📚 Mastering Inequalities for WJEC A-Level Maths: A Complete Guide | A-Level WJEC 数学：不等式考点精讲
Inequalities are a cornerstone of A-Level Mathematics, essential for modelling real-world constraints and solving a wide variety of problems. In the WJEC specification, you need to master solving linear, quadratic, rational, and absolute value inequalities, as well as interpreting them graphically and handling parameters. This guide walks you through every key topic with clear explanations, worked examples, and common pitfalls, all tailored to the WJEC exam style.
不等式是 A-Level 数学的基石，对于建立现实世界约束的模型和解决各种问题至关重要。在 WJEC 考纲中，你需要掌握求解线性不等式、二次不等式、有理不等式和绝对值不等式，以及如何用图像解释它们和处理参数。本指南将带你逐一梳理每个关键主题，提供清晰的解释、例题和常见错误，完全贴合 WJEC 考试风格。
1. What Are Inequalities? | 什么是不等式？
An inequality compares two expressions using one of the symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to). The solution to an inequality is a set of values that make the statement true, often expressed in set notation or interval notation, e.g. {x : x > 3} or (3, ∞). Remember that multiplying or dividing both sides by a negative number reverses the inequality sign.
不等式使用以下符号之一来比较两个表达式：<（小于）、>（大于）、≤（小于或等于）、≥（大于或等于）。不等式的解是使该陈述成立的值的集合，通常用集合符号或区间表示法表示，例如 {x : x > 3} 或 (3, ∞)。请记住，当两边乘以或除以一个负数时，不等号方向要改变。
2. Linear Inequalities | 线性不等式
These involve the variable only to the first power, such as 3x − 5 ≤ 7. Solve them by isolating x using inverse operations, just like equations, but flipping the inequality when multiplying or dividing by a negative. For example, −2x < 8 becomes x > −4. The solution can be shown on a number line with an open or closed circle and shading.
这类不等式只包含一次方的变量，例如 3x − 5 ≤ 7。通过逆运算分离 x 来求解，就像解方程一样，但当乘以或除以负数时要改变不等号。例如，−2x < 8 变为 x > −4。解可以在数轴上用空心或实心圆和阴影表示。
3. Quadratic Inequalities | 二次不等式
For an inequality like x² − x − 6 > 0, first find the critical values by solving the corresponding quadratic equation x² − x − 6 = 0, giving x = −2 or x = 3. Sketch the graph of y = x² − x − 6, a positive parabola crossing the x‑axis at these points. The region where y > 0 lies outside the roots, so the solution is x < −2 or x > 3. Always consider the shape of the graph and whether the inequality is strict or non‑strict.
对于像 x² − x − 6 > 0 这样的不等式，首先通过解对应的二次方程 x² − x − 6 = 0 找到临界值，得到 x = −2 或 x = 3。画出 y = x² − x − 6 的图像，这是一条开口向上的抛物线，在这些点穿越 x 轴。y > 0 的区域在两根之外，因此解为 x < −2 或 x > 3。始终要考虑图像的形状以及不等式是严格还是非严格的。
4. Polynomial Inequalities | 多项式不等式
To solve a cubic or higher-degree inequality like (x + 1)(x − 2)(x − 4) ≤ 0, mark the roots −1, 2, 4 on a number line. Test a value from each interval to determine the sign of the product, or use the fact that the sign alternates if all factors are linear with positive coefficients. Include endpoints where the expression equals zero if the inequality is non‑strict. The solution here is x ≤ −1 or 2 ≤ x ≤ 4.
要解像 (x + 1)(x − 2)(x − 4) ≤ 0 这样的三次或更高次不等式，在数轴上标出根 −1、2、4。从每个区间测试一个值以确定乘积的符号，或者利用如果所有因子都是线性且系数为正则符号交替变化这一事实。如果不等式非严格，则包括表达式等于零的端点。这里的解是 x ≤ −1 或 2 ≤ x ≤ 4。
5. Rational Inequalities | 有理不等式
For an inequality involving fractions, such as (x − 1)/(x + 3) ≥ 0, identify points where the numerator or denominator is zero: x = 1 and x = −3. The expression can change sign only at these critical values. Draw a number line, test intervals, and remember that the denominator can never be zero, so x = −3 is always excluded, often indicated by an open circle. The solution is x < −3 or x ≥ 1.
对于涉及分式的不等式，例如 (x − 1)/(x + 3) ≥ 0，找出分子或分母为零的点：x = 1 和 x = −3。表达式只能在这些临界值处改变符号。画出数轴，测试各个区间，并记住分母绝不能为零，因此 x = −3 总会被排除，通常用空心圆表示。解为 x < −3 或 x ≥ 1。
6. Absolute Value Inequalities | 绝对值不等式
Inequalities with |x| are solved by rewriting them without the modulus. |x| < a (a > 0) means −a < x < a. |x| > a means x < −a or x > a. For more complex expressions like |2x − 3| ≤ 5, translate to −5 ≤ 2x − 3 ≤ 5, then solve the double inequality to get −1 ≤ x ≤ 4. Always check the conditions after squaring if you use that method.
含有 |x| 的不等式可以通过去掉模长重写求解。 |x| < a (a > 0) 意味着 −a < x < a。 |x| > a 意味着 x < −a 或 x > a。对于更复杂的表达式，如 |2x − 3| ≤ 5，转化为 −5 ≤ 2x − 3 ≤ 5，然后解这个双重不等式得到 −1 ≤ x ≤ 4。如果使用平方方法，务必检查条件。
7. Inequalities Involving Parameters | 含参不等式
WJEC exams sometimes include a parameter, e.g. solve ax + 2 > 4 for x, where a is a constant. You must consider separate cases depending on the sign of a. If a > 0, x > 2/a; if a < 0, x < 2/a; if a = 0, the inequality becomes 2 > 4, which is false for all x. Always state the condition alongside the solution.
WJEC 考试有时会包含参数，例如解关于 x 的不等式 ax + 2 > 4，其中 a 是常数。你必须根据 a 的符号分情况讨论。如果 a > 0，则 x > 2/a；如果 a < 0，则 x < 2/a；如果 a = 0，不等式变为 2 > 4，对所有 x 都不成立。始终要在解旁边说明条件。
8. Graphical Solutions of Inequalities | 不等式图解
You can solve an inequality by considering the graphs of two functions. For example, to solve x² < 2 − x, rearrange to x² + x − 2 < 0 and sketch the parabola, or graph y = x² and y = 2 − x to see where the curve is below the line. Graphical methods are especially useful when algebraic manipulation is messy or when a question explicitly asks for a sketch.
你可以通过考虑两个函数的图像来解不等式。例如，要解 x² < 2 − x，可以整理为 x² + x − 2 < 0 并画出抛物线，或者画出 y = x² 和 y = 2 − x 的图像，观察曲线在直线下方的位置。当代数运算复杂或题目明确要求画图时，图像法特别有用。
9. Inequalities and Regions on the Coordinate Plane | 平面区域不等式
An inequality such as y > 2x + 1 defines a half-plane. Shade the region above the line y = 2x + 1 (dashed if strict) and test a point to confirm. Systems like y ≤ x + 2 and x + y < 4 produce an intersection region, fundamental for linear programming. WJEC often expects you to label the feasible region and find its vertices.
像 y > 2x + 1 这样的不等式定义了一个半平面。对直线 y = 2x + 1 上方的区域进行着色（严格不等式用虚线），并测试一个点加以确认。联立不等式组 y ≤ x + 2 和 x + y < 4 会产生一个交集区域，这是线性规划的基础。WJEC 通常要求你标出可行区域并找出其顶点。
10. Solving Systems of Inequalities | 不等式组求解
A system combines multiple inequalities that must hold simultaneously. Solve each inequality separately, then find the overlap of the solution sets. For example, solve 2x + 1 > 3 and x − 4 ≤ 0. The first gives x > 1, the second x ≤ 4, so the combined solution is 1 < x ≤ 4. Express it in set notation or on a number line.
不等式组将多个必须同时成立的不等式组合在一起。分别解每个不等式，然后求各解集的交集。例如，解 2x + 1 > 3 和 x − 4 ≤ 0。第一个得 x > 1，第二个得 x ≤ 4，因此组合解为 1 < x ≤ 4。用集合表示法或数轴表示。
11. Algebraic Manipulation of Inequalities | 不等式的代数操作
When simplifying an inequality, you can add or subtract any number from both sides without changing the direction. Multiplication or division by a positive number preserves the inequality, but by a negative number reverses it. Squaring both sides is only safe when both sides are non‑negative and you maintain the logical equivalence. Always state any restrictions on the variable, especially when clearing denominators.
在简化不等式时，你可以在两边加或减任意数而不改变方向。乘以或除以正数保持不等号方向，但乘以或除以负数会使其反转。只有当两边都非负且保持逻辑等价时，两边平方才是安全的。务必说明对变量的限制条件，特别是在去掉分母时。
12. Common Mistakes and Tips | 常见错误与技巧
One frequent error is forgetting to reverse the inequality when multiplying by a negative number. Another is mishandling the denominator in rational inequalities – always remember the exclusions. In quadratic inequalities, students often write the solution in the wrong region; sketching the graph prevents this. When dealing with parameters, missing the zero case loses marks. Practise past WJEC papers, as the phrasing often asks for solutions in set notation or as exact inequalities.
一个常见错误是乘以负数时忘记翻转不等号。另一个是在有理不等式中错误处理分母——始终记住要排除分母为零的点。在二次不等式中，学生常把解写错区间；画草图可以避免这种错误。处理参数时，遗漏零系数的情况会丢分。练习 WJEC 历年真题，因为题目的措辞经常要求用集合表示法或精确的不等式形式给出答案。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply