📚 Inequalities on Graphs | 图上的不等式
Interpreting inequalities visually using graphs is a fundamental technique in A-Level Mathematics. It involves sketching the boundary of the inequality and shading the region that satisfies the condition. This topic consolidates your understanding of coordinate geometry, functions and algebraic manipulation, and it is essential for tackling more advanced concepts such as linear programming and modelling constraints.
利用图形直观地解释不等式是 A-Level 数学中的一项基本技巧。它包括勾勒不等式的边界并对满足条件的区域进行着色。这个主题巩固了你对坐标几何、函数和代数运算的理解，对于解决更高级的概念（如线性规划与约束建模）至关重要。
1. What a Graphical Inequality Represents | 图形不等式表示什么
A graphical inequality does not describe a single point; it describes an entire region of the Cartesian plane. The boundary curve or line divides the plane into two parts: one part where the inequality is true and one part where it is false. When the boundary is a straight line, the two parts are called half-planes. When the boundary is a curve, such as a parabola or a circle, one part is the inside of the curve and the other is the outside.
图形不等式描述的不是一个点，而是笛卡尔平面上的一整片区域。边界曲线或边界直线把平面分成两部分：一部分使不等式成立，另一部分使不等式不成立。当边界是直线时，这两部分称为半平面；当边界是曲线（如抛物线或圆）时，一部分是曲线内部，另一部分是曲线外部。
by every point strictly above the line y = 2x + 1, and by no point on or below that line.
例如，不等式 y > 2x + 1 被直线 y = 2x + 1 严格上方的每一个点满足，而直线上的点和直线下方的点都不满足它。
y > 2x + 1 ⇔ the region strictly above the line y = 2x + 1
Because the solution set contains infinitely many points, examiners normally expect you to indicate the region clearly with shading or with a letter label such as R, rather than listing coordinates.
由于解集包含无穷多个点，考官通常期望你用阴影或字母标记（例如 R）来清楚地标出该区域，而不是列出坐标。
2. Solid and Dashed Boundaries | 实线边界与虚线边界
the boundary is included and must be drawn as a solid line. For strict inequalities using < or >, the boundary is excluded and must be drawn as a dashed or dotted line.
画任何不等式时，第一个决定是边界本身是否属于解集。对于使用 ≤ 或 ≥ 的包含型不等式，边界被包含在内，必须画成实线。对于使用 < 或 > 的严格不等式，边界被排除在外，必须画成虚线或点线。
| Inequality | 不等式 | Boundary line | 边界线型 | Boundary included? | 边界是否包含 |
| y ≥ x + 2 | Solid | 实线 | Yes | 是 |
| y ≤ x + 2 | Solid | 实线 | Yes | 是 |
| y > x + 2 | Dashed | 虚线 | No | 否 |
| y < x + 2 | Dashed | 虚线 | No | 否 |
A common error is to draw a solid line for a strict inequality, or to shade a region and then leave the boundary dashed even though the inequality is inclusive. Both mistakes lose marks because they change the mathematical meaning of the answer.
一个常见错误是把严格不等式的边界画成实线，或者明明是不等式包含边界却仍然把边界留成虚线。这两种错误都会改变答案的数学含义，因而会被扣分。
3. Shading the Correct Region | 给正确的区域着色
Once the boundary has been drawn correctly, you must shade the side that satisfies the inequality. It is conventional to use light shading or hatching, and to label the region, for example with the letter R, so that the examiner can see exactly which area you mean.
边界画好之后，你必须给满足不等式的那一侧着色。通常使用浅色阴影或斜线阴影，并给该区域加标记（例如字母 R），这样考官就能清楚看到你指的是哪一片区域。
If a question asks you to find the region that is not required, some students find it easier to shade the unwanted region and then label the required region clearly. Whatever convention you choose, the labelling must be unambiguous.
如果题目要求你找出不需要的区域，有些同学会觉得先把不需要的区域涂掉、再清楚标出需要的区域更容易。无论采用哪种约定，标记都必须明确无歧义。
Keep your shading light. Heavy shading makes it difficult to see whether a boundary is solid or dashed, and it also hides the vertices where two boundaries meet, which you often need later in the question.
阴影要画得轻一些。涂得太重会让人看不清边界是实线还是虚线，也会遮住两条边界的交点顶点，而这些顶点在后续问题中往往要用到。
4. The Test Point Method | 检验点法
When you are unsure which side of a boundary to shade, use a test point that is not on the boundary. The origin (0, 0) is almost always the easiest choice, provided the boundary does not pass through it. Substitute the coordinates of the test point into the inequality and see whether the statement is true.
当你无法确定该给边界的哪一侧着色时，使用一个不在边界上的检验点。原点 (0, 0) 几乎总是最容易的选择，只要边界不经过它即可。把检验点的坐标代入不等式，看看命题是否成立。
For instance, to shade y < 3x - 4, substitute (0, 0): the left-hand side is 0 and the right-hand side is -4, so 0 < -4 is false. The origin therefore lies on the wrong side, and you must shade the other side of the line.
例如，要给 y < 3x - 4 着色，代入 (0, 0)：左边是 0，右边是 -4，所以 0 < -4 不成立。因此原点位于错误的一侧，你必须给直线的另一侧着色。
Test point (0, 0): 0 < 3 × 0 - 4 ⇔ 0 < -4 is false
If the boundary passes through the origin, choose a different simple point, such as (1, 0) or (0, 1). Always state the test point you used and the conclusion you drew from it, because that reasoning earns method marks.
如果边界经过原点，就换一个简单的点，例如 (1, 0) 或 (0, 1)。一定要写出你使用的检验点以及由此得出的结论，因为这一推理过程可以获得方法分。
5. Straight-Line Boundaries and Rearranging | 直线边界与变形
an equals sign and rearrange into the form y = mx + c, which gives the gradient and the y-intercept directly.
许多不等式给出的形式不便于作图，例如 3x + 2y ≤ 12。要找出边界，先把不等号换成等号，再变形为 y = mx + c 的形式，这样可以直接读出斜率和 y 轴截距。
s through (0, 6) and (4, 0). Joining these two points with a solid line (because the inequality is inclusive) and testing the origin gives the required region.
另一种方法是求出两个截距。令 x = 0 得 y = 6，令 y = 0 得 x = 4，所以边界经过 (0, 6) 与 (4, 0)。由于是不等式包含边界，用实线连接这两点，再用原点检验，即可得到所需区域。
Students frequently misplace these two because they try to rearrange them into y = mx + c.
还要记住特殊情形：x ≥ a 是经过 (a, 0) 的竖直实线，y < b 是经过 (0, b) 的水平虚线。很多同学因为想把它们变形为 y = mx + c 而把这两种线画错位置。
6. Quadratic Inequalities and Parabolas | 二次不等式与抛物线
When the boundary is a parabola, the region is described relative to the curve rather than relative to a straight line. For y > ax² + bx + c with a > 0, the required region is the area above the parabola, excluding the parabola itself, so the curve is drawn dashed.
当边界是抛物线时，区域是相对于曲线（而不是相对于直线）来描述。对于 a > 0 的 y > ax² + bx + c，所需区域是抛物线以上的部分，且不包含抛物线本身，因此曲线画成虚线。
on lies below the parabola and includes the curve, so the parabola is drawn as a solid line. If a < 0 the parabola opens downwards and the words "above" and "below" swap their positions relative to the shape.
对于 a > 0 的 y ≤ ax² + bx + c，区域位于抛物线下方并包含曲线本身，因此抛物线画成实线。若 a < 0，抛物线开口向下，此时相对于图形形状，"上方"与"下方"的位置会互换。
ts (Δ > 0), one repeated root (Δ = 0) or no real roots (Δ < 0), which affects how the region looks.
你可能还需要通过解 ax² + bx + c = 0 求出抛物线与 x 轴的交点。判别式 Δ = b² - 4ac 告诉你方程有两个根（Δ > 0）、一个重根（Δ = 0）还是没有实根（Δ < 0），这会影响区域的形状。
7. Circles and Their Regions | 圆及其区域
, the region is the closed disc: all points inside the circle plus the circle itself. If the inequality is strict, the circle is excluded and drawn dashed.
圆心为 (a, b)、半径为 r 的圆的方程是 (x - a)² + (y - b)² = r²。若不等式为 (x - a)² + (y - b)² ≤ r²，区域是闭圆盘：圆内所有点加上圆周本身。若为严格不等式，则圆周不属于区域，应画成虚线。
For the reverse inequality (x - a)² + (y - b)² > r², the region is the exterior of the circle: every point whose distance from the centre is greater than r. Testing the centre point is a quick way to confirm which side is which, because the centre gives 0, which is less than r².
对于反向不等式 (x - a)² + (y - b)² > r²，区域是圆的外部：所有到圆心距离大于 r 的点。检验圆心是一种快速确认哪一侧的方法，因为圆心处左边等于 0，小于 r²。
e overlap of the disc and the half-plane forms a shape whose vertices and intersections may be required.
圆区域经常出现在把圆盘与直线结合的题目中，例如同时满足 y ≤ 2x + 1 与 x² + y² ≤ 25 的点集。圆盘与半平面的重叠部分构成一个图形，题目可能要求求出它的顶点与交点。
8. Systems of Inequalities and Feasible Regions | 不等式组与可行域
A system of inequalities is solved graphically by drawing every boundary on the same axes and identifying the region that satisfies all of them at once. This common region is called the feasible region, and it is usually a polygon bounded by the least restrictive parts of the boundary lines.
解不等式组的方法是在同一坐标系中画出所有边界，并找出同时满足全部不等式的区域。这个公共区域称为可行域，它通常是由各边界线限制最紧的部分围成的多边形。
The most reliable technique is to shade the unwanted side of each boundary, so that the unshaded area left over is automatically the feasible region. Alternatively, shade each region separately with a different style and find the overlap. Whichever method you use, label the feasible region clearly.
最可靠的方法是把每条边界的不需要一侧涂掉，这样剩下未被涂到的区域就自动是可行域。另一种方法是分别用不同样式涂出各区域，再找重叠部分。无论用哪种方法，都要清楚标出可行域。
ints obvious, and check whether the variables are restricted to non-negative values, since "x ≥ 0, y ≥ 0" statements are extremely common in modelling questions.
一定要把不等式整理成能清楚看出约束条件的形式，并检查变量是否被限制为非负值，因为 "x ≥ 0, y ≥ 0" 这类表述在建模题中极其常见。
9. Linear Programming and Optimisation | 线性规划与最优化
maximise or minimise. The key theorem is that a linear objective function attains its optimum value at a vertex of the feasible region, so evaluating P at each vertex is enough.
线性规划把一个可行域与一个目标函数（例如 P = 3x + 5y）结合起来，要求你求其最大值或最小值。关键定理是：线性目标函数的最优值一定在可行域的某个顶点处取得，因此只需在每个顶点处计算 P 的值即可。
Repeating the objective function with a numerical value gives a set of parallel lines sometimes called level lines or iso-profit lines. Sliding such a line across the feasible region until it just touches the last vertex is a visual way of confirming the answer you obtained by substitution.
把目标函数赋予不同的数值会得到一组平行线，有时称为等值线或等利润线。让这条直线在可行域上平移到最后一个接触的顶点，就能用图形直观地验证你通过代入得到的结果。
Draw the level line as a dashed line so that it is not confused with the constraints, and remember that if the feasible region is unbounded, the objective function may have no maximum even though the vertices are finite.
把等值线画成虚线，以免与约束条件混淆；同时记住，如果可行域是无界的，那么即使顶点有限，目标函数也可能没有最大值。
10. Finding Intersections and Vertices | 求交点与顶点
ht-hand sides equal gives 2x + 1 = -x + 7, so 3x = 6, x = 2 and y = 5. The vertex is (2, 5).
可行域的顶点通过联立求解成对的边界方程得到。对于两条直线 y = 2x + 1 与 y = -x + 7，令右边相等得 2x + 1 = -x + 7，所以 3x = 6，x = 2，y = 5。顶点为 (2, 5)。
When one boundary is a curve, substitute the linear equation into the curve equation to obtain a quadratic, then solve it by factorising, completing the square or using the quadratic formula. Only the intersection points that actually lie on the feasible region are relevant to the final answer.
当一条边界是曲线时，把直线方程代入曲线方程，得到一个二次方程，再用因式分解、配方或求根公式求解。只有真正位于可行域上的交点才与最终答案相关。
Vertex test: P = 3x + 5y at (2, 5) gives P = 6 + 25 = 31
A well-labelled diagram showing each vertex coordinate is worth method marks even if a later arithmetic slip produces the wrong optimum value, so always show the substitution for every vertex.
即使后面出现计算失误导致最优值错误，一张标出每个顶点坐标的规范图形仍能拿到方法分，所以一定要写出每个顶点的代入过程。
11. Common Mistakes to Avoid | 需要避免的常见错误
The most frequent mistake is shading the wrong side of a boundary. This happens when students rely on memory rather than testing a point, especially with inequalities whose gradient is negative. Always test a point unless you are completely certain.
最常见的错误是给边界的错误一侧着色。这通常发生在学生凭记忆判断、而不去检验点时，尤其是当直线斜率为负时更容易出错。除非你完全确定，否则一定要检验一个点。
e the inequality sign when multiplying or dividing by a negative number, and leaving the boundary unlabelled so that two overlapping regions cannot be distinguished.
其他错误包括：对 ≤ 或 ≥ 画出虚线；不等号两边乘以或除以负数时忘记变号；以及没有标注边界，导致两片重叠区域无法区分。
A further trap is a strict inequality that must be satisfied alongside an inclusive one: the feasible region must respect the stricter condition. If in doubt, check each vertex by substituting back into all the original inequalities.
另一个陷阱是严格不等式与包含型不等式同时出现：可行域必须满足更严格的那个条件。如果有疑问，就把每个顶点代回所有原不等式逐一验证。
12. Exam Technique and Summary | 应试技巧与总结
In an exam, follow a fixed routine: identify the boundary, decide solid or dashed, sketch it, test a point, then shade or label the correct region. Working in this order reduces the chance of a careless slip and makes your reasoning easy for the examiner to follow.
考试时采用固定流程：确定边界，判断实线还是虚线，画出图形，检验一个点，然后给正确区域着色或加标记。按这个顺序作答能减少粗心失误，也让考官容易看懂你的思路。
give solid boundaries, < and > give dashed boundaries; the test point (0, 0) usually settles which side to shade; the feasible region of a system is the intersection of all individual regions; and a linear objective function is optimised at a vertex.
记住这些要点：≤ 与 ≥ 对应实线边界，< 与 > 对应虚线边界；检验点 (0, 0) 通常就能确定该涂哪一侧；不等式组的可行域是所有单个区域的交集；线性目标函数的最优值在顶点处取得。
Finally, keep diagrams large and accurate. A neat sketch with clear intercepts, labelled boundaries and a marked feasible region is worth far more marks than a hurried set of overlapping lines.
最后，图形要画得大而准确。一张截距清晰、边界有标注、可行域有标记的整洁草图，比一团匆忙画成、彼此重叠的直线能拿到多得多的分数。
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