Graphing Inequalities: Region Representation Techniques | 不等式图像：区域表示的技巧

📚 Graphing Inequalities: Region Representation Techniques | 不等式图像：区域表示的技巧

Graphing an inequality is a core skill in A-Level coordinate geometry. The graph of y > 2x + 1 is not a line; it is a half-plane. You must draw the correct boundary and then shade or label the correct side. This article explains the techniques in simple, exam-focused steps.

绘制不等式图像是 A-Level 坐标几何中的核心技能。y > 2x + 1 的图像不是一条直线，而是一个半平面。你需要画出正确的边界，然后涂阴影或标出正确的一侧。本文用简明、紧扣考点的方式讲解相关技巧。

1. What an Inequality Region Means | 理解不等式区域

An inequality in x and y defines a set of points. The line or curve obtained by replacing the inequality sign with ‘=’ is called the boundary. The solution region is the collection of points that satisfy the original inequality.

x 与 y 的不等式定义了一个点集。把不等号换成“=”后得到的直线或曲线称为边界。解区域就是满足原不等式的点的集合。

For y ≥ 2x − 1, every point on y = 2x − 1 is included, as is every point above it.

对于 y ≥ 2x − 1，直线 y = 2x − 1 上的每个点以及它上方的每个点都属于解集。

2. From the Equation to the Boundary Line | 从等式到边界线

To sketch any inequality, begin by sketching the corresponding boundary equation. Example: to graph y ≤ 3x − 2, first draw the boundary line y = 3x − 2.

画任何不等式图像时，都要先画出对应的边界方程。例如，要画 y ≤ 3x − 2，先画边界线 y = 3x − 2。

y = 3x − 2

Use the intercepts or gradient and y-intercept. For the boundary, you need at least two accurate points.

可利用截距，或斜率和 y 截距来画。至少需要两个精确的点才能画出边界。

3. Solid or Dashed Boundary | 实线还是虚线边界

The first decision when drawing the boundary is line style. Use a solid line for ≤ or ≥, because the boundary is part of the solution. Use a dashed line for < or >, because points exactly on the boundary are not included.

绘制边界时首先要决定线型。对于 ≤ 或 ≥，用实线，因为边界属于解集；对于 < 或 >，用虚线，因为恰好位于边界上的点不包含在解集中。

The table below summarises the convention used in exam mark schemes.

下表总结了考试评分标准中使用的约定。

4. The Test Point Method | 测试点法

A test point tells you which side of the boundary to shade. Choose a point that is clearly not on the boundary, substitute it into the inequality, and check whether the statement is true. If it is true, shade the side containing that point; if false, shade the other side.

测试点帮助你判断应涂边界哪一侧。选一个明显不在边界上的点，代入不等式，判断不等式是否成立。若成立，涂包含这个点的一侧；若不成立，涂另一侧。

For y < 2x + 1, using (0,0) gives 0 < 1, which is true. Therefore shade the side that contains (0,0). If the boundary passes through (0,0), choose another point such as (0,1) or (1,0).

对于 y < 2x + 1，代入 (0,0) 得 0 < 1，成立。因此涂包含 (0,0) 的一侧。如果边界过 (0,0)，就改选 (0,1) 或 (1,0) 等点。

5. Quick Shading for y-Isolated Inequalities | y 已分离时快速涂色

If the inequality is written as y > mx + c, shade above the line; if it is written as y < mx + c, shade below the line. This is faster than testing a point.

如果不等式写成 y > mx + c，涂直线上方；如果写成 y < mx + c，涂直线下方。这比测试点更快。

Be careful when rearranging. To turn 2y ≥ 4x − 6 into y ≥ 2x − 3, you divide by a positive number, so the sign stays the same. If you multiply or divide by a negative number, the inequality sign must be reversed.

重排时要小心。例如把 2y ≥ 4x − 6 化为 y ≥ 2x − 3，除以正数，不等号方向不变。如果乘以或除以负数，需要把不等号反向。

6. Systems of Inequalities | 不等式组区域

For a system, you must find the region that satisfies every inequality at the same time. Graph each boundary, shade lightly or use arrows, and keep only the common overlap. In A-Level answers it is often better to shade the unwanted region and label the required region with the letter R.

对于不等式组，需要找出同时满足所有不等式的区域。画出每条边界，轻微涂色或使用箭头

Published by TutorHao | Mathematics Revision Series | aleveler.com

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