A-Level Maths: Linear Inequalities and the Shaded Regions | 线性不等式表示的平面区域

📚 A-Level Maths: Linear Inequalities and the Shaded Regions | 线性不等式表示的平面区域

Linear inequalities are mathematical statements that relate two expressions using <, >, ≤ or ≥. When the variables represent coordinates, each inequality defines a region in the coordinate plane. This topic is central to linear programming and graphical methods in A-Level mathematics.

线性不等式是通过 <、>、≤ 或 ≥ 比较两个表达式的数学关系。当变量表示坐标时，每个不等式都在坐标平面上定义了一个区域。这一主题是 A-Level 数学中线性规划和图解法的核心内容。

1. Single Linear Inequality | 单个线性不等式的表示

A linear inequality in two variables x and y can be written in the general form ax + by < c, ax + by > c, ax + by ≤ c or ax + by ≥ c, where a, b and c are constants, and a and b are not both zero.

关于变量 x 和 y 的线性不等式通常写成：ax + by < c、ax + by > c、ax + by ≤ c 或 ax + by ≥ c，其中 a、b、c 为常数，且 a 和 b 不同时为零。

For example, y > 2x + 1 is a linear inequality. Its graph is the set of all points (x, y) that make the inequality true. Because the inequality is linear, this set is always a half-plane (one side of a straight line).

例如，y > 2x + 1 是一个线性不等式。它的图像是使不等式成立的所有点 (x, y) 的集合。由于不等式是线性的，这个集合总是一个半平面（位于一条直线的一侧）。

2. Drawing the Boundary Line | 绘制边界线

To sketch the region, we first draw the corresponding boundary line by replacing the inequality sign with an equals sign. For y > 2x + 1, the boundary line is y = 2x + 1.

要画出区域，我们首先将不等式中的不等号改为等号，得到对应的边界线。对于 y > 2x + 1，边界线是 y = 2x + 1。

The line itself is included in the region only if the inequality is non-strict. Use a solid line for ≤ or ≥, and a dashed line for < or >.

只有当不等式是“非严格”的（≤ 或 ≥）时，直线本身才属于区域。画图时，≤ 或 ≥ 用实线，< 或 > 用虚线。

y = 2x + 1 → solid if y ≥ 2x + 1, dashed if y > 2x + 1

y = 2x + 1 → 若 y ≥ 2x + 1 用实线，若 y > 2x + 1 用虚线

To draw the line accurately, find two points that satisfy the equation. For y = 2x + 1, choose x = 0 (y = 1) and x = 1 (y = 3). Plot these points and join them with a straight line.

为了精确绘制直线，可以找出满足方程的两个点。对于 y = 2x + 1，可取 x = 0（得 y = 1）和 x = 1（得 y = 3）。描出这两点并用直线连接。

3. Choosing the Correct Half-Plane | 确定正确的半平面

Once the boundary line is drawn, we need to decide which side of the line represents the solution. A reliable method is to test a point that is not on the boundary line.

边界线画好后，需要判断直线的哪一侧代表解集。一种可靠的方法是测试一个不在边界线上的点。

For example, test the origin (0, 0) whenever it is not on the line. Substitute x = 0 and y = 0 into the inequality y > 2x + 1:

例如，只要原点 (0, 0) 不在直线上，就可以测试它。将 x = 0、y = 0 代入不等式 y > 2x + 1：

0 > 2(0) + 1 → 0 > 1 (false)

Because the statement is false, the origin is not in the solution region. Therefore we shade the side opposite to the origin. If it were true, we would shade the side containing the origin.

因为该语句不成立，所以原点不在解区域内。因此我们应画出原点相对一侧的区域。若语句成立，则画出包含原点的那一侧。

Always choose a simple test point, preferably (0, 0).
If the boundary passes through the origin, choose another point such as (1, 0) or (0, 1).

始终选择简单的测试点，最好用 (0, 0)。
若边界线过原点，则选择其他点，如 (1, 0) 或 (0, 1)。

4. Inequalities x ≥ 0 and y ≥ 0 | 不等式 x ≥ 0 与 y ≥ 0

Special linear inequalities such as x ≥ 0 and y ≥ 0 are common in linear programming. The boundary line x = 0 is the y-axis; the region x ≥ 0 is the right side of the y-axis. Similarly, y ≥ 0 is the upper side of the x-axis.

特殊的不等式 x ≥ 0 和 y ≥ 0 在线性规划中很常见。边界线 x = 0 就是 y 轴；区域 x ≥ 0 是 y 轴的右侧。类似地，y ≥ 0 是 x 轴的上方。

These inequalities often restrict the feasible region to the first quadrant, which is realistic in many business and economic problems where quantities cannot be negative.

这些不等式常常把可行域限制在第一象限，这在许多商业和经济问题中是合理的，因为数量通常不能为负。

5. Regions from Multiple Inequalities | 多个不等式定义的区域

In examination questions, you are usually asked to identify the region that satisfies several linear inequalities simultaneously. This region is the intersection of all the individual half-planes.

在考试题中，通常会要求你找出同时满足多个线性不等式的区域。这个区域是各个半平面的交集。

For instance, consider:

例如，考虑：

y ≥ 2x + 1, y ≤ 4 − x, x ≥ 0, y ≥ 0

Draw each boundary line, decide which side satisfies each inequality, and then shade only the region that satisfies all of them. This common shaded region is called the feasible region.

先画出每条边界线，判断每个不等式所满足的一侧，然后只对同时满足所有不等式的区域进行阴影填充。这个公共阴影区域称为可行域。

Feasible region = set of points satisfying every inequality

可行域 = 满足所有不等式的点集

6. Worked Example: Finding the Shaded Region | 例题：求阴影区域

Let us shade the region defined by the inequalities:

我们来画出由下列不等式定义的区域：

y ≥ 2x + 1, y < 5 − 2x, x ≥ 0

Step 1: Draw the boundary lines. For y = 2x + 1, use a solid line because the inequality is ≥. For y = 5 − 2x, use a dashed line because the inequality is <. The line x = 0 is the y-axis, drawn solid because x ≥ 0.

第一步：绘制边界线。y = 2x + 1 因不等式为 ≥ 用实线；y = 5 − 2x 因不等式为 < 用虚线；x = 0 是 y 轴，因 x ≥ 0 用实线。

Step 2: Test the origin (0, 0) for each inequality. For y ≥ 2x + 1, 0 ≥ 1 is false, so the region is above that line. For y < 5 − 2x, 0 < 5 is true, so the region is below that line. For x ≥ 0, the region is to the right of the y-axis.

第二步：对每个不等式测试原点 (0, 0)。对于 y ≥ 2x + 1，0 ≥ 1 不成立，所以区域在该直线上方；对于 y < 5 − 2x，0 < 5 成立，所以区域在该直线下方；对于 x ≥ 0，区域在 y 轴右侧。

Step 3: Shade the region that satisfies all three conditions. This region is a triangle whose vertices are the intersection points of the boundary lines.

第三步：画出同时满足三个条件的区域。该区域是一个三角形，其顶点是边界线的交点。

7. Finding Vertices of the Region | 求区域的顶点

Vertices are the corner points of the feasible region. They are found by solving the equations of the intersecting boundary lines simultaneously.

顶点是可行域的角点。通过联立求解相交边界线的方程，可以得到顶点坐标。

In the example above, the vertices are:

在上述例子中，顶点为：

Intersection of y = 2x + 1 and the y-axis (x = 0): substitute x = 0 gives y = 1, so (0, 1).
Intersection of y = 5 − 2x and the y-axis (x = 0): y = 5, so (0, 5).
Intersection of y = 2x + 1 and y = 5 − 2x: solve 2x + 1 = 5 − 2x.

y = 2x + 1 与 y 轴（x = 0）的交点：代入 x = 0 得 y = 1，即 (0, 1)。
y = 5 − 2x 与 y 轴（x = 0）的交点：y = 5，即 (0, 5)。
y = 2x + 1 与 y = 5 − 2x 的交点：解 2x + 1 = 5 − 2x。

The third vertex is found as follows:

第三个顶点计算如下：

2x + 1 = 5 − 2x → 4x = 4 → x = 1

Substituting x = 1 into y = 2x + 1 gives y = 3. So the vertex is (1, 3).

将 x = 1 代入 y = 2x + 1，得 y = 3。因此顶点为 (1, 3)。

8. Using Inequalities in Linear Programming | 线性规划中的应用

Linear programming uses feasible regions to solve optimisation problems. An objective function, such as P = 3x + 2y, is maximised or minimised subject to a set of linear constraints.

线性规划使用可行域来解决最优化问题。目标函数（如 P = 3x + 2y）在一组线性约束下求最大值或最小值。

The key theorem states that the optimum value of a linear objective function over a convex feasible region occurs at a vertex, or along an entire edge if the objective line is parallel to that edge. Therefore, evaluating the objective function at each vertex is usually sufficient.

一个重要定理指出：在线性目标函数中，最优值在凸可行域的顶点处取得；若目标直线与某条边平行，也可能在整条边上取得。因此，通常只需在每个顶点处计算目标函数值即可。

Vertex \| 顶点	P = 3x + 2y
(0, 1)	3(0) + 2(1) = 2
(0, 5)	3(0) + 2(5) = 10
(1, 3)	3(1) + 2(3) = 9

Here P is maximised at (0, 5) with maximum value 10. This simple process demonstrates the power of graphical methods.

这里 P 在 (0, 5) 处达到最大值，最大值为 10。这一简单过程体现了图解法的威力。

9. Common Mistakes and Exam Tips | 常见错误与应试技巧

Students often make mistakes when deciding whether the boundary line should be solid or dashed, or when shading the wrong side of the line. Always test a point after drawing the line.

学生在判断边界线应使用实线还是虚线，或者画错直线哪一侧时，常常犯错。画完直线后务必测试一个点。

Read the inequality carefully: < and > are strict, so the boundary line is dashed; ≤ and ≥ are non-strict, so the boundary line is solid.
When shading multiple inequalities, label the feasible region with an R or a clear shade pattern.
If the feasible region is unbounded, state this clearly and check whether a maximum or minimum exists.

仔细阅读不等式：< 和 > 是严格的，边界线用虚线；≤ 和 ≥ 是非严格的，边界线用实线。
画多个不等式时，用字母 R 或清晰的阴影标出可行域。
若可行域无界，应明确说明，并判断最大值或最小值是否存在。

10. Conclusion | 总结

Linear inequalities provide a visual method for solving systems of constraints. By drawing boundary lines, testing a point, and shading the correct half-plane, you can accurately represent feasible regions. This skill is essential for linear programming and appears frequently in A-Level examinations.

线性不等式为求解约束系统提供了直观的方法。通过绘制边界线、测试一个点并正确覆盖半平面，你可以准确表示可行域。这一技能对于线性规划至关重要，在 A-Level 考试中频繁出现。

Practice with a variety of inequalities, including those with negative coefficients, to build confidence and speed. Remember to always check your shaded region against the original inequalities.

建议练习各种不等式，包括含负系数的情形，以增强信心和速度。请始终用原始不等式检验你的阴影区域。

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A-Level Maths: Linear Inequalities and the Shaded Regions | 线性不等式表示的平面区域