Linear Inequalities | 线性不等式

📚 Linear Inequalities | 线性不等式

Linear inequalities are one of the fundamental topics in A-Level Mathematics. They extend the concept of linear equations by using inequality symbols such as >, <, ≥ and ≤. Mastering how to solve and represent inequalities is essential for tackling more advanced areas like quadratic inequalities and linear programming. In this guide, we will cover all the key methods for solving linear inequalities, representing solutions, and tackling systems of inequalities required for the Edexcel A-Level syllabus.

线性不等式是A-Level数学的基础主题之一。它通过使用诸如 >、<、≥ 和 ≤ 等不等号，对线性方程的概念进行了扩展。掌握如何求解和表示不等式，对于攻克二次不等式和线性规划等更高阶的内容至关重要。在本指南中，我们将涵盖Edexcel A-Level教学大纲所要求的所有关键方法，包括求解线性不等式、表示解以及处理不等式组。

1. Introduction to Linear Inequalities | 线性不等式导论

A linear inequality compares two algebraic expressions using one of the symbols <, >, ≤ or ≥. For example, 4x – 7 < 5 states that the expression 4x – 7 is strictly less than 5. These inequalities can be solved to find a range of values for the variable, rather than a single solution.

线性不等式使用 <、>、≤ 或 ≥ 之一来比较两个代数表达式。例如，4x – 7 < 5 表示表达式 4x – 7 严格小于 5。这些不等式可以求解，得出变量的一个取值范围，而非单一的解。

The solution set of a linear inequality typically consists of infinitely many real numbers. It can be represented on a number line, using set notation, or with interval notation.

线性不等式的解集通常由无限多个实数组成。它可以用数轴、集合符号或区间表示法来表示。

Understanding the direction of the inequality is crucial. Graphic representations and algebraic manipulation both rely on a clear grasp of what each symbol means.

理解不等号的方向至关重要。图形表示和代数操作都依赖于对每个符号含义的清晰把握。

2. Solving Basic Linear Inequalities | 解简单线性不等式

To solve a linear inequality, we use the same techniques as solving linear equations: we can add, subtract, multiply or divide both sides by the same positive number without changing the direction of the inequality. The goal is to isolate the variable on one side.

要解一个线性不等式，我们使用与解线性方程相同的技巧：两边可以加上、减去、乘以或除以同一个正数，而不改变不等号的方向。目标是将变量孤立在一边。

For example, solve 3x – 5 ≤ 10. Add 5 to both sides to get 3x ≤ 15. Then divide both sides by 3 to obtain x ≤ 5. This means any value of x that is less than or equal to 5 satisfies the original inequality.

例如，解 3x – 5 ≤ 10。两边加 5 得 3x ≤ 15。然后两边除以 3，得到 x ≤ 5。这意味着任何小于或等于 5 的 x 值都满足原不等式。

3x – 5 ≤ 10 → 3x ≤ 15 → x ≤ 5

Always present the solution in its simplest form, and remember to check by substituting a value from the solution set back into the original inequality.

始终以最简形式呈现解，并记得从解集中取一个值代回原不等式进行验证。

3. Multiplying or Dividing by a Negative Number | 乘以或除以负数

The most important rule when dealing with inequalities is that if you multiply or divide both sides by a negative number, you must reverse the direction of the inequality sign. This is because multiplying by a negative flips the order of numbers on the number line.

处理不等式时最重要的规则是：如果两边乘以或除以一个负数，就必须将不等号的方向反转。这是因为乘以负数会翻转数轴上的大小顺序。

For instance, solve -2x > 6. Divide both sides by -2 and reverse the > sign to <, yielding x < -3. A common mistake is to forget the reversal, leading to an incorrect solution set.

例如，解 -2x > 6。两边除以 -2，并将 > 号反转为 <，得到 x < -3。一个常见错误是忘记反转，从而导致错误的解集。

-2x > 6 → x < -3

Another example: 5 – x ≥ 2. Subtract 5 from both sides to get -x ≥ -3. Multiply both sides by -1 and reverse the inequality to obtain x ≤ 3. Always double-check when a negative coefficient is involved.

另一个例子：5 – x ≥ 2。两边减去 5 得 -x ≥ -3。两边乘以 -1 并反转不等号，得到 x ≤ 3。当涉及负系数时，务必仔细检查。

4. Representing Solutions on a Number Line | 在数轴上表示解

Linear inequalities are often illustrated on a number line. For strict inequalities (< or >), we use an open circle at the boundary point to show that the value itself is not included. For inclusive inequalities (≤ or ≥), a closed (solid) circle is drawn.

线性不等式常常在数轴上表示。对于严格不等式（< 或 >），我们在边界点处使用空心圆来表示该值本身不包含在内。对于包含边界的不等式（≤ 或 ≥），则画一个实心圆。

After marking the circle, shade the line to the left or to the right according to the inequality direction. For example, x > -1 is shown by an open circle at -1 and a shaded arrow extending to the right. For x ≤ 4, use a closed circle at 4 and shade to the left.

标记圆圈后，根据不等号的方向将数轴向左或向右涂抹阴影。例如，x > -1 在 -1 处画空心圆，并向右涂抹阴影箭头。对于 x ≤ 4，则在 4 处画实心圆并向左涂抹阴影。

Double inequalities such as 2 < x ≤ 5 are represented by an open circle at 2, a closed circle at 5, and shading the segment between them.

像 2 < x ≤ 5 这样的双重不等式，在 2 处画空心圆，在 5 处画实心圆，并涂抹两者之间的线段。

5. Set Notation and Interval Notation | 集合符号与区间表示法

Instead of drawing a number line, we can use formal notation. In set-builder notation, the solution is written as { x : condition }. For example, { x : x > 4 } reads “the set of all x such that x is greater than 4”.

除了画数轴外，我们还可以使用形式化的表示法。在集合构建符号中，解写作 { x : 条件 }。例如，{ x : x > 4 } 读作“所有使得 x 大于 4 的 x 的集合”。

Interval notation provides a compact alternative using brackets. A round bracket ( or ) indicates the endpoint is excluded, while a square bracket [ or ] means it is included. Infinity symbols always take a round bracket.

区间表示法提供了一种使用括号的紧凑替代方案。圆括号 ( 或 ) 表示端点不包含在内，而方括号 [ 或 ] 表示包含端点。无穷符号总是与圆括号搭配。

Common conversions are shown below:

常见的转换如下：

Inequality x ≥ 2 → Interval [2, ∞)
Inequality -1 < x ≤ 3 → Interval (-1, 3]
Inequality x < 0 → Interval (-∞, 0)

中文对照：不等式 x ≥ 2 对应区间 [2, ∞)；-1 < x ≤ 3 对应区间 (-1, 3]；x < 0 对应区间 (-∞, 0)。练习在不同表示法之间转换是考试中的常见要求。

It is important to be fluent in switching between inequality, number line, set-builder, and interval notations, as exam questions frequently test this skill.

熟练地在不等式、数轴、集合构建和区间表示法之间转换非常重要，因为考试题目经常考察这一技能。

6. Solving Double Inequalities | 解双重不等式

A double inequality combines two inequality statements into one continuous expression, such as -4 < 3x + 2 ≤ 11. The goal is to isolate x in the middle by performing the same operation on all three parts of the inequality.

双重不等式将两个不等式合并为一个连续表达式，例如 -4 < 3x + 2 ≤ 11。解决的方法是对不等式的所有三个部分执行相同的运算，从而将中间的 x 孤立出来。

Start by subtracting 2 from all parts: -6 < 3x ≤ 9. Then divide every part by 3: -2 < x ≤ 3. The solution is all values of x strictly greater than -2 and less than or equal to 3.

首先所有部分减去 2：-6 < 3x ≤ 9。然后每个部分除以 3：-2 < x ≤ 3。解是所有严格大于 -2 且小于或等于 3 的 x 值。

-4 < 3x + 2 ≤ 11 → -6 < 3x ≤ 9 → -2 < x ≤ 3

If you need to multiply or divide by a negative number, remember to reverse both inequality signs. For example, solving -6 ≤ -2x < 4 requires dividing by -2 and flipping both signs, resulting in 3 ≥ x > -2, which is often rewritten as -2 < x ≤ 3.

如果你需要乘以或除以负数，记得将两个不等号均反转。例如，解 -6 ≤ -2x < 4 需要除以 -2 并同时反转两个不等号，得到 3 ≥ x > -2，通常重写为 -2 < x ≤ 3。

7. Linear Inequalities with Brackets | 带括号的线性不等式

When an inequality contains brackets, the first step is always to expand them using the distributive law. Then collect like terms and solve as usual.

当不等式中含有括号时，第一步总是使用分配律将其展开。然后合并同类项，并像往常一样求解。

Example: Solve 2(3x – 1) > 4x + 10. Expand to get 6x – 2 > 4x + 10. Subtract 4x from both sides: 2x – 2 > 10. Add 2: 2x > 12. Finally, divide by 2 to obtain x > 6.

例子：解 2(3x – 1) > 4x + 10。展开得 6x – 2 > 4x + 10。两边减 4x：2x – 2 > 10。加 2：2x > 12。最后除以 2 得 x > 6。

2(3x – 1) > 4x + 10 → 6x – 2 > 4x + 10 → x > 6

It is helpful to work step by step, keeping the inequality sign aligned. Unlike equations, you must stay alert for any multiplication or division by a negative during expansion or simplification, though that rarely happens merely from expanding brackets.

逐步操作会很有帮助，保持不等号对齐。与方程不同，虽然在展开或化简过程中很少发生，但你必须警惕在展开或化简过程中任何乘以或除以负数的操作。

8. Inequalities with Variables on Both Sides | 两边都有变量的不等式

When the variable appears on both sides of the inequality, bring all variable terms to one side and constants to the other, just as you would with an equation. The inequality sign is preserved as long as you only add or subtract.

当变量出现在不等式两边时，将所有变量项移到一边，常数项移到另一边，就像处理方程一样。只要只做加减运算，不等号的方向就会保留。

Solve: 5x – 3 ≤ 2x + 9. Subtract 2x from both sides: 3x – 3 ≤ 9. Add 3: 3x ≤ 12. Divide by 3 (positive, no reversal): x ≤ 4. The solution is any number less than or equal to 4.

解：5x – 3 ≤ 2x + 9。两边减去 2x：3x – 3 ≤ 9。加 3：3x ≤ 12。除以 3（正数，不需要反转）：x ≤ 4。解是任何小于或等于 4 的数。

If moving terms results in a negative coefficient for the variable, you can either multiply/divide by that negative and flip the sign, or rearrange differently to keep the coefficient positive. Both methods are valid; choose the one you find more straightforward.

如果移项后变量的系数为负，你可以乘以/除以该负数并反转不等号，或者重新排列以确保系数为正。两种方法都有效；选择你认为更直接的方法。

9. Systems of Linear Inequalities | 线性不等式组

A system of linear inequalities requires finding all values that satisfy two or more inequalities simultaneously. Solve each inequality individually, then determine the intersection of their solution sets.

线性不等式组要求找到同时满足两个或更多不等式的所有值。分别求解每个不等式，然后确定它们解集的交集。

Example: Solve x + 2 > 5 and 2x – 1 < 9. From the first, x > 3. From the second, 2x < 10 → x < 5. The intersection is 3 < x < 5, meaning x is strictly between 3 and 5.

例子：解 x + 2 > 5 和 2x – 1 < 9。由第一个得 x > 3；由第二个得 2x < 10 → x < 5。交集为 3 < x < 5，即 x 严格介于 3 和 5 之间。

Graphing each solution on a number line helps visualise the overlapping region. The final solution can be expressed as a double inequality or in interval notation (3, 5).

在数轴上画出每个解有助于直观地看出重叠区域。最终解可以表示为双重不等式或区间 (3, 5)。

When a system has no overlapping region, it has no solution. This occurs, for example, with x > 4 and x < 2, because no number can be simultaneously greater than 4 and less than 2.

当一个不等式组没有重叠区域时，它无解。例如，x > 4 和 x < 2 就没有重叠，因为没有数能同时大于 4 又小于 2。

10. Graphical Representation of Linear Inequalities | 线性不等式的图形表示

On a 2D coordinate plane, a linear inequality like y < 2x + 1 divides the plane into two regions. The boundary line is y = 2x + 1. For strict inequalities (< or >), the line is drawn dashed to indicate that points on the line are not included. For ≤ or ≥, the line is solid.

在二维坐标平面上，像 y < 2x + 1 这样的线性不等式将平面分为两个区域。边界线为 y = 2x + 1。对于严格不等式（< 或 >），直线画为虚线，以表示直线上的点不包含在内。对于 ≤ 或 ≥，则画为实线。

To determine which side of the line to shade, pick a test point not on the line (commonly (0,0) if the line does not pass through the origin) and see if it satisfies the inequality. If it does, shade the side containing the test point; otherwise, shade the opposite side.

为了确定直线的哪一侧需要涂阴影，选择一个不在直线上的测试点（通常如果直线不经过原点，可选(0,0)），看它是否满足不等式。如果满足，就涂包含测试点的一侧；否则涂另一侧。

For y < 2x + 1, testing (0,0) gives 0 < 2(0) + 1 → 0 < 1, which is true. So we shade the region below the dashed line. Graphs of multiple inequalities create a feasible region where all shadings overlap.

对于 y < 2x + 1，测试 (0,0) 得到 0 < 2(0) + 1 → 0 < 1，成立。因此我们涂虚线下方区域。多个不等式的图形会产生一个所有阴影重叠的可行域。

11. Applications: Introduction to Linear Programming | 应用：线性规划入门

Linear programming is a method that uses systems of linear inequalities to model real-world constraints and find the best outcome — such as maximising profit or minimising cost. The constraints are

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Linear Inequalities | 线性不等式