📚 Graphical Inequalities – Shading the Unwanted Region | 图形不等式 – 阴影区域划分
In IGCSE Mathematics, being able to represent inequalities on a graph is a fundamental skill that links algebra with geometry. Edexcel's topic 18.4 focuses specifically on drawing linear inequalities and shading the region that does not satisfy the condition – often called the 'unwanted region'. Mastering this topic not only helps you solve single inequalities but also prepares you to tackle systems of inequalities where you must identify a feasible region bounded by several lines. Let's explore every step, from drawing the boundary line to shading the correct side and dealing with multiple constraints.
在IGCSE数学中，将不等式表示在图形上是一项连接代数与几何的基本技能。Edexcel 第18.4专题着重于绘制线性不等式并对不满足条件的区域进行阴影处理——通常称为「不需要的区域」。掌握这一主题不仅能帮助你解决单个不等式，还能为处理多个不等式所围成的可行区域做好准备。让我们从绘制边界线开始，逐步到正确的阴影侧，再到处理多个约束条件。
1. What Are Graphical Inequalities? | 什么是图形不等式？
ate is strictly greater than twice the x-coordinate plus 1. Your task is to draw the line y = 2x + 1, decide whether it is solid or dashed, and then shade the region that fails the inequality.
图形不等式涉及在坐标网格上着色，以显示哪些点满足一个或多个关于 x 和 y 的线性不等式。不同于数轴，你现在面对的是整个笛卡尔平面。例如，y > 2x + 1 表示所有满足 y 坐标严格大于 2 倍 x 坐标加 1 的点。你的任务是画出直线 y = 2x + 1，判断它是实线还是虚线，然后给不满足不等式的区域涂上阴影。
2. Drawing the Boundary Line | 绘制边界线
eating the inequality as an equation: replace >, <, ≥ or ≤ with an equals sign. For y ≤ 3x – 2, draw the line y = 3x – d the line across the whole grid unless the question specifies a restricted domain. This line divides the plane into two halves.
首先将不等式当作方程处理：将 >, <, ≥ 或 ≤ 替换为等号。对于 y ≤ 3x – 2，绘制直线 y = 3x – 2。至少找出两个点，通常是 y 截距和利用斜率的另一点。准确描点并用直尺画线。除非题目限定了定义域，否则直线要贯穿整个网格。这条线将平面分成两个半平面。
3. Solid vs Dashed Lines | 实线与虚线
Use a solid line uality: ≤ or ≥. This shows that points on the line are included in the solution set. Use a dashed (or dotted) line for strict inequalities: < or >. The dashed line indicates the boundary itself is not part of the region. This small distinction often carries marks in exams, so be precise.
当不等式包含等号（≤ 或 ≥）时使用实线。这表示直线上的点属于解集。对于严格不等式（< 或 >），使用虚线。虚线表明边界本身不属于所求区域。这个小区别在考试中常常占分，因此务必准确。
4. Choosing Which Side to Shade | 选择哪一侧着色
Once the line is drawn, pick a test point not on the line. The origin (0, 0) is ideal if the line does not pass through it. Substitute the coordinates into the original inequality. If the test point satisfies the inequality, it lies in the 'wanted' region – so you shade the other side. If it does not satisfy, shade the side containing the test point. Edexcel IGCSE requires you to shade the unwanted region, so always think: which side fails the test?
画出直线后，选择一个不在直线上的测试点。如果直线不经过原点，(0, 0) 是最佳选择。将坐标代入原不等式。若测试点满足不等式，则它位于「想要的」区域——因此你应着色另一侧。若不满足，则着色包含测试点的一侧。Edexcel IGCSE 要求对不需要的区域进行着色，所以始终思考：哪一侧未通过测试？
5. Shading the Unwanted Region – Single Inequality | 为单个不等式绘制不需要的区域
For y > –x + 1, draw a dashed line through (0,1) and (1,0). Test (0,0): 0 > –0 + 1 is false. So (0,0) is in the unwanted region. Shade the entire half-plane containing (0,0). Leave the opposite side unshaded; this white area (together with the dashed line if it were inclusive) represents the solution set. Always label your shaded region as 'R' if requested.
对于 y > –x + 1，绘制经过 (0,1) 和 (1,0) 的虚线。测试 (0,0)：0 > –0 + 1 为假。因此 (0,0) 属于不需要的区域。着色包含 (0,0) 的整个半平面。另一侧留白；这片白色区域（若包含等号则连同直线）代表解集。如果题目要求，始终将着色区域标为「R」。
6. Dealing with Vertical and Horizontal Lines | 处理垂直与水平线
Inequalities like x < 4 or y ≥ – so (0,0) is in the wanted region. Shade the side where x > 4 (the unwanted side). For y ≥ –2, draw a solid horizontal line at y = –2. Testing (0,0) gives 0 ≥ –2, true, so shade below the line (where y < –2). Always apply the same logic: test, then shade the opposite if the test point works.
形如 x < 4 或 y ≥ –2 的不等式很常见。对于 x < 4，在 x = 4 处画一条虚线。测试 (0,0)：0 < 4 为真，因此 (0,0) 在想要的区域。着色 x > 4 的一侧（不需要的一侧）。对于 y ≥ –2，在 y = –2 处画一条实水平线。测试 (0,0) 得到 0 ≥ –2 为真，所以着色直线下方（y < –2 的区域）。始终运用同样逻辑：测试，若测试点满足则着色对侧。
7. Systems of Inequalities – The Feasible Region | 不等式组 – 可行区域
When several inequalities must hold at once, the solution is the intersection of the individual wanted regions, i.e. the area that remains unshaded after shading all unwanted parts. Each inequality contributes its own unwanted half-plane. The unshaded region common to all is called the feasible region. It is often a polygon whose vertices are points of intersection of the boundary lines. Exam questions may ask you to find these vertices or the maximum/minimum value of an expression within this region.
当多个不等式必须同时成立时，解为各个想要区域的交集，也就是在对所有不需要的部分着色后仍然留白的区域。每个不等式贡献其自身的不需要的半平面。所有不等式共同留下的未着色区域称为可行区域。它通常是一个多边形，其顶点是边界线的交点。考题可能要求你找出这些顶点，或在该区域内求某表达式的最大/最小值。
8. Step-by-Step Method for Multiple Inequalities | 多不等式的逐步解法
Begin by drawing all boundary lines using the correct line style. Next, for each inequality choose a test point and determine which side is unwanted; shade it lightly so that overlapping shading becomes darker. Alternatively, you can shade each unwanted region with lines or a distinct pattern. Once all are drawn, the region left unshaded (or with the lightest shading) is your feasible region. Finally, clearly outline this region and label it R. Remember: the feasible region includes any solid boundary segments that border it.
首先用正确的线型画出所有边界线。接着，对每个不等式选取一个测试点，确定哪一侧是不需要的；轻轻着色，使重叠的阴影变深。或者，你可以用斜线或不同图案对每个不需要的区域进行着色。当所有不等式都处理完后，未被着色（或着色最浅）的区域就是你的可行区域。最后，清晰勾勒该区域并标注为 R。记住：可行区域包括与其相邻的实线边界段。
9. Worked Example – Two Inequalities | 示例讲解 – 两个不等式
Problem: Shade the unwanted region for the inequalities y ≤ 2x + 3 and y > –x – 1.
解答：
Step 1: Draw y = 2x + 3 as a solid line (passing through (0,3) and (–1.5,0)). Test (0,0): 0 ≤ 2(0)+3 true, so wanted side. Shade the region above the line (where y > 2x+3).
Step 2: Draw y = –x – 1 as a dashed line (through (0,–1) and (–1,0)). Test (0,0): 0 > –0 –1 true, so wanted side. Shade the region below the line (where y < –x–1).
lap near the lower-right is the feasible region. You can find its vertex by solving 2x+3 = –x–1, giving x = –4/3, y = 1/3. Label the clear region R.
步骤1：将 y = 2x + 3 画成实线（经过 (0,3) 和 (–1.5,0)）。测试 (0,0)：0 ≤ 2(0)+3 成立，故为想要侧。着色直线上方区域（y > 2x+3 部分）。
步骤2：将 y = –x – 1 画成虚线（经过 (0,–1) 和 (–1,0)）。测试 (0,0)：0 > –0 –1 成立，想要侧。着色直线下方区域（y < –x–1 部分）。
步骤3：右下方的未重叠部分为可行区域。通过解 2x+3 = –x–1 可得顶点 x = –4/3，y = 1/3。将空白区域标为 R。
10. Finding Vertices and Optimisation | 寻找顶点与优化
Once the feasible region is identified, you may be asked to find the coordinates of its vertices. Solve the pair of simultaneous equations for each intersection of boundary lines that are part of the region. If the question involves maximising or minimising an expression like 3x + 5y, evaluate this expression at each vertex. The highest or lowest value gives the optimum. This links directly to linear programming concepts, which appear in higher-tier IGCSE papers.
确定可行区域后，你可能需要求其顶点坐标。对构成区域边界的每一对边界线求交点，解方程组即可。如果题目涉及最大化或最小化 3x + 5y 这样的表达式，要在每个顶点处计算该表达式。最大值或最小值即是最优解。这直接联系到线性规划的概念，在 IGCSE 高级卷中会出现。
11. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Many students shade the wanted region instead of the unwanted one, losing easy marks. Always confirm whether the question says 'shade the region that satisfies' or 'shade the unwanted region' – Edexcel normally uses the latter. Another common error is using a solid line for < or >, or forgetting to test when the line passes through the origin; in that case, pick (0,1) or (1,0). Also, double-check the direction of vertical line inequalities: x < 5 shades the right side, not the left.
许多学生会着色的想要区域而非不需要的区域，从而丢失简单分数。务必确认题目是「shade the region that satisfies」还是「shade the unwanted region」——Edexcel 通常使用后者。另一个常见错误是对 < 或 > 使用实线，或当直线经过原点时忘记测试；此时可选取 (0,1) 或 (1,0)。此外，要仔细检查垂直不等式的方向：x < 5 应着色右侧，而非左侧。
12. Summary and Final Tips | 总结与最后提示
Graphical inequalities tie together coordinate geometry and algebraic reasoning. Remember the golden rule: draw accurately, use solid/dashed lines correctly, test a point, and shade the unwanted side. For systems, shade each unwanted region lightly; the final unshaded area is your answer. Practice with past Edexcel IGCSE questions, and always label your region with R if required. With these skills, you can confidently tackle any inequalities question and also lay the groundwork for further topics like linear programming.
图形不等式将坐标几何与代数推理紧密结合。记住黄金法则：准确绘图、正确使用实/虚线、测试一点、并对不需要的一侧着色。对于不等式组，轻轻涂绘每个不需要的区域；最终未着色的部分就是答案。利用 Edexcel IGCSE 历年真题进行练习，如果需要，始终将区域标为 R。掌握这些技能后，你不仅能自信应对任何不等式问题，还能为后续的线性规划等内容打下坚实基础。
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