📚 A-Level Maths: Representing Inequalities on Graphs | A-Level 数学：不等式在图像上的表示方法
Inequalities are a fundamental part of A-Level Mathematics, and representing them graphically provides a powerful visual tool for understanding and solving problems. Unlike equations, which correspond to a single line or curve, inequalities correspond to entire regions on a graph. Mastery of this topic is essential for success in topics like linear programming and optimization.
不等式是 A-Level 数学中的一个基础部分，用图像表示不等式为我们理解和解决问题提供了一个强大的可视化工具。与对应单一直线或曲线的方程不同，不等式对应的是图上的整个区域。掌握这个知识点对于后续在线性规划和优化等主题中取得成功至关重要。
1. Reviewing Lines and Curves | 回顾直线与曲线
Before we shade regions, we must be comfortable graphing the boundary lines. The equation y = mx + c represents a straight line. The graph of a quadratic equation, like y = ax² + bx + c, is a parabola. These graphs act as the “fences” that separate the coordinate plane into distinct regions.
在给区域着色之前，我们必须熟练地画出边界线（即方程对应的图像）。方程 y = mx + c 表示一条直线。二次方程，如 y = ax² + bx + c，的图像是抛物线。这些图像就像“篱笆”，将坐标平面划分成不同的区域。
To plot a straight line, you can find the x- and y-intercepts. For example, for the line 2x + y = 4, substituting x = 0 gives y = 4, and substituting y = 0 gives x = 2. Plot the points (0, 4) and (2, 0) and draw a straight line through them.
要绘制一条直线，你可以找到它与 x 轴和 y 轴的交点。例如，对于直线 2x + y = 4，代入 x = 0 得到 y = 4，代入 y = 0 得到 x = 2。在图上标出点 (0, 4) 和 (2, 0)，并通过它们画一条直线。
The table below shows the intercepts for some common linear equations.
下表展示了一些常见线性方程的截距。
| Equation 方程 | x-intercept x截距 | y-intercept y截距 |
| 2x + y = 4 | 2 | 4 |
| y = x – 3 | 3 | -3 |
| x = 2 | 2 | 无 None |
2. Solid vs. Dashed Lines | 实线 vs. 虚线
The first crucial decision when drawing an inequality is the type of line to use. If the inequality is strict (using < or >), the boundary line itself is NOT included in the solution set. We represent this with a dashed line.
绘制不等式时，第一个关键步骤是决定使用什么类型的线。如果不等式是严格的（使用 < 或 >），那么边界线本身不包含在解集中。我们用虚线来表示这种情况。
If the inequality is non-strict (using ≤ or ≥), the boundary line IS included. We represent this with a solid line. This distinction is critical because misrepresenting it loses marks and changes the geometric meaning of the solution.
如果不等式是非严格的（使用 ≤ 或 ≥），则包含边界线。我们用实线来表示这种情况。这个区别至关重要，因为画错不仅会扣分，还会改变解在几何意义上的正确性。
For example, the inequality y > 2x + 1 requires a dashed line, whereas y ≥ 2x + 1 requires a solid line.
例如，不等式 y > 2x + 1 需要使用虚线，而 y ≥ 2x + 1 则需要使用实线。
3. Plotting a Linear Inequality | 绘制线性不等式
To shade the correct region for an inequality like y < 2x + 3, follow these steps. First, draw the line y = 2x + 3 as a dashed line because the inequality is strict. Second, choose a test point not on the line; the origin (0, 0) is usually the best. Substitute the point into the inequality.
要为像 y < 2x + 3 这样的不等式选择正确的阴影区域，请遵循以下步骤。首先，由于不等式是严格的，将 y = 2x + 3 绘制为虚线。其次，选择一个不在直线上（即边界线）上的测试点；原点 (0, 0) 通常是最好的选择。将该点代入不等式。
For the point (0, 0), we substitute x = 0 and y = 0 into y < 2x + 3, giving 0 < 3. This is true. Therefore, we shade the side of the line that contains the origin.
对于点 (0, 0)，我们将 x = 0 和 y = 0 代入 y < 2x + 3，得到 0 < 3。这是正确的。因此，我们给包含原点的这一侧区域涂上阴影。
If the statement were false, we would shade the opposite side. This test works for any inequality, provided the chosen point is not on the line.
如果该语句为假，我们就给相反的一侧涂阴影。只要所选点不在线上，这个测试对任何不等式都适用。
4. The Origin Test (Shortcut) | 使用原点测试法（快捷方式）
The (0, 0) test is a shortcut that can save valuable time. For a line in the form y = mx + c or ax + by = c, simply substituting zero often reveals which side is which. However, be extremely careful if the boundary line passes directly through the origin.
(0, 0) 测试是一种可以节省宝贵时间的快捷方法。对于 y = mx + c 或 ax + by = c 形式的直线，通常只需代入零即可判断哪边是解区域。但是，如果边界线正巧通过原点，则需要格外小心。
If the line passes through the origin, such as y > 3x, the origin test cannot be used because 0 > 0 is false, yet the origin is on the line. In this case, choose another convenient point, such as (0, 1) or (1, 0).
如果直线通过原点，例如 y > 3x，则无法使用原点测试，因为 0 > 0 为假，但原点又恰好在线上。在这种情况下，请选择另一个方便的点，例如 (0, 1) 或 (1, 0)。
Let’s quickly verify this with the inequality y + 2x ≥ 0. Passing through the origin, we test (0, 1): 1 + 0 = 1 ≥ 0, which is true. So the region containing (0, 1) is shaded.
让我们用不等式 y + 2x ≥ 0 快速验证一下。因为它经过原点，我们测试 (0, 1)：1 + 0 = 1 ≥ 0，这是正确的。因此，包含 (0, 1) 的区域就是阴影区域。
5. Multiple Inequalities and the Feasible Region | 多个不等式与可行区域
A classic A-Level question asks for the region that satisfies multiple inequalities simultaneously. This requires us to find the intersection of all shaded regions. Let’s consider the three inequalities: y < x + 2, y ≥ 0, and x ≥ 0.
一个经典的 A-Level 题目要求找出同时满足多个不等式的区域。这要求我们找出所有阴影区域的交集。让我们考虑三个不等式：y < x + 2, y ≥ 0, 和 x ≥ 0。
First, draw the line y = x + 2 as a dashed line and shade below it. Then, draw the line y = 0 (the x-axis) as a solid line and shade above it. Finally, draw the line x = 0 (the y-axis) as a solid line and shade to the right of it.
首先，将直线 y = x + 2 绘制为虚线，并在其下方涂阴影。然后，将直线 y = 0（即 x 轴）绘制为实线，并在其上方涂阴影。最后，将直线 x = 0（即 y 轴）绘制为实线，并在其右侧涂阴影。
The “triple-shaded” region, where all three conditions overlap, is the solution set. In many exam questions, this region is labeled with the letter R.
“三层阴影”重合的区域，即三个条件重叠的区域，就是解集。在许多考题中，这个区域会用字母 R 标记。
This overlapping region forms a polygon, often a triangle. Finding the vertices of this polygon is a common requirement, as they are often used to find maximum or minimum values in linear programming problems.
这个重叠区域形成一个多边形，通常是三角形。找出这个多边形的顶点是一个常见要求，因为在线性规划问题中，它们通常用于求最大值或最小值。
6. Plotting Quadratic Inequalities | 绘制二次不等式
Inequalities involving quadratic terms, such as y ≥ x² – 1, are represented using the graph of the parabola. First, plot the curve y = x² – 1. Since the inequality is ≥, the curve itself is drawn with a solid line.
涉及二次项的不等式，例如 y ≥ x² – 1，用抛物线的图像来表示。首先，画出曲线 y = x² – 1。由于不等式是 ≥，曲线本身要用实线绘制。
To determine the region, substitute a test point that is not on the curve. Using the origin (0, 0) in this case yields 0 ≥ -1, which is true. Therefore, the required region is inside the parabola (above the curve).
为了确定区域，代入一个不在曲线上的测试点。在这种情况下，将原点 (0, 0) 代入得到
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