📚 G - Inequalities (inequations) | 不等式
In mathematics, an inequality compares two expressions using symbols such as <, >, ≤, ≥. Unlike equations, inequalities define a range of possible values rather than a single solution. In the IB Mathematics curriculum, inequalities appear across algebra, functions, and calculus, requiring students to solve, graph, and interpret both single and compound inequalities, quadratic inequalities, absolute value inequalities, and rational inequalities. Mastery of inequalities is essential for understanding domains, optimisation, and modelling real-world constraints.
在数学中，不等式使用 <、>、≤、≥ 等符号比较两个表达式。与方程不同，不等式定义的是一系列可能的值，而非单一解。在IB数学课程中，不等式贯穿代数、函数和微积分，要求学生求解、绘制图像并解释一次不等式、复合不等式、二次不等式、绝对值不等式和分式不等式。掌握不等式对于理解定义域、最优化和实际建模至关重要。
1. Introduction to Inequality Symbols and Basic Principles | 不等式符号与基本性质
An inequality expresses a relationship of order: a < b means a is less than b; a > b means a is greater than b; a ≤ b means a is less than or equal to b; and a ≥ b means a is greater than or equal to b. When solving inequalities, addition and subtraction preserve the inequality direction. Multiplication or division by a positive number also preserves the direction, but multiplying or dividing by a negative number reverses the inequality sign. For example, if x > 3, then −2x < −6.
不等式表达顺序关系：a < b 表示 a 小于 b；a > b 表示 a 大于 b；a ≤ b 表示小于或等于；a ≥ b 表示大于或等于。解不等式时，加法和减法保持不等号方向不变。乘以或除以正数也保持方向，但乘以或除以负数会反转不等号。例如，若 x > 3，则 −2x < −6。
2. Notation and Number Lines | 记法与数轴表示
Solution sets for inequalities can be expressed using interval notation, set-builder notation, and graphically on a number line. An open circle indicates a strict inequality (< or >), while a closed (filled) circle indicates inclusivity (≤ or ≥). For instance, x > 2 is represented by an open circle at 2 with shading to the right, and in interval notation as (2, ∞). Set-builder notation writes {x ∈ ℜ : x > 2}. The symbol ℜ denotes the set of real numbers.
不等式的解集可用区间记号、集合构造器记号和数轴上的图形表示。空心圆表示严格不等式（< 或 >），实心圆表示包含等号（≤ 或 ≥）。例如，x > 2 在数轴上用2处的空心圆和向右的阴影表示，区间记号为 (2, ∞)。集合构造器记号为 {x ∈ ℜ : x > 2}。符号 ℜ 表示实数集。
3. Solving Linear Inequalities | 解一元一次不等式
A linear inequality involves a variable raised only to the first power. Solving follows the same steps as solving a linear equation with the critical condition of sign reversal when multiplying/dividing by a negative number. Example: Solve 3x − 5 ≤ 7. Adding 5 gives 3x ≤ 12; dividing by 3 (positive) yields x ≤ 4. The solution set is (−∞, 4] or {x ∈ ℜ : x ≤ 4}.
一元一次不等式仅含一次变量。其解法与线性方程相同，但必须注意乘以或除以负数时反转不等号。例如：解 3x − 5 ≤ 7。两边加5得 3x ≤ 12；除以正数3，得 x ≤ 4。解集为 (−∞, 4] 或 {x ∈ ℜ : x ≤ 4}。
For inequalities with variables on both sides, collect all variable terms on one side. Solve 5 − 2x > x + 7. Subtract x and 5: −3x > 2. Divide by −3 and reverse: x < −2/3.
对于变量在两边的不等式，将所有变量项移至一边。解 5 − 2x > x + 7。移项得 −3x > 2。除以 −3 并反转不等号：x < −2/3。
4. Compound (Double) Inequalities | 复合（双重）不等式
A compound inequality expresses two constraints simultaneously, e.g. −3 < 2x + 1 ≤ 5. This can be split into two separate inequalities or solved as a single three-part inequality by performing operations on all three parts. Solve: −3 < 2x + 1 ≤ 5. Subtract 1 from all parts: −4 < 2x ≤ 4. Divide by 2: −2 < x ≤ 2. The solution is the intersection of the two inequalities.
复合不等式同时表达两个条件，例如 −3 < 2x + 1 ≤ 5。可拆分成两个独立不等式，或在整个三元不等式上执行运算。解法：减1：−4 < 2x ≤ 4。除以2：−2 < x ≤ 2。解集为两个不等式的交集。
5. Quadratic Inequalities | 二次不等式
A quadratic inequality has the form ax² + bx + c > 0 (or with <, ≤, ≥). The preferred method is to move all terms to one side, factorise the quadratic if possible, find critical values (roots), and use a sign diagram to determine intervals where the expression is positive or negative. Alternatively, you can sketch the graph of the quadratic function and identify where it lies above or below the x-axis.
二次不等式形如 ax² + bx + c > 0（或 <、≤、≥）。推荐方法是将所有项移到一边，若可能则因式分解，找出临界值（根），并用符号表确定表达式为正或负的区间。也可绘制二次函数图像，直接找出位于 x 轴上方或下方的部分。
Example: Solve x² − 5x + 6 < 0. Factorise: (x − 2)(x − 3) < 0. Critical values: x = 2 and x = 3. Sign test: for x < 2, both factors negative → product positive; for 2 < x < 3, (x − 2) positive, (x − 3) negative → product negative; for x > 3, both positive → product positive. The inequality is satisfied when product < 0, so solution is 2 < x < 3, or (2, 3).
例题：解 x² − 5x + 6 < 0。因式分解：(x − 2)(x − 3) < 0。临界值：x = 2 和 x = 3。符号检验：x < 2 时两因式均负→积为正；2 < x < 3 时 (x − 2) 正、(x − 3) 负→积为负；x > 3 时两因式均正→积为正。不等式要求积 < 0，因此解为 2 < x < 3，或 (2, 3)。
6. Graphical Approach to Quadratic Inequalities | 二次不等式的图像解法
Consider the quadratic function f(x) = ax² + bx + c. Its graph is a parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward. The inequality f(x) > 0 corresponds to the parts of the graph above the x-axis, and f(x) < 0 to parts below. Find the x-intercepts (roots) by solving ax² + bx + c sual approach is especially useful when the quadratic does not factorise neatly, requiring the quadratic formula or discriminant analysis to locate roots.
考虑二次函数 f(x) = ax² + bx + c。其图像为抛物线。若 a > 0，抛物线开口向上；若 a < 0，则开口向下。不等式 f(x) > 0 对应图像位于 x 轴上方的部分，f(x) < 0 对应 x 轴下方的部分。通过求解 ax² + bx + c = 0 找到 x 截距（根），然后绘制抛物线。当二次式不能整洁因式分解时，需用求根公式或判别式分析确定根的位置，这种直观方法尤为有效。
Example: Solve −x² + 4x − 3 ≥ 0. Multiply by −1 (reverse sign): x² − 4x + 3 ≤ 0. Roots: x = 1, 3. Parabola opens upward. The graph is below the x-axis between the roots. Solution: 1 ≤ x ≤ 3, or [1, 3].
例题：解 −x² + 4x − 3 ≥ 0。乘以 −1（反转不等号）：x² − 4x + 3 ≤ 0。根：x = 1, 3。抛物线开口向上。在两根之间图像位于 x 轴下方。解集：1 ≤ x ≤ 3，或 [1, 3]。
7. Inequalities Involving Absolute Values | 含绝对值的不等式
The absolute value inequality |x| < k (with k > 0) means the distance from x to 0 is less than k, which is equivalent to −k < x < k. The inequality |x| > k means x < −k or x > k. For linear expressions: to solve |2x − 3| ≤ 5, write −5 ≤ 2x − 3 ≤ 5. Solve the compound inequality: add 3: −2 ≤ 2x ≤ 8; divide by 2: −1 ≤ x ≤ 4.
绝对值不等式 |x| < k（k > 0）表示 x 到 0 的距离小于 k，等价于 −k < x < k。不等式 |x| > k 表示 x < −k 或 x > k。对于一次表达式：解 |2x − 3| ≤ 5，写成 −5 ≤ 2x − 3 ≤ 5。求解复合不等式：加3得 −2 ≤ 2x ≤ 8；除以2得 −1 ≤ x ≤ 4。
If the absolute value is compared to a negative number, special care is needed. For example, |x + 1| < −2 has no solution because absolute values are always non-negative. Conversely, |x + 1| > −2 is true for all real x.
若绝对值与负数比较，需特别处理。例如，|x + 1| < −2 无解，因为绝对值始终非负。反知，|x + 1| > −2 对所有实数 x 恒成立。
8. Rational Inequalities | 分式不等式
Rational inequalities involve fractions with polynomials, e.g. (x − 1)/(x a sign table over intervals separated by these values. The denominator must never be zero, so those points are excluded from the solution set.
分式不等式涉及多项式分式，例如 (x − 1)/(x + 2) > 0。关键是避免直接乘以分母，因为其正负未知。应将所有项移至一边化为单个分式与零比较，找出临界值（分子=0或分母=0），并以这些值为界划分区间，制作符号表。分母不可为零，因此那些点必须从解集中排除。
Example: Solve (x − 3)/(x + 1) ≤ 2. Move 2: (x − 3)/(x + 1) − 2 ≤ 0. Combine: (x − 3 − 2x − 2)/(x + 1) ≤ 0 → (−x − 5)/(x + 1) ≤ 0. Critical values: x = −5 (numerator zero), x = −1 (denominator zero). Sign test for intervals: x < −5, −5 < x < −1, x > −1. The expression is ≤ 0 when x ∈ [−5, −1). Note −1 is excluded.
例题：解 (x − 3)/(x + 1) ≤ 2。移项通分：(x − 3 − 2x − 2)/(x + 1) ≤ 0 → (−x − 5)/(x + 1) ≤ 0。临界值：x = −5（分子为零），x = −1（分母为零）。区间符号检验：x < −5，−5 < x < −1，x > −1。表达式 ≤ 0 的解为 x ∈ [−5, −1)。注意 −1 被排除。
9. Systems of Linear Inequalities | 线性不等式组
A system of linear inequalities consists of two or more linear inequalities that must be satisfied simultaneously. The solution region is the intersection of the half-planes defined by each inequality. When graphing, first draw the boundary line for each inequality (dashed for strict < or >, solid for ≤ or ≥), then shade the appropriate side. The feasible region is the overlap of all shaded areas. Vertices of this region often represent optimal solutions in linear programming problems.
线性不等式组包含两个或更多必须同时满足的一次不等式。解区域是每个不等式所定义半平面的交集。作图时，先绘出每个不等式的边界线（严格不等式 < 或 > 用虚线，包含等号的 ≤ 或 ≥ 用实线），然后涂染符合要求的一侧。可行区域是所有阴影区域的重叠部分。该区域的顶点通常代表线性规划问题中的最优解。
Example: Graph the system y ≤ 2x + 1, y > −x + 2, x ≥ 0, y ≥ 0. The solution is a polygonal region in the first quadrant bounded by the lines. Testing a point like (1,1) confirms whether it lies within the region.
例题：绘制不等式组 y ≤ 2x + 1，y > −x + 2，x ≥ 0，y ≥ 0 的图像。解为第一象限内由直线围成的多边形区域。通过检验点 (1,1) 可确认其是否位于该区域之内。
10. Applications of Inequalities: Word Problems and Modelling | 不等式的应用：文字题与建模
Inequalities frequently model constraints in optimisation problems, such as budget restrictions, production limits, or geometric conditions. Translate the verbal description into algebraic inequalities. For instance, a company producing chairs and tables: wood and labour constraints give inequalities like 2x + 3y ≤ 60 (wood) and 4x + 2y ≤ 80 (labour), where x and y are quantities. The objective is to maximise profit, which is evaluated at vertices of the feasible region.
不等式常用于最优化问题中建模约束条件，如预算限制、生产上限或几何条件。将文字描述转化为代数不等式。例如，某公司生产椅子和桌子：木材和劳动力约束给出不等式 2x + 3y ≤ 60（木材）和 4x + 2y ≤ 80（劳动力），其中 x 和 y 是数量。目标是最大化利润，通常在可行区域的顶点处评估。
Non-linear constraints may appear in areas such as geometry (area > certain value) or physics (projectile motion above a given height). For example, the height of a ball h(t) = −5t² + 20t + 1. When is h > 10? Solve −5t² + 20t + 1 > 10, leading to a quadratic inequality.
非线性约束可出现在几何（面积 > 某值）或物理（抛体运动高于给定高度）等领域。例如，球的高度 h(t) = −5t² + 20t + 1。何时 h > 10？求解 −5t² + 20t + 1 > 10，化为二次不等式。
11. Common Mistakes and Strategies | 常见错误与应对策略
Many errors arise from forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always pay attention to the direction change, especially in compound inequalities. Another common mistake is squaring both sides without considering sign restrictions. Always test critical points and check endpoints. For rational inequalities, multiplying by the denominator without accounting for its sign is a frequent source of incorrect solutions. Use sign tables or graphical methods to avoid this pitfall.
许多错误源于乘以或除以负数时忘记反转不等号。务必注意方向变化，特别是在复合不等式中。另一个常见错误是不考虑符号限制就对两边平方。始终保持临界点的检验并检查端点。对于分式不等式，未考虑分母正负就直接乘分母是错误解法的常见根源。使用符号表或图像法可避免这一陷阱。
Another pitfall: writing solutions that do not satisfy the original inequality due to invalid transformations, such as cancelling terms incorrectly. Always check your solution by substituting test values from each interval. Be precise with open and closed dots on number lines and in interval notation.
另一陷阱：因无效变形（如错误约分）而写出不满足原不等式的解。始终通过代入各区间检验值来验证解答。在数轴和区间记号中精确使用空心、实心点。
12. IB Exam Tips for Inequalities | IB考试中不等式的应试技巧
Examiners often mark the reasoning process, not just the final answer. When solving inequalities, show steps clearly: rearrangement, critical value identification, sign table or sketch, and final answer in correct notation. For questions asking to "hence solve" an inequality, look for prior parts of the question that may provide factorisations or roots. When dealing with absolute values, explicitly state the two cases. Always check the domain when working with square roots, logarithms, or denominators. In Paper 2, you may use a GDC to graph and find intersection points, but you must still explain the solution set in algebraic notation.
评分官通常注重推理过程，而非仅最终答案。解不等式时，清晰展示步骤：移项、确定临界值、符号表或草图、并以正确记号给出最终答案。对于要求“由此解不等式”的题目，注意利用题目前面部分可能已给出的因式分解或根。处理绝对值时，明确列出两种情况。涉及平方根、对数或分母时，务必检查定义域。在卷二考试中，可使用图形计算器绘制并寻找交点，但仍须用代数记号解释解集。
Finally, practise with a variety of inequality types to build fluency. Recognise that some inequalities can be solved by squaring (if both sides are non-negative) or by using substitution for repeated expressions. Always stay aware of the fundamental principle: the inequality symbol's direction is preserved under addition, subtraction, multiplication by a positive constant, but reversed when multiplied or divided by a negative constant.
最后，通过练习多种类型的不等式来培养熟练度。要明白某些不等式可通过平方（若两边非负）或用代换处理重复表达式来求解。始终保持对基本原理的认知：不等号的方向在加法、减法和乘以正常数时保持不变，但在乘以或除以负常数时反转。
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