📚 Linear Inequalities | 线性不等式
Linear inequalities extend the idea of linear equations by replacing the equals sign with an inequality symbol. This topic appears throughout Edexcel A-Level Mathematics, from solving simple one-variable inequalities to shading regions defined by systems of inequalities.
线性不等式将一次方程中的等号替换为不等号，从而推广了一次方程的思想。该主题贯穿 Edexcel A-Level 数学，从求解简单的一元一次不等式到绘制由不等式组所定义的区域。
1. What Is a Linear Inequality? | 什么是线性不等式？
A linear inequality in one variable can be written in one of the forms ax + b < c, ax + b ≤ c, ax + b > c or ax + b ≥ c, where a, b and c are real constants and a ≠ 0. It compares an affine expression to a constant rather than stating that two expressions are equal.
一元一次线性不等式可以写成 ax + b < c、ax + b ≤ c、ax + b > c 或 ax + b ≥ c 的形式，其中 a、b、c 为实数且 a ≠ 0。它比较一次表达式与常数的大小，而不是表示两个表达式相等。
On a number line, the solution set is a half-line or a segment, not just a single point. Understanding linear inequalities is essential for later topics such as quadratic inequalities, regions in coordinate geometry and linear programming.
在数轴上，解集是一条射线或线段，而不仅仅是一个点。理解线性不等式是后续二次不等式、坐标几何区域和线性规划等主题的基础。
2. Notation and Symbols | 符号与记号
The four main symbols are < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). A strict inequality such as x < 4 excludes the endpoint x = 4; an inclusive inequality such as x ≤ 4 includes it.
四种主要符号是 <（小于）、>（大于）、≤（小于或等于）和 ≥（大于或等于）。严格不等式如 x < 4 不包含端点 x = 4；可包含不等式如 x ≤ 4 则包含端点。
We also use set-builder notation such as {x : x > 2}, interval notation such as (2, ∞), and number-line diagrams. In interval notation, round brackets ( ) exclude an endpoint and square brackets [ ] include it.
我们还使用集合构造记号如 {x : x > 2}、区间记号如 (2, ∞) 以及数轴图。区间记号中，圆括号 ( ) 表示不包含端点，方括号 [ ] 表示包含端点。
x < a means all values less than a; x ≥ b means b and all values greater than b.
x < a 表示所有小于 a 的值；x ≥ b 表示 b 以及所有大于 b 的值。
3. Solving One-Step and Multi-Step Inequalities | 解一步和多步不等式
To solve a linear inequality, isolate the variable using inverse operations, exactly as in linear equations. For example, to solve 2x + 3 ≤ 11, subtract 3 from both sides to get 2x ≤ 8, then divide both sides by 2 to obtain x ≤ 4.
解一元一次不等式时，与解一元一次方程一样，用逆运算将变量分离。例如，解 2x + 3 ≤ 11，两边先减 3 得 2x ≤ 8，再两边除以 2 得 x ≤ 4。
2x + 3 ≤ 11 → 2x ≤ 8 → x ≤ 4
If the unknown appears on both sides, collect like terms first. Given 5x − 2 > 2x + 7, subtract 2x from both sides to obtain 3x − 2 > 7, add 2 to both sides to get 3x > 9, then divide by 3: x > 3.
如果未知数出现在两边，先合并同类项。例如 5x − 2 > 2x + 7，两边减 2x 得 3x − 2 > 7，两边加 2 得 3x > 9，再除以 3：x > 3。
5x − 2 > 2x + 7 → 3x > 9 → x > 3
4. Multiplying or Dividing by a Negative Number | 乘以或除以负数
The key difference from equations is that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example, −3x < 12 becomes x > −4 after dividing both sides by −3.
与方程的关键区别在于：当不等式两边同乘或同除以一个负数时，必须反转不等号。例如 −3x < 12 两边除以 −3 后变为 x > −4。
−3x < 12 → x > −4
A common explanation is that multiplying by −1 reflects the number line, so smaller numbers become larger and the order reverses. Always check the sign of the coefficient of x before dividing; if it is negative, reverse the inequality symbol.
一个常见的解释是乘以 −1 相当于在数轴上反射，因此较小的数变成较大的数，顺序发生反转。除以 x 的系数之前一定要检查其符号；如果系数为负，要反转不等号。
If a < 0, then ax
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