Limited Government: Constraints and Inequalities in Edexcel A-Level Maths | 有限政府：Edexcel A-Level数学中的约束与不等式

📚 Limited Government: Constraints and Inequalities in Edexcel A-Level Maths | 有限政府：Edexcel A-Level数学中的约束与不等式

In politics, limited government means power is bounded by rules. In Edexcel A-Level Mathematics, the same word captures a core idea: variables are usually not free; they are limited by inequalities and constraints. This revision article treats “limited government” as a mathematical metaphor and covers the inequalities, feasible regions and optimisation techniques that appear throughout the Edexcel specification.

在政治中，有限政府意味着权力受到规则约束。在 Edexcel A-Level 数学中，同一个词也体现了一个核心思想：变量通常不是自由的，它们受到不等式和约束的限制。这篇复习文章把“有限政府”当作数学隐喻，讲解贯穿 Edexcel 考试大纲的不等式、可行区域和最优化方法。

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1. From Limited Government to Limited Variables | 从有限政府到有限变量

In political theory, a government is limited when its authority is constrained by a constitution. In mathematics, a variable is limited when its value is constrained by an inequality. For example, the condition x + 3 < 10 limits x to values less than 7.

在政治理论中，当一个政府的权力受到宪法约束时，它就是“有限政府”。在数学中，当一个变量的取值受到不等式约束时，它就是“有限变量”。例如，条件 x + 3 < 10 将 x 限制为小于 7 的值。

A constraint describes the boundary between what is allowed and what is not allowed. In Edexcel questions, you will often be asked to find the set of values of x that satisfy one or more inequalities.

约束描述的是允许与不允许之间的边界。在 Edexcel 考题中，你经常需要求满足一个或多个不等式的 x 的取值集合。

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2. Inequality Notation and Interval Form | 不等式记号与区间形式

The basic symbols are used to compare two algebraic expressions. You must be able to move confidently between inequality notation and interval notation.

基本符号用于比较两个代数表达式。你必须能够熟练地在不等式记号和区间记号之间转换。

Inequality	Meaning	Interval
x < a	x is less than a	(−∞, a)
x ≤ a	x is less than or equal to a	(−∞, a]
x > a	x is greater than a	(a, ∞)
x ≥ a	x is greater than or equal to a	[a, ∞)
a < x < b	x lies strictly between a and b	(a, b)
a ≤ x ≤ b	x lies between a and b inclusive	[a, b]

Use round brackets for strict inequalities and square brackets for inclusive endpoints. Infinity is always written with a round bracket.

严格不等式使用圆括号，包含端点时使用方括号。无穷大总是使用圆括号。

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3. Solving Linear Inequalities | 解一元一次不等式

Linear inequalities are solved in the same way as linear equations, with one extra rule: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

一元一次不等式的解法与一元一次方程相同，但有一条额外规则：如果将两边同时乘以或除以一个负数，必须反转不等号方向。

Example: Solve 3x + 5 ≤ 14.

示例：解 3x + 5 ≤ 14。

3x + 5 ≤ 14
3x ≤ 9
x ≤ 3

So the solution is x ≤ 3, or in interval form (−∞, 3].

因此解为 x ≤ 3，或用区间形式表示为 (−∞, 3]。

Adding or subtracting the same number from both sides does not change the inequality. Multiplying or dividing both sides by a positive number also does not change it.

在不等式两边同时加上或减去同一个数，不改变不等号方向。两边同时乘以或除以一个正数，也不改变不等号方向。

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4. Reversing the Inequality Sign | 反转不等号方向

The most common error in this topic is forgetting to reverse the sign when multiplying or dividing by a negative number. This is because the order of real numbers reverses under multiplication by a negative.

本专题中最常见的错误就是在乘以或除以负数时忘记反转不等号。这是因为实数的大小顺序在乘以负数后会发生反转。

Example: Solve 5 − 2x < 11.

示例：解 5 − 2x < 11。

5 − 2x < 11
−2x < 6
x > −3

Notice that dividing by −2 changed < to >. The solution is x > −3.

注意除以 −2 后 < 变成了 >。解为 x > −3。

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5. Quadratic Inequalities | 一元二次不等式

For a quadratic inequality such as x² − 5x + 6 < 0, first factorise the quadratic and identify the critical values where the expression equals zero.

对于一元二次不等式，例如 x² − 5x + 6 < 0，先对二次式进行因式分解，并找出表达式等于零的临界值。

x² − 5x + 6 < 0
(x − 2)(x − 3) < 0

The critical values are x = 2 and x = 3. The product (x − 2)(x − 3) is negative between these two values, so the solution is 2 < x < 3.

临界值为 x = 2 和 x = 3。积 (x − 2)(x − 3) 在这两个值之间为负，因此解为 2 < x < 3。

Sketching the parabola can help you see whether the quadratic is positive or negative outside or between the roots.

画出抛物线草图可以帮助你判断二次式在两根之间或两根之外是正还是负。

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6. Graphical Inequalities and Feasible Regions | 图形不等式与可行区域

A linear inequality in two variables, such as y > 2x + 1, represents a region on the coordinate plane. Use a dashed line for strict inequalities and a solid line for inclusive inequalities.

含有两个变量的线性不等式，例如 y > 2x + 1，表示坐标平面上的一个区域。严格不等式使用虚线，包含端点的不等式使用实线。

To decide which side of the line satisfies the inequality, substitute a test point such as (0, 0) into the inequality. If the test point makes the inequality true, shade that side.

要判断直线的哪一侧满足不等式，可将测试点（如 (0, 0)）代入不等式。如果测试点使不等式成立，就涂色那一侧。

The intersection of all shaded regions is called the feasible region. This is the set of all points that satisfy every constraint at the same time.

所有涂色区域的交集称为可行区域。它是同时满足所有约束的点的集合。

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7. Linear Programming: Optimising under Constraints | 线性规划：在约束下求最优

Linear programming uses inequalities to model limited resources. The feasible region is bounded by constraints, and the optimal value of an objective function occurs at a vertex of this region.

线性规划使用不等式来建模有限资源。可行区域由约束围成，目标函数的最优值出现在该区域的顶点处。

Suppose you need to maximise P = 2x + 3y subject to x ≥ 0, y ≥ 0, x + y ≤ 6 and x ≤ 4. First draw the feasible region, then evaluate P at each vertex.

假设在 x ≥ 0、y ≥ 0、x + y ≤ 6 和 x ≤ 4 的约束下，需要最大化 P = 2x + 3y。首先画出可行区域，然后在每个顶点处计算 P 的值。

The vertex coordinates are found by solving the equations of the boundary lines. The largest value of P is the maximum.

顶点坐标通过解边界直线的方程组得到。P 的最大值就是目标函数的最大值。

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8. Modelling

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