IB Math: Inequalities – Exam Focus | IB 数学：不等式 考点精讲

📚 IB Math: Inequalities – Exam Focus | IB 数学：不等式 考点精讲

Inequalities are a core component of the IB Mathematics curriculum, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI) courses. They require a blend of algebraic manipulation, graphical understanding, and logical reasoning. From simple linear inequalities to quadratic, rational, and absolute value forms, mastering the techniques and common pitfalls is essential for success in Paper 1 and Paper 2 exams. This guide breaks down the key concepts you must know, with strategies tailored for IB-style questions.

不等式是 IB 数学课程的核心内容，无论在分析与方法（AA）还是应用与解释（AI）课程中都频繁出现。它需要代数运算、图像理解和逻辑推理的综合运用。从简单的线性不等式到二次、有理和绝对值不等式，掌握解题技巧并避开常见陷阱对 Paper 1 和 Paper 2 的考试至关重要。本指南将逐一解析你必须掌握的关键知识点，并提供针对 IB 风格考题的应考策略。

1. Understanding Inequality Symbols | 理解不等号

Inequalities use the symbols > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). A crucial rule is that multiplying or dividing both sides by a negative number reverses the inequality direction. For example, -2x < 6 becomes x > -3. Always pay attention to whether the boundary point is included (using closed dot or square bracket) or excluded (open dot or round bracket) when expressing solution sets.

不等号包括 >（大于）、<（小于）、≥（大于等于）和 ≤（小于等于）。一条关键规则是：当不等式两边同乘或同除以一个负数时，不等号方向必须改变。例如 -2x < 6 变为 x > -3。在表示解集时，务必注意边界点是否包含在内（用实心点或方括号）还是排除在外（用空心点或圆括号）。

2. Solving Linear Inequalities | 解线性不等式

Linear inequalities such as 3x – 5 > 7 are solved by isolating the variable, exactly like linear equations, but with the extra care of the sign reversal rule. The solution is often expressed in set notation, for instance {x ∈ ℝ : x > 4}, or in interval notation (4, ∞). On a number line, this is shown with an open circle at 4 and an arrow to the right. IB questions may combine two linear inequalities into a compound form, which we will cover later.

像 3x – 5 > 7 这样的线性不等式，解法与线性方程一样，只需将变量分离，但额外要注意变号规则。解集通常用集合符号表示，例如 {x ∈ ℝ : x > 4}，或用区间符号 (4, ∞) 表示。在数轴上，用空心圆标出 4 并向右画箭头。IB 考题常会将两个线性不等式结合成复合不等式，后文会详细说明。

3. Solving Quadratic Inequalities | 解二次不等式

To solve a quadratic inequality like x² – 4x + 3 < 0, first find the roots by solving x² – 4x + 3 = 0. Factorisation gives (x – 1)(x – 3) = 0, so roots are x = 1 and x = 3. Since the coefficient of x² is positive, the parabola opens upward. The expression is negative between the roots, so the solution is 1 < x < 3, or (1, 3) in interval notation. If the inequality had been ≥ 0, the solution would be x ≤ 1 or x ≥ 3, including the boundary points.

解二次不等式如 x² – 4x + 3 < 0，首先解对应方程 x² – 4x + 3 = 0 求根。因式分解得 (x – 1)(x – 3) = 0，根为 x = 1 和 x = 3。由于 x² 系数为正，抛物线开口向上。该二次表达式在两根之间为负，因此解为 1 < x < 3，区间记作 (1, 3)。若不等式为 ≥ 0，则解为 x ≤ 1 或 x ≥ 3，边界点包含在内。

4. The Sign Chart Method | 符号表法

When a quadratic does not factorise easily or when dealing with rational inequalities, a sign chart (sign diagram) is invaluable. Mark the real roots on a number line, splitting it into intervals. Test one value from each interval in the factored or original inequality to determine the sign. For example, to solve (x + 2)(x – 4) > 0, the roots are -2 and 4. Testing x = -3 gives (+), x = 0 gives (-), and x = 5 gives (+). The solution is x < -2 or x > 4.

当二次式不易因式分解或处理分式不等式时，符号表（符号图）是非常有用的工具。在数轴上标出所有实根，将数轴分割成若干区间。在每个区间取一个测试值代入原不等式或其因式形式，确定各区间的符号。例如解不等式 (x + 2)(x – 4) > 0，根为 -2 和 4。测试 x = -3 得正号，x = 0 得负号，x = 5 得正号。因此解为 x < -2 或 x > 4。

5. Rational Inequalities | 有理不等式

Rational inequalities involve fractions with polynomials, such as (x+2)/(x-3) ≥ 0. The critical points are where the numerator is zero (x = -2) and where the denominator is zero (x = 3). Note that x = 3 is always excluded from the domain. Make a sign chart with these points. Testing intervals yields: for x < -2, fraction is (+); between -2 and 3, (-); and for x > 3, (+). Since we want ≥ 0 and x ≠ 3, the solution is x ≤ -2 or x > 3. Always pay attention to whether the fraction can be zero (numerator zero gives equality) and never include denominator zeros.

有理不等式涉及含多项式的分式，例如 (x+2)/(x-3) ≥ 0。临界点是分子为零（x = -2）和分母为零（x = 3）的点。注意 x = 3 必须从定义域中剔除。以这些点为界制作符号表。测试各区间得：x < -2 时分式为正；-2 < x < 3 时为负；x > 3 时为正。由于要求 ≥ 0 且 x ≠ 3，解为 x ≤ -2 或 x > 3。务必注意分式可以等于零（分子为零），但分母为零的点绝对不可包含。

6. Absolute Value Inequalities | 绝对值不等式

Absolute value inequalities come in two main forms. For |f(x)| < a (with a > 0), the inequality is equivalent to -a < f(x) < a. For |f(x)| > a, it splits into f(x) < -a or f(x) > a. A typical IB problem might ask to solve |2x – 1| ≥ 5. This leads to 2x – 1 ≤ -5 or 2x – 1 ≥ 5. Solving gives x ≤ -2 or x ≥ 3. When the inequality involves a variable inside and outside the absolute value, graphical interpretations or squaring both sides (if both sides are non-negative) can be used, but always check for extraneous solutions.

绝对值不等式主要有两种形式。对于 |f(x)| < a（a > 0），等价于 -a < f(x) < a。对于 |f(x)| > a，则拆分为 f(x) < -a 或 f(x) > a。一道典型的 IB 题可能要求解 |2x – 1| ≥ 5。这转化为 2x – 1 ≤ -5 或 2x – 1 ≥ 5。解得 x ≤ -2 或 x ≥ 3。当不等式内外部都含有变量时，可通过图像解释或两侧平方（确保两边非负）来求解，但务必检查是否存在增根。

7. Graphical Approach | 图像法

IB exam questions frequently ask you to relate inequalities to graphs. For instance, solving f(x) > g(x) involves finding the x-values where the graph of y = f(x) is above the graph of y = g(x). This is often applied to solving inequalities like x² < 2x + 3 by plotting y = x² and y = 2x + 3, then identifying the interval where the parabola is below the line. Transferring between algebraic and graphical representations is a key skill, and you may be asked to sketch a function and shade the region representing the inequality.

IB 考题常会要求你将不等式与图像联系起来。例如，解 f(x) > g(x) 就意味着找出使得 y = f(x) 图像位于 y = g(x) 之上的 x 值。这常用于解如 x² < 2x + 3 的不等式，可通过绘制 y = x² 和 y = 2x + 3，然后确定抛物线位于直线下方的区间。在代数表达与图像表达之间转换是一项关键技能，你可能会被要求绘制函数草图并涂鸦表示不等式的区域。

8. Compound Inequalities | 复合不等式

Compound inequalities involve a combination of two or more simple inequalities using ‘and’ (intersection) or ‘or’ (union). A typical example is 2 < 3x + 1 ≤ 7, which is an ‘and’ situation. Solve both parts simultaneously: subtract 1 to get 1 < 3x ≤ 6, then divide by 3 to obtain 1/3 < x ≤ 2. For ‘or’ cases such as x < -1 or x > 4, the solution set is the union of the two intervals. Always express these logically and, when asked, use correct notation like (-∞, -1) ∪ (4, ∞).

复合不等式通过“且”（交集）或“或”（并集）将两个或更多简单不等式组合起来。一个典型例子是 2 < 3x + 1 ≤ 7，这属于“且”的情形。可以同时求解：减 1 得 1 < 3x ≤ 6，再除以 3 得 1/3 < x ≤ 2。对于“或”的情形，比如 x < -1 或 x > 4，解集是这两个区间的并集。务必有逻辑地表达这些解集，并根据要求采用正确记号，如 (-∞, -1) ∪ (4, ∞)。

9. Using the Discriminant | 判别式的运用

IB AA questions sometimes involve inequalities that require the discriminant Δ = b² – 4ac. For example, for the quadratic kx² + 2x + k > 0 to hold for all real x, we need a > 0 (so k > 0) and Δ < 0. Here Δ = 4 – 4k² < 0 gives k² > 1, so k > 1 (since k > 0). Combined, k > 1. This connects inequalities with the nature of quadratic functions. Another scenario is finding values of k for which an equation has no real roots, expressed as an inequality Δ < 0.

在 IB AA 课程中，有时不等式的求解需要用到判别式 Δ = b² – 4ac。例如，要使二次式 kx² + 2x + k > 0 对所有实数 x 恒成立，需要 a > 0（即 k > 0）且 Δ < 0。计算 Δ = 4 – 4k² < 0 得 k² > 1，结合 k > 0 得 k > 1。这便将不等式与二次函数的性质联系起来。另一种常见情形是求使方程无实根的 k 值，此时也需列出不等式 Δ < 0。

10. IB Exam Tips | IB 考试技巧

In IB exams, inequality questions often award marks for correct method, appropriate interval notation, and clearly defined boundary treatment. Always show the sign reversal step explicitly. When dealing with rational inequalities, state the domain restrictions. Double-check whether the inequality includes equality and represent it properly with brackets. In Paper 2 calculator questions, you may use graphing to verify your solutions, but you must also show the algebraic process to earn full marks. Practice past paper questions that mix inequalities with logarithms, exponentials, or trigonometric functions, as these are popular in Section B.

在 IB 考试中，不等式题目的评分通常侧重正确的方法、恰当的区间记号和边界处理的明确定义。务必清清楚楚地写出变号的步骤。处理分式不等式时，要声明定义域限制。复查不等式是否包含等号，并用括号正确表示。在 Paper 2 允许使用计算器的问题中，你可以借助图像验证答案，但必须展示代数过程才能获得满分。多练习将不等式与对数、指数或三角函数结合的历年真题，这类综合题在 Section B 中很常见。

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