📚 Inequalities Key Points Review | 不等式考点精讲
Inequalities form a fundamental part of CIE A-Level Mathematics, requiring both algebraic skill and careful logical reasoning. From linear to quadratic, absolute value and rational inequalities, mastering these topics builds a strong foundation for calculus and beyond. This article reviews key concepts, standard solution methods, and common pitfalls, helping you approach any inequality problem with confidence.
不等式是CIE A-Level数学的基础内容，需要代数技巧和严密的逻辑推理。从线性不等式到二次不等式、绝对值不等式和分式不等式，掌握这些知识能为微积分及更高层次的学习打下坚实基础。本文梳理核心概念、标准解法及常见易错点，帮助大家自信应对各类不等式问题。
1. Introduction to Inequalities | 不等式简介
An inequality compares two expressions using the symbols < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). Unlike equations, the solution is typically a range of values, often represented on a number line or in interval notation.
不等式使用符号 <（小于）、>（大于）、≤（小于等于）和 ≥（大于等于）来比较两个表达式。与方程不同，不等式的解通常是一个数值范围，常用数轴或区间表示法来表示。
The solution set of an inequality is the collection of all real numbers that make the inequality true. For example, x > 3 means any number strictly greater than 3 forms the solution, written as (3, ∞) in interval notation.
不等式的解集是所有使不等式成立的实数的集合。例如，x > 3 表示任何严格大于 3 的数都是解，用区间表示为 (3, ∞)。
2. Basic Properties of Inequalities | 不等式的基本性质
Understanding the algebraic rules for inequalities is crucial. If a > b, then a + c > b + c for any real number c. Adding or subtracting the same quantity from both sides does not affect the inequality sign.
理解不等式的代数运算法则至关重要。若 a > b，则对于任意实数 c 有 a + c > b + c。两边同加或同减一个数，不等号方向不变。
If both sides are multiplied or divided by a positive constant, the inequality sign remains unchanged. However, multiplying or dividing by a negative number reverses the inequality: if a > b and c < 0, then ac < bc.
两边同乘或同除以一个正数，不等号方向不变；但若乘或除以一个负数，不等号必须反向：若 a > b 且 c < 0，则 ac < bc。
Be careful when squaring both sides or taking reciprocals. Squaring preserves the inequality only if both sides are non‑negative. For reciprocals, if a > b > 0, then 1/a < 1/b. If the signs differ, the inequality may not hold in the same direction.
两边同时平方或取倒数时需要谨慎。平方仅在两边均为非负数时才保持不等号方向。对于倒数，若 a > b > 0，则 1/a < 1/b。若符号不同，不等号方向可能不成立。
3. Solving Linear Inequalities | 解一元一次不等式
Linear inequalities are solved similarly to linear equations, with the key difference being the reversal of the inequality sign when multiplying or dividing by a negative number. For instance, solve 3x − 5 > 7. Add 5 to both sides to obtain 3x > 12, then divide by 3 to get x > 4.
解线性不等式与解线性方程类似，关键区别在于当乘或除以负数时不等号要反向。例如，解 3x − 5 > 7：两边加 5 得 3x > 12，再除以 3 得到 x > 4。
When the variable appears on both sides, collect like terms first. For −2x + 1 ≤ 3x − 14, add 2x to both sides: 1 ≤ 5x − 14. Then add 14: 15 ≤ 5x, so x ≥ 3. Notice that the inequality sign did not reverse because we divided by a positive 5.
当变量出现在两边时，先合并同类项。对于 −2x + 1 ≤ 3x − 14，两边加 2x 得 1 ≤ 5x − 14，然后加 14 得 15 ≤ 5x，故 x ≥ 3。注意因为除以正数 5，不等号方向没有改变。
Always express the solution as a simplified inequality, and where required, on a number line with an open circle for strict inequalities and a closed circle for inclusive inequalities.
解要表示成最简不等式形式，如果题目要求，还要在数轴上表示：严格不等式用空心圆圈，包含等号用实心圆圈。
4. Solving Quadratic Inequalities | 解一元二次不等式
Quadratic inequalities involve expressions of the form ax² + bx + c > 0, < 0, ≥ 0 or ≤ 0. The most reliable method begins by finding the critical values where the quadratic equals zero. Factorise, if possible, or use the quadratic formula.
二次不等式涉及形如 ax² + bx + c > 0、< 0、≥ 0 或 ≤ 0 的表达式。最可靠的方法是先求出二次式等于零时的关键值（临界点）。如果可以，进行因式分解，或者使用求根公式。
Consider x² − 5x + 6 > 0. Factorising gives (x − 2)(x − 3) > 0. The critical values are x = 2 and x = 3. These divide the real number line into three intervals: (−∞, 2), (2, 3) and (3, ∞). Test a point from each interval to determine the sign of the product.
以 x² − 5x + 6 > 0 为例。因式分解得 (x − 2)(x − 3) > 0。关键值为 x = 2 和 x = 3。它们将数轴分成三个区间：(−∞, 2)、(2, 3) 和 (3, ∞)。从每个区间选一个测试点，判断乘积的符号。
For x < 2, both factors are negative, so the product is positive. For 2 < x < 3, one factor is positive and the other negative, so the product is negative. For x > 3, both are positive. The inequality requires > 0, so the solution is x < 2 or x > 3, which in interval notation is (−∞, 2) ∪ (3, ∞).
当 x < 2 时，两个因子均为负，乘积为正；当 2 < x < 3 时，一个为正一个为负，乘积为负；当 x > 3 时，两者皆正。不等式要求 > 0，所以解为 x < 2 或 x > 3，用区间表示即 (−∞, 2) ∪ (3, ∞)。
When the quadratic has no real roots (discriminant < 0), it is either always positive or always negative depending on the leading coefficient a. For example, x² + x + 1 > 0 is true for all real x because a = 1 > 0 and the discriminant is −3 < 0.
若二次式没有实根（判别式 < 0），则根据首项系数 a 的正负，它要么恒正要么恒负。例如，x² + x + 1 > 0 对所有实数 x 均成立，因为 a = 1 > 0 且判别式为 −3 < 0。
5. Graphical Interpretation of Quadratic Inequalities | 二次不等式的图像解释
Sketching the graph of y = ax² + bx + c can provide a quick visual solution. The parabola crosses the x‑axis at the roots, and the inequality y > 0 corresponds to the portions of the graph above the x‑axis.
画出 y = ax² + bx + c 的图像可提供快速的直观解法。抛物线与 x 轴相交于对应方程的根，而不等式 y > 0 对应图像位于 x 轴上方的部分。
For a positive leading coefficient a > 0, the parabola opens upwards. The expression is positive outside the interval between the roots and negative inside. For a < 0, it opens downwards, and the positive region lies between the roots.
当首项系数 a > 0 时，抛物线开口向上，二次式在两根区间之外为正，两根之间为负。当 a < 0 时，开口向下，为正的区域在两根之间。
This geometric viewpoint helps check algebraic answers. If the inequality is x² − 4x + 3 ≤ 0, the roots are 1 and 3. With a = 1 > 0, the graph is below the x‑axis between x = 1 and x = 3, so the solution is 1 ≤ x ≤ 3.
这种几何视角有助于检验代数答案。若不等式为 x² − 4x + 3 ≤ 0，其根为 1 和 3。由于 a = 1 > 0，图像在 x = 1 到 x = 3 之间位于 x 轴下方，所以解为 1 ≤ x ≤ 3。
6. Inequalities Involving Absolute Values | 绝对值不等式
Absolute value inequalities often appear in the form |x| < a or |x| > a. Geometrically, |x| < a (with a > 0) means the distance from x to 0 is less than a, which gives −a < x < a. Similarly, |x| > a means the distance is greater than a, yielding x < −a or x > a.
绝对值不等式常以 |x| < a 或 |x| > a 的形式出现。几何上讲，|x| < a（a > 0）表示 x 到 0 的距离小于 a，得到 −a < x < a。类似地，|x| > a 表示距离大于 a，得出 x < −a 或 x > a。
For more complex expressions, treat the inside as a compound inequality. Solve |2x − 3| ≤ 5 by writing −5 ≤ 2x − 3 ≤ 5. Add 3 throughout: −2 ≤ 2x ≤ 8, then divide by 2: −1 ≤ x ≤ 4. The solution is the closed interval [−1, 4].
对于更复杂的表达式，将内部当作复合不等式处理。解 |2x − 3| ≤ 5 时，可写成 −5 ≤ 2x − 3 ≤ 5。整体加 3 得 −2 ≤ 2x ≤ 8，再除以 2 得 −1 ≤ x ≤ 4。解为闭区间 [−1, 4]。
When the inequality involves an expression on both sides, such as |x + 1| > 2x, careful case analysis based on the sign of the expression inside the absolute value is required. Alternatively, squaring both sides can be used if we ensure both sides are non‑negative, but always check for extraneous solutions.
当不等式两边都有表达式，例如 |x + 1| > 2x，需根据绝对值内部表达式的符号进行仔细的分类讨论。另一种方法是两边平方，但需确保两边非负，并验证是否有增根。
7. Rational Inequalities | 分式不等式
Rational inequalities involve fractions with polynomials, such as (x − 1)/(x + 2) > 0. One method is to find critical points where the numerator or denominator is zero, then test intervals. Here critical points are x = 1 and x = −2 (note the denominator cannot be zero).
分式不等式包含多项式分式，例如 (x − 1)/(x + 2) > 0。一种方法是找出分子或分母为零的关键点，然后划分区间进行检验。该例的关键点为 x = 1 和 x = −2（注意分母不能为零）。
The sign of the fraction depends on the signs of numerator and denominator. For x < −2, both are negative, giving a positive result. For −2 < x < 1, numerator negative and denominator positive, result is negative. For x > 1, both positive, result positive. The solution to > 0 is x < −2 or x > 1.
分式的符号取决于分子和分母的符号。当 x < −2 时，两者皆负，结果为正；当 −2 < x < 1 时，分子负、分母正，结果为负；当 x > 1 时，两者皆正，结果为正。不等式 > 0 的解为 x < −2 或 x > 1。
A safer algebraic approach for any rational inequality is to bring all terms to one side, combine into a single fraction, and then analyse the sign using a sign table. For (2x + 1)/(x − 3) ≤ 1, rewrite as (2x + 1)/(x − 3) − 1 ≤ 0, simplify to (x + 4)/(x − 3) ≤ 0. Critical points are x = −4 and x = 3. Testing intervals yields −4 ≤ x < 3, remembering to exclude x = 3 where the denominator is zero.
对于任何分式不等式，更稳妥的代数方法是把所有项移到一边，合并成一个分式，然后利用符号表分析符号变化。例如解 (2x + 1)/(x − 3) ≤ 1，改写为 (2x + 1)/(x − 3) − 1 ≤ 0，化简得 (x + 4)/(x − 3) ≤ 0。关键点为 x = −4 和 x = 3。检验区间得到 −4 ≤ x < 3，注意排除使分母为零的 x = 3。
8. Systems of Inequalities | 不等式组
A system of inequalities requires finding all values that satisfy every inequality simultaneously. The solution set is the intersection of the individual solution sets. For a linear system like x > 2 and x ≤ 5, the solution is 2 < x ≤ 5, or (2, 5].
不等式组要求找出同时满足所有不等式的数值范围，其解集是各个不等式解集的交集。对于诸如 x > 2 且 x ≤ 5 的线性不等式组，解为 2 < x ≤ 5，即 (2, 5]。
More complex systems may involve quadratic and linear inequalities together. For instance, x² − 4 > 0 and x + 1 > 0. The first gives x < −2 or x > 2. The second gives x > −1. Intersecting them, only x > 2 survives. The solution is (2, ∞).
更复杂的不等式组可能同时包含二次和线性不等式。例如 x² − 4 > 0 且 x + 1 > 0。第一个解得 x < −2 或 x > 2，第二个解得 x > −1。取交集后，只有 x > 2 保留下来。解为 (2, ∞)。
When solving systems graphically, shade the regions that satisfy each inequality on a number line (for one variable) or on a coordinate plane (for two‑variable inequalities). The overlapping region represents the final solution set.
用图像法解不等式组时，在数轴（一元）或坐标平面（二元）上画出每个不等式的区域，重叠部分即为最终解集。对于一元情形，在数轴上标示各不等式的解区间，然后取公共部分。
9. Interval Notation and Number Lines | 区间表示与数轴
Interval notation is the standard way to express continuous sets of real numbers. Parentheses ( ) indicate open endpoints (strict inequality), while square brackets [ ] indicate closed endpoints (inclusive). The union symbol ∪ joins disjoint intervals.
区间表示法是表达实数连续集合的标准方式。圆括号 ( ) 表示开区间（严格不等式），方括号 [ ] 表示闭区间（包含端点）。并集符号 ∪ 用于连接不相交的区间。
Common intervals include (a, b) for a < x < b, [a, b] for a ≤ x ≤ b, (a, ∞) for x > a, and (−∞, b] for x ≤ b. Infinity symbols always take parentheses because infinity is not a number.
常见区间如 (a, b) 表示 a < x < b，[a, b] 表示 a ≤ x ≤ b，(a, ∞) 表示 x > a，(−∞, b] 表示 x ≤ b。无穷符号总是配圆括号，因为无穷不是具体数值。
On a number line, rigorous representation uses an open circle for excluded values and a filled circle for included values. Arrows indicate intervals extending to infinity. This visual aid helps avoid mistakes when writing the final answer.
在数轴上，用空心圆表示不包含的点，实心圆表示包含的点，用箭头指向无穷远。这种视觉辅助工具有助于避免在书写最终答案时出错。
10. Common Mistakes and Tips | 常见错误与技巧
One frequent error is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always check the sign of the coefficient of x before dividing, and reverse if necessary.
一个常见错误是在乘或除以负数时忘记将不等号反向。在除以 x 的系数之前，务必检查该系数的符号，若为负则需反向。
In quadratic inequalities, writing the solution as a single compound inequality is a trap. The answer to x² > 4 is not −2 < x < 2; it should be x < −2 or x > 2. Similarly, note that x² < 4 gives −2 < x < 2.
在二次不等式中，误将解写成一个单一的复合不等式是一个陷阱。x² > 4 的解并非 −2 < x < 2，而应是 x < −2 或 x > 2。同样，注意 x² < 4 的解才是 −2 < x < 2。
For rational inequalities, never cross‑multiply without considering the sign of the denominator. Instead, bring all terms to one side and combine into a single fraction. Also, always explicitly state the restriction that denominators cannot be zero in the final answer.
对于分式不等式，切勿在不考虑分母符号的情况下进行交叉相乘。正确做法是将所有项移到一边并合并成一个分式。此外，最终答案中要明确指出分母不为零的限制。
When dealing with absolute values, remember that |x| < a translates to a single interval, while |x| > a gives two separate regions. Mistaking one for the other can cost marks. Draw a quick number line to confirm the logical structure.
处理绝对值时，记住 |x| < a 转化为一个单一的区间，而 |x| > a 给出两个分离的区域。混淆二者可能导致失分。快速画一条数轴以确认逻辑结构。
Finally, always double‑check boundary values by substituting them back into the original inequality. Pay special attention to whether endpoints are included or excluded, as this detail often distinguishes between full marks and a partially correct solution.
最后，务必将边界值代回原不等式进行检验。特别注意端点是否包含在内，这一点往往决定了答案是完全正确还是部分正确。
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