📚 Mastering Inequalities for GCSE WJEC Maths | GCSE WJEC 数学 不等式 考点精讲
Inequalities are a fundamental part of GCSE WJEC Mathematics, appearing across number, algebra, and graphical topics. They require you to compare values, solve algebraic statements, and represent solution sets clearly on number lines and graphs. Understanding the symbols, the manipulation rules, and the visual representations is essential for success in both the non-calculator and calculator papers. Whether you are preparing for foundation or higher tier, inequalities will test your logical reasoning and precision.
不等式是GCSE WJEC数学的基础部分，贯穿于数、代数和图形等多个主题。你需要比较数值大小、求解代数式，并清晰地在数轴和图像上表示解集。理解不等号及其运算规则，掌握图像表示方法，是应对非计算器和计算器试卷的关键。无论你准备基础级还是更高级别的考试，不等式都将检验你的逻辑推理和答题的准确性。
1. Understanding the Symbols | 理解不等号
The four basic inequality symbols form the foundation of this topic. The symbol < means ‘less than’, so x < 5 reads as ‘x is less than 5’. The symbol > means ‘greater than’, for example x > 2 means ‘x is greater than 2’. The symbol ≤ means ‘less than or equal to’, and ≥ means ‘greater than or equal to’. In WJEC questions, you must also recognise phrases like ‘at most’ (≤) and ‘at least’ (≥).
四个基本不等号是这一专题的基石。符号 < 表示“小于”，因此 x < 5 读作“x 小于 5”。符号 > 表示“大于”，例如 x > 2 表示“x 大于 2”。符号 ≤ 表示“小于或等于”，而 ≥ 表示“大于或等于”。在WJEC考题中，你还必须识别“最多”（≤）和“至少”（≥）等措辞。
2. Number Line Representation | 数轴表示法
On a number line, a strict inequality like x < 3 is shown with an open circle at 3 and a line extending to the left. An inequality with an inclusive bound, such as x ≥ –1, uses a filled circle at –1 and a line to the right. WJEC examiners expect neat, clear diagrams: circles exactly on the number, a clear direction line, and a consistent scale. When you have a compound inequality like –2 < x ≤ 4, the region between –2 (open circle) and 4 (filled circle) is shaded.
在数轴上，像 x < 3 这样的严格不等式用 3 处的空心圆点和向左延伸的线表示。包含边界的不等式，例如 x ≥ –1，则在 –1 处用实心圆点并向右画线。WJEC考官要求图示整洁清晰：圆点精确对准数字，方向线清楚，比例一致。当你遇到复合不等式如 –2 < x ≤ 4 时，需要在 –2（空心圆）到 4（实心圆）之间的区域涂上阴影。
3. Solving Linear Inequalities | 解一元一次不等式
To solve a linear inequality, treat it like an equation but remember the golden exception: if you multiply or divide both sides by a negative number, you must reverse the inequality direction. For example, solve 3 – 2x > 7. Subtract 3 from both sides to get –2x > 4. Now divide by –2 and reverse the sign: x < –2. Always check your answer by substituting a test value. WJEC problems often combine multiple steps and may involve brackets or fractional terms.
解一元一次不等式时，可将其当作方程处理，但要牢记一条黄金例外：若对不等式两边同乘或同除一个负数，必须反转不等号方向。例如，解 3 – 2x > 7。两边减 3 得 –2x > 4。然后除以 –2 并反转符号：x < –2。始终通过代入测试值来检查答案。WJEC的问题通常包含多个步骤，并可能涉及括号或分数项。
4. Compound Inequalities | 复合不等式
A compound inequality combines two simple inequalities into one statement, such as –3 ≤ 2x + 1 < 5. To solve, break it into two separate inequalities: –3 ≤ 2x + 1 and 2x + 1 < 5. Solve each to find x ≥ –2 and x < 2. The combined solution is –2 ≤ x < 2. Always present the final answer in the same form, neatly overlapping the two solution sets. WJEC higher-tier papers might include negative coefficients within the compound structure, requiring careful sign reversal.
复合不等式将两个简单不等式合并为一个语句，例如 –3 ≤ 2x + 1 < 5。求解时，将其拆分为两个独立不等式：–3 ≤ 2x + 1 和 2x + 1 < 5。分别求解得到 x ≥ –2 和 x < 2。合并后的解为 –2 ≤ x < 2。始终以相同形式呈现最终答案，并确保两个解集的重叠部分整齐。WJEC高年级试卷可能会在复合结构中包含负系数，需小心地反转符号。
5. Quadratic Inequalities | 二次不等式
For quadratic inequalities, such as x² – 4x – 5 > 0, first solve the corresponding quadratic equation x² – 4x – 5 = 0 to find critical values. Factorising gives (x – 5)(x + 1) = 0, so x = 5 or x = –1. These split the number line into three regions. Test a value from each region in the original inequality to determine where the statement is true. The solution is x < –1 or x > 5. WJEC expects both algebraic method and graphical illustration, often linking to parabola sketches.
对二次不等式，例如 x² – 4x – 5 > 0，首先求解相应的二次方程 x² – 4x – 5 = 0 以找到临界值。因式分解得 (x – 5)(x + 1) = 0，所以 x = 5 或 x = –1。这些临界值将数轴分成三个区间。从每个区间取一个测试值代入原不等式，以确定哪些区间使不等式成立。解为 x < –1 或 x > 5。WJEC既要求代数方法，也要求图像解释，通常与抛物线草图相联系。
6. Representing Quadratic Inequalities Graphically | 二次不等式的图像表示
Sketching the graph y = x² – 4x – 5 helps visualise the inequality x² – 4x – 5 > 0. The parabola opens upwards and crosses the x-axis at –1 and 5. The inequality asks where y > 0, meaning above the x-axis. The graph is positive outside the interval [–1, 5], confirming x < –1 or x > 5. For ≤ or ≥ inequalities, include the critical values on the x-axis with solid dots and a solid shading region. WJEC examiners look for clearly labelled axes, intercepts, and a smooth curve.
通过绘制 y = x² – 4x – 5 的图像，可以直观理解不等式 x² – 4x – 5 > 0。该抛物线开口向上，与 x 轴交于 –1 和 5。不等式问的是 y > 0 的区域，即 x 轴上方部分。在区间 [–1, 5] 之外，函数图像为正值，证实了 x < –1 或 x > 5。对于 ≤ 或 ≥ 型不等式，需在 x 轴上用实点标示临界值，并填充对应的实心阴影区域。WJEC考官看重坐标轴标注清晰、截距正确以及平滑的曲线。
7. Inequalities in Two Variables | 二元线性不等式
An inequality like y ≥ 2x + 1 defines a region on the coordinate plane. Graph the boundary line y = 2x + 1 as a solid line because the inequality includes equals. Choose a test point not on the line, often (0,0), to determine which side to shade: if 0 ≥ 2(0) + 1 is false, shade the opposite side. For strict inequalities (> or <), use a dashed line. WJEC questions often ask you to satisfy multiple inequalities simultaneously, identifying a feasible region in linear programming contexts.
像 y ≥ 2x + 1 这样的不等式在坐标平面上定义了一个区域。画出边界线 y = 2x + 1，由于包含等于，应使用实线。选择一个不在直线上的测试点（通常为 (0,0)）来判断哪一侧需要涂阴影：若 0 ≥ 2(0) + 1 为假，则涂另一侧。对严格不等式（> 或 <），使用虚线。WJEC题目常要求同时满足多个不等式，从而在线性规划背景下找出可行域。
8. Shading Regions Satisfying Multiple Inequalities | 为满足多个不等式的区域涂阴影
When you have a system like y < x + 2, y ≥ –1, and x ≤ 3, draw all boundary lines on the same grid, using dashed or solid as appropriate. Shade each inequality’s unwanted region lightly, or shade the required region clearly as per the question’s instruction. The intersection of all shaded regions is the final answer. Label any vertices and ensure the region is visibly distinct. WJEC marks for accuracy in plotting, line style, and correct identification of the overlap.
当你面对 y < x + 2、y ≥ –1 和 x ≤ 3 这样的不等式组时，在同一网格上画出所有边界线，并根据情况使用虚线或实线。可以轻轻涂出每个不等式不需要的区域，或依照题目要求清晰标示所求区域。所有阴影区域的交集即为最终答案。标记所有顶点，确保区域明显可辨。WJEC的评分点包括描点的准确性、线型运用，以及正确识别重叠区域。
9. Solving Word Problems with Inequalities | 用不等式解应用题
WJEC often embeds inequalities in real-life contexts, such as cost constraints, age limits, or measurement errors. First, define a variable and write a mathematical inequality based on the wording. For instance, ‘A base costs £3, a mid-range ticket £5, and the budget is at most £60’ becomes 3b + 5m ≤ 60. Solve or graph the inequality to answer the specific question. Always interpret your solution back in context and check whether rounding affects the answer, especially for discrete quantities.
WJEC常将不等式融入实际生活场景，如费用限制、年龄条件或测量误差。首先定义变量，并根据题意写出数学不等式。例如，“一张基础票 3 英镑，一张中档票 5 英镑，预算最多 60 英镑”可写为 3b + 5m ≤ 60。通过求解或绘制不等式来回答具体问题。务必回归情境解读答案，并检查取整是否影响结果，特别是对于离散量。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
Many students forget to flip the inequality when multiplying or dividing by a negative. Others confuse open and filled circles on number lines, or use the wrong line style in graphs. A frequent WJEC pitfall is misreading ‘at least’ and ‘at most’, leading to reversed symbols. Practice writing solutions in set notation or as intervals where required. Always check your work by substituting boundary values and a point from the solution set. Clear presentation, including labelled diagrams, can earn method marks even if the final answer is slightly off.
很多学生在乘除负数时忘记反转不等号。另一些人混淆数轴上的空心圆与实心圆，或者在图像中使用错误的线型。WJEC常见的陷阱是误读“至少”和“最多”，导致符号用反。按要求练习用集合符号或区间形式书写解集。始终通过代入边界值和解集内的一个点来检验。即使最终答案稍有偏差，清晰的表达（包括带标注的示意图）也能帮助你赢得步骤分。
11. Linking Inequalities to Other Topics | 不等式与其他专题的联系
Inequalities are not isolated; they connect to sequences, functions, and even trigonometry in WJEC Maths. You might be asked to find the range of values for which a sequence term exceeds a given number, or to state the domain where a function is positive. In higher tier, you could meet inequalities with algebraic denominators, requiring consideration of sign changes. Treat every inequality as a chance to demonstrate your broader mathematical reasoning. Revise by drawing together number, algebra, and graphing skills.
不等式并非孤立的知识点；在WJEC数学中，它们与数列、函数甚至三角学都有联系。题目可能要求你找出数列某项大于给定值的取值范围，或陈述函数为正的定义域。在高年级阶段，你还可能遇到带代数分母的不等式，需要考虑符号变化。把每一次解不等式当作展示你综合数学推理能力的机会。复习时，要有机结合数字运算、代数处理和图形绘制等技能。
12. Quick-Reference Inequality Properties | 不等式性质速查表
A summary table can reinforce key rules. Below is a concise reference for WJEC candidates:
下面这个总结表可以帮你巩固关键规则，供WJEC考生快速查阅：
| Operation / 运算 | Rule / 规则 | Example / 示例 |
| Add/Subtract / 加减 | Keep sign / 保持不等号方向 | x + 2 > 5 → x > 3 |
| Multiply/Divide by positive / 乘以/除以正数 | Keep sign / 保持不等号方向 | 2x < 10 → x < 5 |
| Multiply/Divide by negative / 乘以/除以负数 | Reverse sign / 反转不等号方向 | –3x ≥ 9 → x ≤ –3 |
| Squaring both sides / 两边平方 | Only if both sides ≥ 0 / 仅当两边均 ≥ 0 | √x < 2 → 0 ≤ x < 4 |
| Reciprocal / 倒数 | Reverse sign if same sign / 同号则反转不等号 | 1/x > 2 → 0 < x < 1/2 (x positive) |
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