📚 Regions Defined by Inequalities | 由不等式定义的区域
In A-Level mathematics, the concept of ‘regions’ is a powerful visual tool for solving inequalities. A region is simply a set of points in the coordinate plane that satisfy one or more algebraic conditions. By shading these areas on a graph, we can instantly identify all possible solutions, whether for a single linear inequality, a system of constraints, or a quadratic boundary. This topic bridges algebraic reasoning with geometric intuition, and it is a staple in Edexcel’s Pure Mathematics papers, often appearing alongside straight-line graphs, curve sketching, and even optimisation problems.
在A-Level数学中，‘区域’的概念是解不等式的一件强大视觉工具。区域就是坐标平面上满足一个或多个代数条件的点集。通过在图形上描影这些区域，我们可以立即识别出所有可能的解，无论是对于单一线性不等式、一组约束条件，还是二次曲线边界。这一主题将代数推理与几何直觉连接起来，是Edexcel纯数学试卷中的重要内容，经常与直线图形、曲线素描甚至最优化问题一同出现。
1. What Is a Region in Coordinates? | 坐标中的区域是什么？
A region is the two-dimensional area on a Cartesian plane that is defined by an inequality. For example, the inequality y > 2x + 1 tells us that for every point (x, y) in the region, the y-coordinate is strictly greater than twice the x-coordinate plus one. The boundary line is y = 2x + 1, and the region lies entirely on one side of it. Understanding which side to shade is the core skill.
区域是笛卡尔平面上由不等式定义的二维面积。例如，不等式 y > 2x + 1 告诉我们，对于区域内的每一个点 (x, y)，其 y 坐标严格大于 x 坐标的两倍加一。边界线是 y = 2x + 1，区域完全位于该线的某一侧。理解该描哪一侧是核心技能。
2. Representing Linear Inequalities | 表示线性不等式
To represent a linear inequality, first draw the boundary line as if it were an equation. Use your knowledge of y = mx + c to plot it accurately. If the inequality is strict (< or >), the line is dashed to show it is not included. If the inequality includes equality (≤ or ≥), draw a solid line. After drawing the boundary, choose a test point not on the line — often (0, 0) if it lies away from the line — and substitute it into the inequality. If the test point satisfies the inequality, shade the side containing that point; otherwise, shade the opposite side.
要表示线性不等式，首先画出边界线，就像处理方程一样。运用 y = mx + c 的知识准确绘出直线。若不等式是严格不等号（< 或 >），则用虚线表示边界不包含在内。若不等式包含等号（≤ 或 ≥），则画实线。画出边界后，选择一个不在线上的测试点——如果 (0, 0) 不在线上，通常就用它——将其代入不等式。如果测试点满足不等式，就描影该点所在的一侧；否则，描影对侧。
3. Shading the Required Region | 描影所需区域
Always shade the region that satisfies the inequality. In Edexcel exams, you must clearly indicate which side is your answer; often a question will instruct you to shade the unwanted region instead. Read the question carefully. If asked to shade the region where a condition is true, label it R. Use neat, light shading or hatching — heavy shading can obscure coordinates that you might need to refer to later.
始终描影满足不等式的区域。在Edexcel考试中，你必须清晰表明哪一侧是你的答案；通常题目会要求你描影不需要的区域。仔细读题。如果要求描影满足条件的区域，就标上 R。使用干净、轻浅的阴影或斜线——过重的阴影可能遮盖掉你后续可能需要参考的坐标。
4. Using Solid and Dashed Lines | 使用实线与虚线
The distinction between solid and dashed boundaries is crucial. A solid line means the points on the line belong to the region (inclusive), while a dashed line means they do not. For a system of inequalities, you might have a mix. Always use a ruler for straight boundaries. In questions involving curved boundaries such as parabolas, you must draw them smoothly and still apply the solid/dashed rule according to the inequality signs.
实线与虚线边界的区别至关重要。实线表示线上的点属于该区域（包含），而虚线表示不属于。对于不等式组，你可能混合使用两者。画直线边界时务必使用直尺。在涉及抛物线等曲线边界的问题中，你必须平滑绘出，并仍根据不等号使用实线或虚线。
5. Systems of Linear Inequalities | 线性不等式组
Many real-world and exam problems involve two or more linear inequalities. The feasible region is the intersection of the individual shaded regions — the area where all conditions hold simultaneously. To find it, graph each inequality one by one, shading lightly in different directions or using different colours in rough work, then identify the overlap. In the final answer, clearly shade the feasible region and label its vertices.
许多现实问题和考试题目涉及两个或更多线性不等式。可行域就是各个单独描影区域的交集——即所有条件同时成立的区域。要找到它，逐一画出每个不等式的图形，在草稿中用不同方向或不同颜色描影，然后识别重叠部分。在最终答案中，清晰描影可行域并标注其顶点。
6. Feasible Regions and Vertices | 可行域与顶点
Once you have shaded the feasible region, it is often a polygon. The vertices (corners) of this polygon are found by solving the equations of the boundary lines in pairs. These vertices are essential because, in linear programming, the optimal value of an objective function occurs at a vertex. To record them, set up simultaneous equations for each intersecting pair of boundary lines and solve. List all vertex coordinates in a small table for clarity.
一旦你描影出可行域，它通常是一个多边形。该多边形的顶点（角点）通过成对求解边界线方程得出。这些顶点至关重要，因为在线性规划中，目标函数的最优值出现在顶点处。要记录它们，为每一对相交的边界线建立联立方程并求解。将所有的顶点坐标列在一个整洁的小表格中以清晰呈现。
7. Quadratic Inequalities and Curved Boundaries | 二次不等式与曲线边界
Not all regions are bounded by straight lines. Quadratic inequalities like y ≥ x² − 4 or y < −x² + 2x + 3 produce regions with parabolic boundaries. The method remains similar: draw the curve (solid or dashed as appropriate), pick a test point, and shade the correct side. For a vertical parabola, the region 'inside' or 'outside' depends on the inequality direction. A common mistake is to confuse y > x² (above the parabola) with x > y² (to the right of a sideways parabola). Practice both forms.
并非所有区域都由直线围成。诸如 y ≥ x² − 4 或 y < −x² + 2x + 3 这样的二次不等式会产生具有抛物线边界的区域。方法仍然类似：画出曲线（适当使用实线或虚线），选取测试点，并描影正确一侧。对于开口向上的抛物线，‘内部’或‘外部’区域取决于不等号方向。一个常见错误是将 y > x²（抛物线上方）与 x > y²（开口向右的抛物线右侧）混淆。两种形式都需要练习。
8. Region Satisfying Multiple Conditions | 满足多个条件的区域
It is common to mix linear and non-linear conditions. For instance, define region R as { (x, y) : y ≥ 2x − 1, y ≤ x² + 2, x ≥ 0 }. Start by sketching the line y = 2x − 1 (solid, since ≥), then the parabola y = x² + 2 (solid, since ≤), and finally the vertical line x = 0 (the y-axis, solid). Use test points to shade each region and then highlight the intersection. In such problems, the curved boundary may cut the linear boundary in two points, and the feasible region might be a non-polygonal shape.
混合线性与非线性条件的情况很常见。例如，定义区域 R 为 { (x, y) : y ≥ 2x − 1, y ≤ x² + 2, x ≥ 0 }。首先画出直线 y = 2x − 1（实线，因为 ≥），然后是抛物线 y = x² + 2（实线，因为 ≤），最后是垂直线 x = 0（y 轴，实线）。用测试点描影每个区域，随后突出显示交集。在这类问题中，曲线边界可能与直线边界相交于两点，可行域可能是一个非多边形的形状。
9. Practical Applications | 实际应用
Regions defined by inequalities are not just an abstract exercise; they model constraints in business, logistics, and science. For example, a company might need to produce x units of product A and y units of product B, subject to time, material, and demand constraints such as 3x + 4y ≤ 240, 2x + y ≤ 100, x ≥ 10, y ≥ 10. The feasible region shows all possible production plans. From there, one can find the maximum profit by evaluating an objective function at the vertices.
由不等式定义的区域并非只是抽象练习；它们为商业、物流和科学中的约束条件建模。例如，一家公司需要生产 x 单位产品 A 和 y 单位产品 B，并受时间、物料和需求的约束，如 3x + 4y ≤ 240, 2x + y ≤ 100, x ≥ 10, y ≥ 10。可行域显示了所有可能的生产计划。由此，可以通过在顶点处评估目标函数来找到最大利润。
10. Common Mistakes to Avoid | 应避免的常见错误
Even confident students lose marks on regions. Watch out for these pitfalls: (1) Forgetting to draw a boundary as dashed when the inequality is strict. (2) Shading the wrong side because the test point was on the boundary or too close. Always pick a simple, far-off test point like (0,0) when possible and check the arithmetic. (3) Not labelling the final region clearly. If the question says ‘label the region R’, make sure it is large and clear. (4) Overlooking the integer-only constraints — if x and y must be whole numbers, solutions are only those grid points within the shaded region.
即使是自信的学生也会在区域题上失分。注意这些易错点：(1) 当不等式为严格不等时，忘记将边界画成虚线。(2) 因为测试点位于边界上或太近而导致描错侧。总是尽可能选取简单且远离边界的测试点，如 (0,0)，并检查计算。(3) 没有清晰标注最终区域。如果题目要求‘标注区域 R’，确保标注大而清晰。(4) 忽略了仅限整数的约束——如果 x 和 y 必须是整数，那么解只能是阴影区域内的那些整数格点。
11. Exam Technique for Edexcel | Edexcel考试技巧
In an Edexcel A-Level paper, region questions often carry 4–7 marks. To secure full marks: read the inequality signs aloud as you copy them to avoid reversing them. Use a table to show how you found vertex coordinates, for example by solving y = 2x − 1 and y = −x + 5 simultaneously. Display the calculation step by step. For quadratic boundaries, sketch the curve clearly and mark any axis intercepts. Always answer the specific demand, whether it is to shade R, to find vertices, or to state whether a given point lies in the region.
在Edexcel A-Level试卷中，区域题通常占4–7分。要拿到满分：抄写不等式时读出不等号，以避免将其方向弄反。使用表格展示你如何求得顶点坐标，例如通过联立求解 y = 2x − 1 和 y = −x + 5。逐步展示计算过程。对于二次边界，清晰画出曲线并标出任何轴截距。始终回答具体要求，不论是描影 R、求顶点，还是判断某给定点是否位于区域内。
12. Key Summary and Link to Further Topics | 核心总结与延伸主题的联系
Mastering regions equips you with a visual problem-solving strategy that extends far beyond single inequalities. It directly feeds into the linear programming section of Decision Mathematics and into the loci and regions of complex numbers in Further Pure. At the Pure Mathematics level, being able to accurately sketch inequalities and find feasible regions is a transferable skill that supports curve analysis and even integration ‘area between curves’ questions. Practice using past Edexcel papers; the graph-drawing routine will soon become second nature.
掌握区域为你提供了一种视觉化解题策略，其应用远不止于单一不等式。它直接服务于决策数学中的线性规划部分，以及进阶纯数学中复数的轨迹与区域。在纯数学层面，能准确描绘不等式并求出可行域是一项可迁移的技能，能够支持曲线分析，乃至积分中的‘曲线间面积’问题。请使用Edexcel历年真题进行演练；图形绘制步骤将很快成为本能。
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