"（严格不等，端点不包含）。
- > 表示"大于"（严格不等，端点不包含）。
- ≤ 表示"小于或等于"（非严格，端点包含）。
- ≥ 表示"大于或等于"（非严格，端点包含）。
The table below summarises how a simple inequality is written in interval notation and how it is drawn on a number line.
下表总结了简单不等式如何写成区间表示法，以及如何在数轴上画出图形。
| Inequality | Interval Notation | Number Line |
|---|---|---|
| x > a | (a, ∞) | Open circle at a, arrow to the right |
| x < b | (-∞, b) | Open circle at b, arrow to the left |
| x ≥ a | [a, ∞) | Filled circle at a, arrow to the right |
| x ≤ b | (-∞, b] | Filled circle at b, arrow to the left |
| a < x < b | (a, b) | Open circles at both ends, segment between them |
| a ≤ x ≤ b | [a, b] | Filled circles at both ends, segment between them |
| a < x ≤ b | (a, b] | Open at a, filled at b, segment between them |
| x < a or x > b | (-∞, a) ∪ (b, ∞) | Two arrows pointing outwards from open circles |
2. Set-Builder Notation, Intervals and Infinity | 集合表示法、区间与无穷大
We denote solutions using set-builder notation {x | included. The symbol ∞ (infinity) is never a number, so it is always written with a round bracket, as in (-∞, 3) or [2, ∞).
我们用集合构建符号 {x | 条件} 或区间表示法如 (a, b)、[a, b]、(a, b] 来表示解集。圆括号表示端点不包含，方括号表示端点包含。符号 ∞（无穷大）不是数，因此永远与圆括号搭配，例如 (-∞, 3) 或 [2, ∞)。
Set-builder notation is preferred when the solution is described by a property rather than a simple range, for example {x : x ∈ ℝ, x ≠ 2}. The symbol ∈ means 'is an element of' and ℝ denotes the set of real numbers. When a solution consists of two separate pieces, the union symbol ∪ joins the intervals.
当解集由某种性质而非简单范围描述时，更适合用集合表示法，例如 {x : x ∈ ℝ, x ≠ 2}。符号 ∈ 表示"属于"，ℝ 表示实数集。当解集由两段互不相连的部分组成时，用并集符号 ∪ 连接区间。
{x | x < -2} ∪ {x | x > 3} = (-∞, -2) ∪ (3, ∞)
The intersection symbol ∩ is used when a solution must satisfy two conditions at once. Solving a pair of simultaneous inequalities always produces an intersection, never a union, unless the word 'or' appears in the question.
当解集必须同时满足两个条件时，使用交集符号 ∩。求解联立不等式组得到的一定是交集而非并集，除非题目中出现了"或"字。
3. The Golden Rules for Manipulating Inequalities | 处理不等式的基本法则
Inequalities behave much like equations, but with one crucial difference. You may add or subtract the same quantity on both sides without changing the direction of the inequality sign. You may also multiply or divide both sides by the same positive number.
不等式的变形与方程十分相似，但有一个关键区别。两边同时加上或减去同一个量，不等号方向不变；两边同时乘以或除以同一个正数，不等号方向也不变。
However, multiplying or dividing both sides by a negative number reverses the direction of the inequality. This is the single most common source of lost marks in the whole topic, and it applies to every inequality type, not only linear ones.
但是，两边同时乘以或除以一个负数时，不等号方向必须翻转。这是整个专题中最常见的失分点，而且它适用于所有类型的不等式，不仅限于线性不等式。
-2x > 6 ⇒ x < -3
Three further rules protect you from invalid algebra. First, never multiply both sides by an expression whose sign you do not know, such as (x - 2). Second, never square both sides unless both sides are known to be non-negative. Third, if you take reciprocals, the inequality direction reverses when both sides have the same sign.
另有三条规则可避免无效运算。第一，绝不要两边同时乘以一个符号未知的表达式，例如 (x - 2)。第二，除非已知两边都非负，否则不要两边同时平方。第三，取倒数时，若两边同号，不等号方向翻转。
- Adding the same term: sign unchanged.
- Multiplying by a positive constant: sign unchanged.
- Multiplying by a negative constant: sign reverses.
- Taking reciprocals of same-signed quantities: sign reverses.
- 同加一项：符号不变。
- 乘以正常数：符号不变。
- 乘以负常数：符号翻转。
- 同号取倒数：符号翻转。
4. Linear Inequalities in One Variable | 一元一次不等式
A linear inequality contains the variable to the first power only. The method is identical to solving a linear equation, with the sign-flip rule applied whenever you divide by a negative coefficient. Always finish by writing the solution in both inequality and interval form, since examiners often award a mark for correct notation.
一元一次不等式中，变量只出现一次幂。解法与解一元一次方程完全相同，只在除以负系数时应用翻转法则。最后务必同时写出不等式形式与区间形式的答案，因为评分标准常常为正确记号单独给分。
k is to substitute a value from the solution, such as x = 0, back into the original inequality.
例题：解 3x - 5 ≤ 7。两边加 5 得 3x ≤ 12，再除以 3 得 x ≤ 4。区间表示为 (-∞, 4]。一个有效的检验方法是从解集中取一个值，如 x = 0，代回原不等式验证。
g the x terms on the side where the coefficient is positive avoids a negative division altogether.
第二个例子展示翻转：解 8 - 2x < 3x - 7。把变量集中到一侧得 15 < 5x，即 x > 3，也就是 (3, ∞)。注意：尽量把 x 项集中到系数为正的一侧，就可以完全避免除以负数。
5. Quadratic Inequalities | 二次不等式
Quadratic inequalities are solved by a fixed three-step routine. Rearrange the inequality so that one side is zero, factorise the quadratic completely, then identify the critical values where each factor is zero. These critical values divide the number line into intervals, and you test the sign of the product in each interval.
二次不等式用固定的三步法求解。先把不等式整理成一侧为零，再将二次式完全因式分解，然后找出使每个因式为零的临界值。这些临界值把数轴分成若干区间，再判断每个区间内乘积的符号。
, so the region between the roots is negative and the solution is x < -2 or x > 3, that is (-∞, -2) ∪ (3, ∞).
例如解 x² - x - 6 > 0。因式分解得 (x - 3)(x + 2) > 0，临界值为 x = 3 与 x = -2。取 x = 0 检验得负值，说明两根之间为负区间，故解为 x < -2 或 x > 3，即 (-∞, -2) ∪ (3, ∞)。
The shape of the parabola explains the pattern. When the coefficient of x² is positive, the curve opens upwards, so '> 0' lies outside the roots and '< 0' lies between them. When the coefficient of x² is negative, the pattern is exactly reversed.
抛物线的形状可以解释这一规律。当 x² 的系数为正时，抛物线开口向上，"> 0" 对应两根之外，"< 0" 对应两根之间。当 x² 的系数为负时，规律恰好相反。
- a > 0 and f(x) > 0: solution is x < r₁ or x > r₂.
- a > 0 and f(x) < 0: solution is r₁ < x < r₂.
- No real roots and a > 0: f(x) > 0 for all real x, and f(x) < 0 has no solution.
- Repeated root r: f(x) ≥ 0 holds for all x, with equality only at x = r.
- a > 0 且 f(x) > 0：解为 x < r₁ 或 x > r₂。
- a > 0 且 f(x) < 0：解为 r₁ < x < r₂。
- 无实根且 a > 0：对所有实数 x 都有 f(x) > 0，而 f(x) < 0 无解。
- 重根 r：f(x) ≥ 0 对所有 x 成立，仅当 x = r 时取等号。
6. Polynomial inequalities and sign diagrams | 多项式不等式与符号图
For cubic and higher-degree inequalities, factorisation plus a sign diagram is the safest approach. Mark every real root on a horizontal axis, then determine the sign of the entire product in each interval by considering the sign of each linear factor. Work from right to left for speed: the rightmost interval always takes the sign of the leading coefficient.
对于三次及更高次不等式，因式分解加符号图是最稳妥的方法。把每个实根标在水平轴上，再通过判断各线性因式的符号来确定整个乘积在每个区间上的正负。从右向左判断最快捷：最右侧区间的符号总与首项系数相同。
aised to an odd power, such as (x + 4)³, crosses the axis as normal. A factor with an even multiplicity therefore produces a 'bounce' in the sign diagram.
重因式需要特别注意。像 (x - 1)² 这样的因式永远不变号，因此图像在 x = 1 处只是触碰坐标轴，并不穿过。奇次幂因式如 (x + 4)³ 则照常穿过坐标轴。因此偶重数的因式会在符号图中造成一次"反弹"。
n the crossing points, giving -4 ≤ x ≤ 2. Note x = -4 is included because of the ≤ sign, even though the factor is squared.
例：解 (x + 4)²(x - 2) ≤ 0。根为 x = -4（重根）与 x = 2。首项系数为正，乘积只在两个穿越点之间为负，得 -4 ≤ x ≤ 2。由于是 ≤ 号，x = -4 需要包含在内，尽管该因式被平方。
7. Rational Inequalities | 有理不等式
A rational inequality involves an algebraic fraction. The golden rule is never to cross-multiply, because the denominator may be negative for some values of x. Instead, move every term to one side, combine the fractions over a single common denominator, and factorise the numerator.
有理不等式中含有代数分式。首要法则是绝不交叉相乘，因为分母在某些 x 取值下可能为负。正确做法是把所有项移到一侧，通分为一个分式，再把分子分解因式。
f the denominator (where the expression is undefined, always excluded). Build a sign diagram using all of these values.
临界值来自两处：分子的零点（使表达式为零，若为 ≤ 或 ≥ 则包含）和分母的零点（使表达式无定义，永远排除）。用这些值构造符号图。
gives the expression positive for x < -1 and for x > 3, so the solution is (-∞, -1] ∪ (3, ∞). The value x = -1 is included, but x = 3 can never be included.
例：解 (x + 1)/(x - 3) ≥ 0。临界值为 x = -1 与 x = 3。检验区间可知，当 x < -1 以及 x > 3 时表达式为正，故解为 (-∞, -1] ∪ (3, ∞)。x = -1 包含在内，而 x = 3 永远不能包含。
8. Inequalities Involving Absolute Value | 含绝对值的不等式
The absolute value |x| means the distance of x from zero, so it is never negative. This geometric idea converts two standard patterns into immediate solutions: |x| < a becomes -a < x < a, while |x| > a becomes x < -a or x > a, valid for a > 0.
绝对值 |x| 表示 x 到零点的距离，因此永不为负。这一几何含义使两种标准形式可以直接写出解：|x| < a 化为 -a < x < a；|x| > a 化为 x < -a 或 x > a（其中 a > 0）。
|x| < a ⇔ -a < x < a |x| > a ⇔ x < -a or x > a
More general expressions are handled the same way. Solve |3x - 6| is [-1, 5]. Always check that the right-hand side is positive before applying the pattern; if it is negative, the inequality |expression| < negative has no solution at all.
更一般的表达式同理处理。解 |3x - 6| ≤ 9，写成 -9 ≤ 3x - 6 ≤ 9，加 6 再除以 3 得 -1 ≤ x ≤ 5，即 [-1, 5]。应用该形式前务必确认右侧为正；若右侧为负，则 |表达式| < 负数 完全无解。
When an absolute value appears on both sides, squaring is legitimate because both sides are non-negative. For instance |x - 1| < |x + 3| squares to (x - 1)² < (x + 3)², which simplifies to x > -1. Remember that |x - a| is also the distance between x and a on the number line, a fact that speeds up many IB questions.
当两边都出现绝对值时，可以平方，因为两边均非负。例如 |x - 1| < |x + 3| 平方得 (x - 1)² < (x + 3)²，化简得 x > -1。还要记住，|x - a| 也等于数轴上 x 与 a 之间的距离，这一事实能大幅加快许多 IB 题目的求解。
9. Simultaneous and Double Inequalities | 联立不等式与双重不等式
f you multiply or divide by a negative number. Here, subtracting 1 gives -3 < 3x ≤ 9, and dividing by 3 gives -1 < x ≤ 3, or (-1, 3].
像 -2 < 3x + 1 ≤ 10 这样的双重不等式可以一次完成求解。对三部分同时施行相同运算；若乘以或除以负数，两个不等号都要翻转。此处减 1 得 -3 < 3x ≤ 9，除以 3 得 -1 < x ≤ 3，即 (-1, 3]。
≤ 7 respectively, so the final answer is 3 < x ≤ 7. If the two conditions contradict each other, the system has no solution and you should say so explicitly.
对于真正意义上的联立不等式，应分别求解后取交集。解 2x - 1 > 5 与 4 - x ≥ -3，分别得 x > 3 与 x ≤ 7，故最终答案为 3 < x ≤ 7。若两个条件互相矛盾，则方程组无解，必须明确写出这一点。
- Solve each inequality independently.
- Draw both solutions on one number line.
- Keep only the overlap (intersection) unless the question says 'or'.
- State 'no solution' if the intervals do not overlap.
- 逐个独立求解每个不等式。
- 把两个解画在同一条数轴上。
- 除非题目为"或"，否则只保留重叠部分（交集）。
- 若区间没有重叠，写明"无解"。
10. Graphical Interpretation and Linear Programming | 图像解释与线性规划
lid line for ≤ or ≥, then shade the required half-plane. Choosing a test point such as (0, 0) is the quickest way to decide which side to shade.
二元不等式描述坐标平面上的一个区域。严格不等式用虚线画边界，≤ 或 ≥ 用实线画边界，然后给所需半平面涂色。选取检验点如 (0, 0) 是判断涂哪一侧的最快方法。
In linear programming problems, several inequalities form the constraints and their overlap is called the feasible region. The optimal value of a linear objective function always occurs at a vertex of this region, so you evaluate the objective at each corner and compare
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