Inequalities on Graphs | 图形中的不等式

📚 Inequalities on Graphs | 图形中的不等式

In Edexcel A-Level Mathematics, inequalities are often linked directly to graphs. Instead of manipulating symbols alone, you can sketch or interpret two curves and use their intersection points to decide where one function lies above, below or between another. This geometric approach is especially useful for quadratic, cubic, modulus and combined inequalities.

在 Edexcel A-Level 数学中，不等式常常与图形直接关联。除了纯代数变形，你还可以通过画图或解读两条曲线，利用它们的交点判断一个函数在何处高于、低于或介于另一个函数。这种几何方法对于二次、三次、绝对值及组合不等式尤其有用。

1. Why Graph Inequalities? | 为什么用图形解不等式

A graph turns a symbolic inequality such as f(x) > g(x) into a visual comparison of heights. For many Edexcel questions, the mark scheme rewards a clear sketch, labelled intersection points, and a shaded or interval answer. Graphs also help you check algebraic solutions quickly.

图形可以把 f(x) > g(x) 这样的符号不等式转化为高度的视觉比较。在许多 Edexcel 考题中，评分标准会奖励清晰的草图、标出交点以及阴影区域或区间答案。图形还能帮助你快速检验代数解。

Graphical methods are not just a fallback; they are often the fastest route when the functions are already given or when the inequality involves modulus or polynomial curves.

图形法不仅是一种备用方案；当函数已经给出，或不等式涉及绝对值、多项式曲线时，它往往是最快的解题路径。

2. Key Principle: Above and Below | 核心原则：上、下关系

The inequality f(x) > g(x) means the graph y = f(x) is strictly above y = g(x). Similarly, f(x) < g(x) means y = f(x) is below y = g(x). The transition points occur where f(x) = g(x), so solving the equality gives the boundaries of the solution regions.

不等式 f(x) > g(x) 表示 y = f(x) 严格位于 y = g(x) 之上。类似地，f(x) < g(x) 表示 y = f(x) 位于 y = g(x) 之下。变化点出现在 f(x) = g(x) 处，因此解方程可得到解区域的边界。

If one curve is above another on an interval, the inequality holds throughout that interval unless a new intersection occurs. Always mark intersections clearly and test one value in each interval if you are unsure.

如果一条曲线在某个区间内位于另一条之上，那么在整个区间内不等式成立，除非出现新的交点。请始终清晰地标出交点；如果不确定，可在每个区间内测试一个值。

3. Reading a Shaded Region | 读取阴影区域

Some Edexcel questions shade a region and ask you to write inequalities that define it. Determine the boundary curves first, then decide whether the region is above or below each boundary. Use solid lines for ≤ or ≥ and dashed lines for < or >.

一些 Edexcel 题目会先给出阴影区域，要求你写出定义该区域的不等式。首先确定边界曲线，然后判断该区域位于每条边界之上还是之下。使用实线表示 ≤ 或 ≥，虚线表示 < 或 >。

When a region is bounded between two curves, you often need a compound inequality such as g(x) < y < f(x), with y-values sandwiched between the lower and upper graphs.

当区域夹在两条曲线之间时，通常需要复合不等式，例如 g(x) < y < f(x)，即 y 值被夹在下方图形与上方图形之间。

4. Quadratic Inequalities | 二次不等式

A quadratic inequality such as ax² + bx + c > 0 is best solved by considering the parabola y = ax² + bx + c. Find its real roots by solving ax² + bx + c = 0. If the coefficient a is positive, the parabola opens upwards; if a is negative, it opens downwards.

二次不等式（如 ax² + bx + c > 0）最好通过抛物线 y = ax² + bx + c 来求解。先解方程 ax² + bx + c = 0 找到实根。若系数 a 为正，抛物线开口向上；若 a 为负，开口向下。

Example: solve x² – 5x + 6 < 0. The roots are x = 2 and x = 3. Since the parabola opens upwards, the graph is below the x-axis between the roots. Therefore the solution is 2 < x < 3.

示例：解 x² – 5x + 6 < 0。根为 x = 2 和 x = 3。由于抛物线开口向上，图形在两根之间位于 x 轴下方。因此解为 2 < x < 3。

x² – 5x + 6 = (x – 2)(x – 3) = 0 ⇒ x = 2, x = 3

If the inequality were

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