线性不等式的图形表示

Q: 对于严格不等式，我应该使用实线还是虚线？

对于<或使用虚线，因为直线上的点不满足该不等式。对于≤或≥使用实线，此时直线上的点是有效解。

Edexcel 国际GCSE 进阶纯数学· 3E· 20 分钟阅读

1. 绘制线性不等式的边界直线★★☆☆☆⏱ 5 min

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📘 定义

边界直线

表示线性不等式对应等式形式的直线，将坐标平面划分为两个半平面。≤/≥使用实线（直线上的点满足不等式），</>使用虚线（直线上的点不满足不等式）。

例:

The inequality $2 x + y \leq 4$ has a solid boundary line $2 x + y = 4$ .

要绘制边界直线，首先将不等式整理为 $y = m x + c$ 的形式，或使用截距法：分别令y=0和x=0，求出x截距和y截距。绘制两个点以画出直线，然后根据不等式符号判断直线应为实线还是虚线。

📐 例题

绘制不等式 $3 x - 2 y > 6$ 的边界直线

1
1. 写出不等式对应的等式形式： $3 x - 2 y = 6$
2
1. 求x截距：令 $y = 0$ ，得 $3 x = 6$ → $x = 2$ ，对应点(2, 0)
3
1. 求y截距：令 $x = 0$ ，得 $- 2 y = 6$ → $y = - 3$ ，对应点(0, -3)
4
1. 该不等式使用>（严格不等号），因此绘制连接(2,0)和(0,-3)的虚线，并标注直线方程 $3 x - 2 y = 6$

Exam tip:

务必为所有边界直线标注完整方程：考试中如果遗漏标注会被扣分。

2. 识别并标注可行域★★★☆☆⏱ 6 min

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📘 定义

可行域

坐标平面上同时满足所有给定线性不等式的所有点的集合。

例:

For constraints $x \geq 0$ , $y \geq 0$ , $x + y \leq 3$ , the feasible region is the triangle with vertices at (0,0), (3,0) and (0,3).

要判断需要标注边界直线的哪一侧，选取一个不在直线上的测试点（如果原点(0,0)不在边界直线上，它是最方便的选择）。将测试点的x和y值代入不等式：如果结果为真，则标注包含该测试点的一侧；如果为假，则标注相反的一侧。如果存在多个不等式，可行域是所有约束条件共同覆盖的区域。

📐 例题

标注满足 $x + y \geq 2$ , $x \leq 3$ , $y \leq 4$ 的可行域

1
1. 绘制所有边界直线： $x + y = 2$ （实线，对应≥）、 $x = 3$ （实线，对应≤）、 $y = 4$ （实线，对应≤）
2
1. 用(0,0)测试 $x + y \geq 2$ ： $0 + 0 = 0 \geq 2$ 不成立，因此标注 $x + y = 2$ 不包含(0,0)的一侧
3
1. 代入(0,0)验证 $x \leq 3$ ： $0 \leq 3$ 成立，因此将 $x = 3$ 上包含(0,0)的一侧区域画上阴影
4
1. 代入(0,0)验证 $y \leq 4$ ： $0 \leq 4$ 成立，因此将 $y = 4$ 上包含(0,0)的一侧区域画上阴影
5
1. 阴影重叠的区域即为可行域：按照题目要求用字母R或交叉阴影清晰标注该区域。

3. 求解简单线性规划问题★★★★☆⏱ 6 min

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📘 定义

线性规划

Objective function is usually written as $P = a x + b y$ , where a and b are constants.

该方法用于在一组线性不等式约束条件下，求解线性目标函数的最大值或最小值。

对于爱德思IGCSE进阶纯数考题，目标函数的最优值（最大值或最小值）始终出现在可行域的某一个顶点（角点）上。求解时首先确定可行域的所有顶点，将每个顶点分别代入目标函数，再选出对应所需最大值或最小值的顶点即可。

📐 例题

某工厂生产椅子（产量为x）和桌子（产量为y），约束条件为 $x + 2 y \leq 10$ 、 $x \geq 0$ 、 $y \geq 0$ 。每把椅子的利润为10美元，每张桌子的利润为15美元，求可获得的最大利润。

1
1. 定义目标函数： $P = 10 x + 15 y$
2
1. 绘制可行域，确定其顶点为：(0,0)、(10, 0)、(0,5)
3
1. 将每个顶点分别代入利润函数：
4
$P(0,0) = 10(0) + 15(0) = 0$
5
$P(10,0) = 10(10) + 15(0) = 100$
6
$P(0,5) = 10(0) + 15(5) = 75$
7
1. 最大利润为100美元，对应生产10把椅子、0张桌子。

Exam tip:

即使你可以通过目测估计出最优点，也必须代入所有顶点进行验证：即便你的绘图存在微小误差，也能获得步骤分数。

4. 图形不等式的考试答题规范★★☆☆☆⏱ 3 min

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遵守答题规范是本考点获得满分的关键，评卷人会对标注缺失或阴影不清晰的作答扣分。

Use a ruler to draw all straight boundary lines: freehand lines will lose marks.
Label every boundary line with its full equation.
Use solid lines for ≤/≥ and dashed lines for </>.
Clearly indicate the feasible region, either by shading it or shading the unwanted regions and marking the feasible region with an R.
Show all test point working when justifying your shaded region.

✓ 快速检测

What type of line should you use for the inequality $4 x - 3 y < 12$ ?
- Solid line
- Dashed line
- Either is acceptable
显示答案
Dashed line —
Strict inequalities < or > use dashed lines, as points on the line do not satisfy the inequality.
Where is the optimal point for a linear objective function located?
- At the centre of the feasible region
- At a vertex of the feasible region
- Along any edge of the feasible region
显示答案
At a vertex of the feasible region —
For all linear programming problems in this syllabus, the optimal value is always at a corner vertex of the feasible region.

5. 常见陷阱

错误做法:

Using a solid line for strict inequalities (< or >)

原因:

Strict inequalities do not include points on the boundary line, so solid lines incorrectly imply points on the line are valid solutions.

正确做法:

Use a dashed line for < and >, solid line only for ≤ and ≥.

错误做法:

Shading the wrong side of the boundary line because you didn't test a point

原因:

Rearranging inequalities incorrectly can lead to reversing the inequality sign by mistake, leading to shading the wrong region.

正确做法:

Always test a point not on the boundary line (preferably (0,0) if possible) to confirm which side to shade.

错误做法:

Forgetting to label boundary lines with their equations

原因:

Exam mark schemes explicitly award marks for correctly labelled boundary lines; missing labels lead to lost marks even if the line is drawn correctly.

正确做法:

Write the full equation of each boundary line next to the line as soon as you draw it.

错误做法:

Calculating the optimal point by eye instead of substituting all vertices

原因:

Slight inaccuracies in graph drawing can make the optimal vertex look closer to the maximum/minimum than it is, leading to incorrect answers.

正确做法:

List all vertices of the feasible region, substitute each into the objective function, and select the value that meets the requirement (max or min).

错误做法:

Forgetting non-negativity constraints (x ≥ 0, y ≥ 0) in linear programming problems

原因:

Most real-world linear programming problems involve quantities that cannot be negative, so omitting these constraints expands the feasible region incorrectly.

正确做法:

Always check if the problem context (number of items, time, money) requires x and y to be non-negative, and add these constraints explicitly.

6. 速查表

Task	Step-by-Step Action
Draw boundary line	Write equality form; 2. Find intercepts; 3. Draw solid (≤/≥) or dashed (</>); 4. Label equation
Shade correct region	Pick test point not on line; 2. Substitute into inequality; 3. Shade side where test point works; 4. Overlap for multiple constraints
Solve linear programming	Define objective function; 2. Identify all feasible region vertices; 3. Substitute each vertex into objective; 4. Select max/min value

7. 常见问题

对于严格不等式，我应该使用实线还是虚线？

对于<或>使用虚线，因为直线上的点不满足该不等式。对于≤或≥使用实线，此时直线上的点是有效解。

我如何判断需要标注边界直线的哪一侧？

选取一个不在直线上的测试点（如果原点(0,0)不在边界上，它是最方便的选择）。将其坐标代入不等式：如果结果为真，则标注包含该点的一侧；如果为假，则标注相反的一侧。

线性目标函数的最优点会位于哪里？

对于本考纲中的所有线性规划问题，最优值（最大值或最小值）始终出现在可行域的顶点（角点）处。

深入阅读

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下一步

Now that you have mastered graphical representation of linear inequalities and basic linear programming, you are ready to move on to more advanced inequality topics in the Edexcel IGCSE Further Pure Math syllabus. This topic forms the foundation for applied problem-solving questions that often combine with coordinate geometry and algebraic manipulation in exam papers. Make sure you practice past paper questions to perfect your graph drawing speed and accuracy, as presentation is key to scoring full marks in this topic. You should also review algebraic methods for solving systems of inequalities to cross-verify your graphical solutions and avoid errors in high-pressure exam settings.