📚 IB WJEC Mathematics: Inequalities Key Points | IB WJEC 数学：不等式 考点精讲
Inequalities form a critical bridge between algebraic manipulation and real-world decision‑making. In both IB and WJEC syllabuses, you are expected to solve linear, quadratic, polynomial, rational, and absolute‑value inequalities, interpret solutions on number lines and coordinate planes, and apply inequalities to practical problems. This guide covers every major inequality topic with bilingual explanations, step‑by‑step methods, and exam‑style tips.
不等式是代数运算与现实应用之间的重要桥梁。在 IB 和 WJEC 的考纲中，你需要熟练求解线性、二次、高次多项式、有理及绝对值不等式，能在数轴和坐标平面上正确表示解集，并运用不等式解决实际问题。本文以中英对照的方式涵盖所有主要不等式考点，提供分步解法与应试技巧。
1. Linear Inequalities | 线性不等式
A linear inequality involves a linear expression, such as ax + b > 0 or ax + b ≤ c. The solution is usually a simple interval.
线性不等式涉及一次表达式，例如 ax + b > 0 或 ax + b ≤ c。其解通常是一个简单区间。
Solving steps: isolate the variable just like a linear equation, but remember to reverse the inequality sign when multiplying or dividing by a negative number.
求解步骤：像解一元一次方程一样分离变量，但当两边同乘或同除以一个负数时，必须反转不等号方向。
Example: Solve 3x − 5 ≥ 7.
Add 5: 3x ≥ 12.
Divide by 3: x ≥ 4.
Solution set: [4, ∞).
示例：解 3x − 5 ≥ 7。
两边加 5：3x ≥ 12。
两边除以 3：x ≥ 4。
解集：[4, ∞)。
When the coefficient of x is negative, e.g., −2x < 6, divide by −2 and flip the sign: x > −3.
当 x 的系数为负时，如 −2x < 6，除以 −2 并反转不等号得 x > −3。
On a number line, use an open circle for strict inequalities (<, >) and a closed circle for inclusive ones (≤, ≥).
在数轴上表示时，严格不等式（<, >）用空心圆点，包含等号的不等式（≤, ≥）用实心圆点。
2. Compound Inequalities | 复合不等式
Compound inequalities link two inequalities with ‘and’ (intersection) or ‘or’ (union). The form a < f(x) < b is a common 'and' situation.
复合不等式通过’且’（交集）或’或’（并集）将两个不等式连接起来。a < f(x) < b 是最常见的'且'型不等式。
Solve a < f(x) < b by splitting into a < f(x) and f(x) < b, then take the intersection of the two solution sets.
求解 a < f(x) < b 时，可拆分为 a < f(x) 与 f(x) < b，再取两个解集的交集。
Example: Solve −1 < 2x + 3 ≤ 7.
Left part: −1 < 2x + 3 ⇒ −4 < 2x ⇒ −2 < x.
Right part: 2x + 3 ≤ 7 ⇒ 2x ≤ 4 ⇒ x ≤ 2.
Intersection: −2 < x ≤ 2, i.e., (−2, 2].
示例：解 −1 < 2x + 3 ≤ 7。
左部：−1 < 2x + 3 ⇒ −4 < 2x ⇒ −2 < x。
右部：2x + 3 ≤ 7 ⇒ 2x ≤ 4 ⇒ x ≤ 2。
取交集：−2 < x ≤ 2，即 (−2, 2]。
For ‘or’ inequalities like x < 2 or x > 5, the solution is the union: (−∞, 2) ∪ (5, ∞).
对于’或’型不等式，如 x < 2 或 x > 5，解集为并集：(−∞, 2) ∪ (5, ∞)。
3. Quadratic Inequalities | 二次不等式
Quadratic inequalities involve expressions of the form ax² + bx + c > 0, < 0, ≥ 0, or ≤ 0. The standard method uses factorisation and a sign chart, or the graphical approach with a parabola.
二次不等式涉及 ax² + bx + c > 0、< 0、≥ 0 或 ≤ 0 等形式。标准解法包括因式分解后作符号表，或利用抛物线图像。
Key steps:
1. Rearrange to have 0 on one side.
2. Factorise the quadratic if possible, or use the quadratic formula to find critical values (roots).
3. Draw a sign chart or sketch the parabola to determine intervals where the expression is positive or negative.
4. Write the solution set based on the inequality sign.
关键步骤：
1. 移项使一边为 0。
2. 尽可能因式分解，或使用求根公式得出临界值（根）。
3. 制作符号表或画出抛物线草图，确定表达式的正负区间。
4. 根据不等号写出解集。
Example: Solve x² − 5x + 6 > 0.
Factorise: (x − 2)(x − 3) > 0.
Critical values: x = 2, x = 3.
Sign chart for (x−2)(x−3):
x < 2 → product positive
2 < x < 3 → product negative
x > 3 → product positive
Since we need > 0, solution: x < 2 or x > 3, i.e., (−∞, 2) ∪ (3, ∞).
示例：解 x² − 5x + 6 > 0。
因式分解：(x − 2)(x − 3) > 0。
临界值：x = 2，x = 3。
符号表：(x−2)(x−3) 的正负：
x < 2 时乘积为正
2 < x < 3 时乘积为负
x > 3 时乘积为正
因为要求 > 0，解为 x < 2 或 x > 3，即 (−∞, 2) ∪ (3, ∞)。
When the quadratic has no real roots (discriminant Δ < 0), the expression is always positive or always negative, depending on the leading coefficient a.
当二次式没有实根（判别式 Δ < 0）时，表达式恒正或恒负，取决于首项系数 a。
4. Discriminant and Nature of Roots in Inequalities | 判别式与不等式根的性质
The discriminant Δ = b² − 4ac tells you how many real roots a quadratic has. For an inequality like ax² + bx + c > 0 with a > 0 and Δ < 0, the parabola lies entirely above the x‑axis, so the inequality is true for all real x.
判别式 Δ = b² − 4ac 能告诉我们二次方程有多少实根。对于 a > 0 且 Δ < 0 的不等式 ax² + bx + c > 0，抛物线完全在 x 轴上方，因此不等式对所有实数 x 恒成立。
Conversely, if a < 0 and Δ < 0, ax² + bx + c > 0 has no real solution, while ax² + bx + c < 0 is true for all x.
反之，如果 a < 0 且 Δ < 0，则 ax² + bx + c > 0 无实数解，而 ax² + bx + c < 0 对所有实数 x 成立。
When Δ = 0, the parabola touches the x‑axis at the vertex. For non‑strict inequalities (≥ or ≤), the critical point is included; for strict inequalities it is excluded.
当 Δ = 0 时，抛物线顶点与 x 轴相切。对于非严格不等式（≥ 或 ≤），临界点包含在解集中；严格不等式则不包含该点。
5. Polynomial Inequalities of Higher Degree | 高次多项式不等式
For cubic or higher‑degree polynomials, the principle is the same: find all real roots, place them on a number line, and test the sign of the polynomial in each interval.
对于三次及更高次多项式，原理相同：求出所有实根，将它们标在数轴上，然后在每个区间内检测多项式的符号。
Factorisation is usually the first step. If a factor is repeated (e.g., (x − 1)²), the sign does not change at that root — it ‘bounces’ off the axis.
因式分解通常是第一步。如果因子有重根（如 (x − 1)²），多项式在该根处不变号——图像在轴处弹回。
Example: Solve (x + 1)(x − 2)²(x − 4) ≤ 0.
Roots: x = −1, x = 2 (double), x = 4.
Sign intervals: (−∞, −1): negative × positive × negative = positive? Let us compute: for x<−1, (x+1) negative, (x−2)² positive, (x−4) negative → product positive. Between −1 and 2: (x+1) positive, (x−2)² positive, (x−4) negative → product negative. Between 2 and 4: positive × positive × negative = negative. x>4: all factors positive → positive.
We want ≤ 0. So solutions: [−1, 4] but note that at x=2 the expression is 0, so it is included. Thus solution: [−1, 4].
示例：解 (x + 1)(x − 2)²(x − 4) ≤ 0。
根：x = −1, x = 2（二重根）, x = 4。
符号区间：x<−1 时，(x+1)负，(x−2)²正，(x−4)负 ⇒ 乘积为正。−1
要求 ≤ 0，解集为 [−1, 4]。注意在 x=2 处表达式为 0，故包含该点。解为 [−1, 4]。
A sign chart or table is extremely helpful here, especially for WJEC and IB exam questions where a structured table earns method marks.
符号表或表格在此处非常有用，尤其在 WJEC 和 IB 考试中，规范的表格能帮助获取过程分。
6. Rational Inequalities | 有理不等式
Rational inequalities involve fractions with polynomials, such as (x + 2)/(x − 3) ≥ 0. The critical values come from both the numerator and the denominator.
有理不等式包含多项式分式，如 (x + 2)/(x − 3) ≥ 0。临界值既来自分子也来自分母。
Steps:
1. Move all terms to one side to obtain a single fraction compared to 0.
2. Factorise numerator and denominator.
3. Identify values where the expression is 0 (numerator = 0) or undefined (denominator = 0). These are the boundary points.
4. Construct a sign chart for the fraction.
5. Select intervals that satisfy the inequality, remembering to exclude values that make the denominator zero.
步骤：
1. 移项使一边为 0，合并成一个分式。
2. 对分子分母进行因式分解。
3. 找出表达式为 0（分子=0）或无定义（分母=0）的点，作为分界点。
4. 制作分式的符号表。
5. 选择满足不等式的区间，同时务必排除使分母为零的临界值。
Example: Solve (x + 2)/(x − 3) > 0.
Numerator zero at x = −2; denominator zero at x = 3.
Intervals: x < −2, −2 < x < 3, x > 3.
Test sign: x = −3 → (−)/(−) = +, so positive; x = 0 → (+)/(−) = −, negative; x = 4 → (+)/(+) = +, positive.
We want > 0, so solution: x < −2 or x > 3, i.e., (−∞, −2) ∪ (3, ∞).
示例：解 (x + 2)/(x − 3) > 0。
分子零点 x = −2；分母零点 x = 3。
区间：x < −2，−2 < x < 3，x > 3。
符号检验：x = −3 → (−)/(−) = +，正；x = 0 → (+)/(−) = −，负；x = 4 → (+)/(+) = +，正。
要求 > 0，解为 x < −2 或 x > 3，即 (−∞, −2) ∪ (3, ∞)。
Never include the denominator zero in the solution set, even for non‑strict inequalities.
即使是非严格不等式，也绝不可将分母零点纳入解集。
7. Absolute Value Inequalities | 绝对值不等式
Absolute value inequalities such as |x| < a or |f(x)| > b can be rewritten as compound inequalities without absolute values.
绝对值不等式如 |x| < a 或 |f(x)| > b 可改写为不含绝对值的复合不等式。
Key equivalent forms:
|f(x)| < k (k > 0) ⇔ −k < f(x) < k
|f(x)| > k (k > 0) ⇔ f(x) < −k or f(x) > k
|f(x)| ≤ k ⇔ −k ≤ f(x) ≤ k
|f(x)| ≥ k ⇔ f(x) ≤ −k or f(x) ≥ k
关键等价形式：
|f(x)| < k (k>0) ⇔ −k < f(x) < k
|f(x)| > k (k>0) ⇔ f(x) < −k 或 f(x) > k
|f(x)| ≤ k ⇔ −k ≤ f(x) ≤ k
|f(x)| ≥ k ⇔ f(x) ≤ −k 或 f(x) ≥ k
Example: Solve |2x − 1| ≤ 5.
Rewrite as −5 ≤ 2x − 1 ≤ 5.
Add 1: −4 ≤ 2x ≤ 6.
Divide by 2: −2 ≤ x ≤ 3.
Solution: [−2, 3].
示例：解 |2x − 1| ≤ 5。
改写为 −5 ≤ 2x − 1 ≤ 5。
加 1：−4 ≤ 2x ≤ 6。
除以 2：−2 ≤ x ≤ 3。
解集：[−2, 3]。
For more complex expressions, isolate the absolute value first, then apply the appropriate rule. Always check for extraneous solutions if the variable appears inside and outside the absolute value.
对于更复杂的表达式，应先将绝对值隔离出来，再应用相应规则。若变量同时出现在绝对值内外，需警惕增根。
8. Inequalities Involving Exponentials and Logarithms | 指数与对数不等式
Exponential inequalities like aˣ > b and logarithmic inequalities like logₐ(x) > c require an understanding of monotonicity: for bases > 1 the function is increasing, for 0 < base < 1 it is decreasing.
指数不等式（如 aˣ > b）和对数不等式（如 logₐ(x) > c）需要利用单调性：当底数大于 1 时函数递增，当 0 < 底数 < 1 时函数递减。
When taking logarithms of both sides or exponentiating, the inequality sign is preserved if the base is > 1, and reversed if 0 < base < 1.
两边取对数或作指数运算时，若底数 > 1 则不等号方向不变；若 0 < 底数 < 1 则需反转不等号。
Example: Solve 2ˣ > 8.
Rewrite 8 as 2³, thus 2ˣ > 2³. Since base 2 > 1, the exponent inequality is preserved: x > 3.
示例：解 2ˣ > 8。
将 8 写作 2³，得 2ˣ > 2³。因为底数 2 > 1，故指数不等式方向不变：x > 3。
For logarithmic inequalities, always check the domain of the logarithm expression: arguments must be positive.
对于对数不等式，务必先检查对数表达式有意义的定义域：真数必须大于 0。
Example: Solve log₂(x + 1) ≤ 3.
Domain: x + 1 > 0 ⇒ x > −1.
Rewrite in exponential form: x + 1 ≤ 2³ = 8 ⇒ x ≤ 7.
Combine with domain: −1 < x ≤ 7.
示例：解 log₂(x + 1) ≤ 3。
定义域：x + 1 > 0 ⇒ x > −1。
改写为指数形式：x + 1 ≤ 2³ = 8 ⇒ x ≤ 7。
与定义域取交集：−1 < x ≤ 7。
9. Graphical Approach to Inequalities | 图形法解不等式
Many inequality problems can be visualised by sketching graphs. For example, solving f(x) > g(x) is equivalent to finding where the graph of y = f(x) lies above that of y = g(x).
许多不等式问题可以通过画图直观解决。例如，解 f(x) > g(x) 相当于寻找 y = f(x) 图像在 y = g(x) 图像上方的区间。
In IB and WJEC exams, you may be asked to use a GDC (graphic display calculator) to solve inequalities like x³ − 4x < 2 numerically or by intersections.
在 IB 和 WJEC 考试中，你可能需要使用图形计算器（GDC）通过数值或交点方式求解如 x³ − 4x < 2 的不等式。
When sketching manually, marking intercepts and asymptotes helps determine where the inequality holds. This method is particularly powerful for rational and modulus functions.
手工画图时，标出截距和渐近线有助于确定不等式成立的范围。这种方法对有理函数和模函数的分析尤为有效。
Remember: the solution set for f(x) < 0 is simply the set of x‑values for which the graph lies below the x‑axis.
请记住：f(x) < 0 的解集就是图像位于 x 轴下方部分所对应的 x 值集合。
10. Systems of Linear Inequalities | 线性不等式组
A system of inequalities, such as { y ≥ 2x + 1, y < −x + 4 }, has a solution region in the coordinate plane. This region is the intersection of the half‑planes defined by each inequality.
不等式组（如 { y ≥ 2x + 1, y < −x + 4 }）在坐标平面上对应一个解区域，它是各个不等式所定义的半平面的交集。
To graph, draw the boundary lines (dashed for strict inequalities, solid for inclusive). Shade the appropriate side for each, then identify the overlapping region.
绘图步骤：画出边界线（严格不等式用虚线，含等号用实线），分别对各个不等式在正确一侧涂色，然后找出所有涂色区域的重叠部分。
Often these systems arise in linear programming problems where you are required to maximise or minimise an objective function subject to constraints.
这类不等式组常出现在线性规划问题中，你需要在一组约束条件下求目标函数的最大值或最小值。
In such cases, the feasible region is a polygon, and the optimal solution occurs at one of its vertices. IB and WJEC may ask you to find these vertices and evaluate the objective function.
此时可行域为一个多边形，最优解必定出现在其某一顶点处。IB 和 WJEC 可能会要求你找出这些顶点并计算目标函数值。
11. Applications and Word Problems | 应用与文字题
Inequality modelling is common in optimisation, finance, and physics. Common scenarios include profit thresholds, distance‑time constraints, concentration mixtures, and geometric conditions.
不等式建模在优化、金融和物理问题中十分常见。典型场景包括利润门槛、距离时间约束、浓度混合问题以及几何条件限制。
Steps for word problems:
1. Identify the variable(s) and define them clearly.
2. Translate the written conditions into inequalities.
3. Solve the inequality or system of inequalities mathematically.
4. Interpret the solution in the context of the problem and check feasibility.
文字题解题步骤：
1. 找出并明确定义变量。
2. 将文字条件转化为不等式。
3. 数学求解不等式或不等式组。
4. 将解代入原问题情境解释，并检验其合理性。
Example: A company sells a product for $40 per unit. Fixed costs are $500 and variable cost per unit is $15. How many units must be sold to make a profit of at least $1000?
Let n be the number of units. Revenue: 40n. Total cost: 500 + 15n. Profit condition: 40n − (500 + 15n) ≥ 1000 ⇒ 25n − 500 ≥ 1000 ⇒ 25n ≥ 1500 ⇒ n ≥ 60. So at least 60 units.
示例：某公司产品售价为每件 40 美元，固定成本 500 美元，变动成本每件 15 美元。要使利润至少达到 1000 美元，需销售多少件？
设件数为 n。收入：40n，总成本：500+15n。利润条件：40n−(500+15n) ≥ 1000 ⇒ 25n−500 ≥ 1000 ⇒ 25n ≥ 1500 ⇒ n ≥ 60。即至少需销售 60 件。
Always note whether the answer requires an integer, and if rounding should be up or down based on the inequality direction.
务必注意答案是否需要取整数，并根据不等号方向决定是向上取整还是向下取整。
12. Common Mistakes and Exam Tips | 常见错误与应试技巧
Here are pitfalls to avoid and strategies to maximise your marks in IB and WJEC inequality questions.
以下是在 IB 和 WJEC 不等式题目中需要避免的陷阱以及提升得分的策略。
Mistake 1: Forgetting to reverse the sign when multiplying/dividing by a negative number.
错误一：乘以或除以负数时忘记反转不等号方向。
Mistake 2: Including denominator zeros in rational inequality solutions.
错误二：将分母零点错误包含进有理不等式解集中。
Mistake 3: Applying the same absolute value rules when k is negative (e.g., |x| < −2 has no solution).
错误三：当 k 为负数时仍机械套用绝对值规则（如 |x| < −2 无解）。
Mistake 4: Ignoring the domain when solving logarithmic inequalities.
错误四：解对数不等式时忽略定义域限制。
Mistake 5: Drawing sign charts for quadratics without first setting the expression to zero; always place all terms on one side.
错误五：未先移项至一边为 0 就画符号表。务必先将表达式化为一边为 0 的形式。
Exam tips:
– Show clearly the sign chart or critical value method; examiners award marks for process.
– Use interval notation correctly: (a, b) for open, [a, b] for closed, (a, b] for half‑open.
– When using a GDC, state the window settings and the method (e.g., “Graph y1 = … and find the x‑intercepts”).
– Double‑check boundary points by substituting back into the original inequality.
– For inequalities involving word problems, conclude with a sentence in context.
应试技巧：
– 清晰展示符号表或临界值法，阅卷人将据此给步骤分。
– 正确使用区间表示法：(a, b) 为开区间，[a, b] 为闭区间，(a, b] 为半开区间。
– 使用图形计算器时，注明窗口设置和所用方法（如“绘制 y1 = … 并求 x 轴交点”）。
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