Mastering IGCSE Edexcel Maths Inequalities | IGCSE Edexcel 数学：不等式 考点精讲

📚 Mastering IGCSE Edexcel Maths Inequalities | IGCSE Edexcel 数学：不等式 考点精讲

Inequalities are a fundamental topic in the IGCSE Edexcel Maths syllabus, bridging basic algebra and graphical analysis. Unlike equations, inequalities express a range of possible values, demanding a shift in how we think about solutions. This guide will systematically walk you through every crucial aspect, from number line representation to solving quadratic inequalities, ensuring you are fully prepared for both Foundation and Higher tier exams.

不等式是 IGCSE Edexcel 数学大纲中的一个基础课题，它连接了基础代数和图形分析。与方程不同，不等式表达的是一系列可能的值，要求我们在思考方式上做出转变。本指南将系统地带你走过每个关键环节，从数轴表示到求解二次不等式，确保你为基础级别和高级别考试做好充分准备。

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1. Understanding Inequality Symbols | 理解不等式符号

Inequalities use symbols to compare two expressions. The four primary symbols are: ‘>’ (greater than), ‘<' (less than), '≥' (greater than or equal to), and '≤' (less than or equal to). It is vital to distinguish between '> 4′ and ‘≥ 4’, as the former excludes 4, while the latter includes 4. When reading an inequality like x < 7, we say 'x is less than 7'.

不等式使用符号来比较两个表达式。四个主要符号是：’>’（大于）、’<'（小于）、'≥'（大于或等于）和 '≤'（小于或等于）。区分 '> 4′ 和 ‘≥ 4’ 至关重要，因为前者不包括 4，而后者包含 4。当我们读到像 x < 7 这样的不等式时，我们读作 'x 小于 7'。

Symbol | 符号	Meaning | 含义	Example | 示例
>	Strictly greater than | 严格大于	x > 5
<	Strictly less than | 严格小于	y < -2
≥	Greater than or equal to | 大于或等于	n ≥ 10
≤	Less than or equal to | 小于或等于	t ≤ 3.5

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2. Representing Inequalities on a Number Line | 在数轴上表示不等式

A key skill is showing the solution set on a number line. For strict inequalities like x > 2, we use an open circle at 2 to show it is not included, and draw an arrow to the right. For inclusive inequalities like x ≤ -1, we use a closed circle at -1, pointing left. Examiners often check for the correct type of circle and the direction of the arrow.

一项关键技能是在数轴上展示解集。对于 x > 2 这样的严格不等式，我们在 2 处画一个空心圆圈表示它不包括在内，并向右画一个箭头。对于 x ≤ -1 这样包含的不等式，我们在 -1 处使用实心圆圈，箭头指向左。考官经常检查圆圈的样式和箭头的方向是否正确。

To represent -3 < x ≤ 4, place an open circle at -3 and a closed circle at 4, then connect them with a solid line. This shows x can be any number strictly greater than -3 and up to and including 4.

要表示 -3 < x ≤ 4，在 -3 处放一个空心圆圈，在 4 处放一个实心圆圈，然后用一条实线将它们连接起来。这表示 x 可以是严格大于 -3 的任何数，直到并包括 4。

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3. Solving Linear Inequalities | 解线性不等式

Solving linear inequalities follows the same steps as solving linear equations: add, subtract, multiply, or divide both sides to isolate the variable. For example, to solve 3x + 5 ≤ 14, subtract 5 from both sides to get 3x ≤ 9, then divide by 3 to get x ≤ 3. The solution set is all real numbers less than or equal to 3.

解线性不等式遵循与解线性方程相同的步骤：加、减、乘或除两边以隔离变量。例如，求解 3x + 5 ≤ 14，两边减去 5 得到 3x ≤ 9，然后除以 3 得到 x ≤ 3。解集是所有小于或等于 3 的实数。

Always present the final answer with the variable on the left, such as x ≤ 3 rather than 3 ≥ x. This makes it easier to interpret on a number line. Remember to check your solution by substituting a value within the range back into the original inequality.

总是将最终答案呈现为变量在左侧的形式，例如 x ≤ 3 而不是 3 ≥ x。这样在数轴上更容易解读。记得通过将范围内的一个值代回原不等式来验证你的解。

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4. The Critical Rule: Multiplying or Dividing by a Negative | 关键规则：乘以或除以负数

This is the most common error in IGCSE inequalities. When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. For instance, -2x > 6 becomes x < -3 after dividing by -2. Forgetting to flip the sign will give a completely wrong solution set.

这是 IGCSE 不等式中最常见的错误。当你对不等式两边乘以或除以一个负数时，你必须改变不等号的方向。例如，-2x > 6 除以 -2 后变成 x < -3。忘记反转符号会得出完全错误的解集。

Consider the problem 5 – 3x ≥ 14. Subtract 5 from both sides: -3x ≥ 9. Now divide by -3, and flip the sign: x ≤ -3. Always scan your working for a negative coefficient of x before performing the final division step.

考虑问题 5 – 3x ≥ 14。两边减去 5：-3x ≥ 9。现在除以 -3，并反转符号：x ≤ -3。在执行最后的除法步骤之前，一定要检查你的计算过程中是否存在 x 的系数为负的情况。

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5. Solving Inequalities with Brackets | 解带括号的不等式

When an inequality contains brackets, expand them first using the distributive law. For example, 2(3x – 1) < 5x + 4 expands to 6x - 2 < 5x + 4. Then subtract 5x from both sides to get x - 2 < 4, and add 2 to both sides to obtain x < 6. The solution is all numbers less than 6.

当不等式包含括号时，首先使用分配律展开它们。例如，2(3x – 1) < 5x + 4 展开得到 6x - 2 < 5x + 4。然后两边减去 5x 得到 x - 2 < 4，两边加 2 得到 x < 6。答案是所有小于 6 的数。

Be careful with negative signs outside brackets: -2(x + 3) > 4 becomes -2x – 6 > 4, then -2x > 10, giving x < -5 after dividing by -2. The combination of expanding brackets and the negative rule tests your attention to detail.

注意括号外的负号：-2(x + 3) > 4 变为 -2x – 6 > 4，然后 -2x > 10，除以 -2 得到 x < -5。展开括号和负数规则的结合考验你对细节的关注。

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6. Forming Inequalities from Word Problems | 从文字题建立不等式

Word problems require translating English phrases into mathematical symbols. ‘At least’ means ≥, ‘at most’ means ≤, ‘more than’ is >, and ‘less than’ is <. For example, 'A taxi charges £2 plus £0.50 per mile. If the fare must be at most £15, find the maximum distance.' The inequality is 2 + 0.5x ≤ 15.

文字题需要将英语短语翻译成数学符号。’至少’ 意味着 ≥，’至多’ 意味着 ≤，’超过’ 是 >，而 ‘少于’ 是 <。例如，'一辆出租车收费 2 英镑加上每英里 0.50 英镑。如果车费必须至多 15 英镑，求最大距离。' 不等式是 2 + 0.5x ≤ 15。

Solving 2 + 0.5x ≤ 15 gives 0.5x ≤ 13, so x ≤ 26 miles. Always define your variable clearly at the start. In exam questions, the context often restricts solutions to positive whole numbers, such as a number of people or items.

解 2 + 0.5x ≤ 15 得到 0.5x ≤ 13，因此 x ≤ 26 英里。务必在开始时明确定义变量。在考试题中，上下文通常限制解为正整数，例如人数或物品数量。

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7. Representing Inequalities Graphically on the Cartesian Plane | 在笛卡尔平面上用图形表示不等式

For inequalities in two variables, such as y > 2x + 1, we draw the boundary line first. If the inequality is strict (> or <), use a dashed line to show the boundary is not included. For ≥ or ≤, use a solid line. Then shade the region that satisfies the inequality. Pick a test point like (0,0) to check which side to shade.

对于两个变量的不等式，例如 y > 2x + 1，我们首先画出边界线。如果不等式是严格的（> 或 <），使用虚线表示边界不包括在内。对于 ≥ 或 ≤，使用实线。然后给满足不等式的区域涂上阴影。选取一个测试点如 (0,0) 来检查涂哪一侧。

If you have multiple inequalities, the solution is the region where all shaded areas overlap. This is often a triangular or quadrilateral region. Label this region clearly with a capital R if asked. Simultaneous graphical inequalities are a staple of Higher tier papers.

如果你有多个不等式，解是所有涂色区域重叠的部分。这通常是一个三角形或四边形区域。如果要求，用大写字母 R 清楚地标记这个区域。联立图形不等式是高级别试卷的必考内容。

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8. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities, such as x² – 4x – 5 > 0, appear on Higher tier IGCSE papers. First, treat it like an equation and solve x² – 4x – 5 = 0 to find the critical values. Factorising gives (x – 5)(x + 1) = 0, so x = 5 or x = -1. These critical values split the number line into three intervals.

二次不等式，例如 x² – 4x – 5 > 0，出现在 IGCSE 高级别试卷中。首先，把它当作方程来解 x² – 4x – 5 = 0 以找到临界值。因式分解得到 (x – 5)(x + 1) = 0，因此 x = 5 或 x = -1。这些临界值将数轴分成三个区间。

Test a value from each interval in the original inequality. For x < -1 (test x = -2): (-2)² - 4(-2) - 5 = 4 + 8 - 5 = 7 > 0, true. For -1 < x < 5 (test x = 0): 0 - 0 - 5 = -5 > 0, false. For x > 5 (test x = 6): 36 – 24 – 5 = 7 > 0, true. The solution is x < -1 or x > 5.

在每个区间中选取一个值代入原不等式进行检验。对于 x < -1（检验 x = -2）：(-2)² - 4(-2) - 5 = 4 + 8 - 5 = 7 > 0，成立。对于 -1 < x < 5（检验 x = 0）：0 - 0 - 5 = -5 > 0，不成立。对于 x > 5（检验 x = 6）：36 – 24 – 5 = 7 > 0，成立。解是 x < -1 或 x > 5。

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9. Sketching Quadratics to Solve Inequalities | 绘制二次函数草图解不等式

An alternative method uses the shape of the quadratic graph. For x² – 4x – 5 > 0, sketch y = x² – 4x – 5. Since the coefficient of x² is positive, the parabola is U-shaped. The graph crosses the x-axis at -1 and 5. We want where the graph is above the x-axis (y > 0). This occurs for x < -1 and x > 5, matching our algebraic solution.

另一种方法是利用二次函数图像的形状。对于 x² – 4x – 5 > 0，绘制 y = x² – 4x – 5 的草图。由于 x² 的系数为正，抛物线呈 U 形。图像在 -1 和 5 处穿过 x 轴。我们需要图像在 x 轴上方的部分（y > 0）。这发生在 x < -1 和 x > 5，与我们的代数解一致。

If the inequality were x² – 4x – 5 < 0, we would take the region below the x-axis, which is between the roots: -1 < x < 5. Visualising the graph helps prevent sign errors and deepens understanding of the solution's structure.

如果不等式是 x² – 4x – 5 < 0，我们会取 x 轴下方的区域，即在两个根之间：-1 < x < 5。将图像可视化有助于防止符号错误，并加深对解结构的理解。

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10. Common Mistakes and How to Avoid Them | 常见错误及如何避免

One frequent error is misreading the inequality symbol. Students sometimes treat ‘x > 3’ as if it includes 3 on a number line, using a closed circle by mistake. Always pause to check: is it strict or inclusive? Another pitfall is mishandling fractions. If you have (2/3)x ≥ 4, multiply both sides by 3/2, not just 3.

一个常见错误是误读不等式符号。学生有时把 ‘x > 3’ 当作在数轴上包含 3，错误地使用实心圆圈。务必停下来检查：它是严格的还是包含的？另一个陷阱是处理分数不当。如果你有 (2/3)x ≥ 4，两边要乘以 3/2，而不仅仅是 3。

For quadratic inequalities, students often forget to consider both regions and only state, for example, x > 5 and forget x < -1. Remember, the product of two factors is positive if both are positive OR both are negative. Also, never write the solution as a single compound inequality like -1 > x > 5; that is mathematically nonsensical.

对于二次不等式，学生们常常忘记考虑两个区域，只陈述例如 x > 5 而忘记 x < -1。记住，两个因式的积为正当且仅当两者都为正或两者都为负。此外，绝不要把解写成类似 -1 > x > 5 这样的单个复合不等式；这在数学上是荒谬的。

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11. Set Notation for Inequality Solutions | 解集的集合符号表示

Higher tier IGCSE introduces set notation to express solutions compactly. The solution to x < 3 can be written as { x : x < 3 }, which reads 'the set of all x such that x is less than 3'. For x < -1 or x > 5, we write { x : x < -1 } ∪ { x : x > 5 }, using the union symbol ∪.

高级别 IGCSE 引入了集合符号来简洁地表示解。x < 3 的解可以写成 { x : x < 3 }，读作 '所有使得 x 小于 3 的 x 的集合'。对于 x < -1 或 x > 5，我们写作 { x : x < -1 } ∪ { x : x > 5 }，使用并集符号 ∪。

For a range like -2 ≤ x ≤ 4, we can use { x : -2 ≤ x ≤ 4 } or interval notation [-2, 4]. The square brackets indicate the endpoints are included. Parentheses ( or ) are used for strict inequalities, so (0, 5) means 0 < x < 5. Practice converting between number lines, inequalities, and set notation.

对于像 -2 ≤ x ≤ 4 这样的范围，我们可以使用 { x : -2 ≤ x ≤ 4 } 或区间记号 [-2, 4]。方括号表示端点包含在内。圆括号（或）用于严格不等式，所以 (0, 5) 意味着 0 < x < 5。练习在数轴、不等式和集合符号之间进行转换。

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12. Exam-Style Practice and Strategy | 考试风格练习与策略

Exam questions often combine multiple skills: solving a linear inequality with brackets, showing the solution on a number line, and interpreting it in context. For example: ‘The perimeter of a rectangle is less than 50 cm. The length is (2x+3) cm and width is (x-1) cm. Find the possible range for x.’

考试题目经常结合多种技能：解一个带括号的线性不等式，在数轴上展示解，并在上下文中解读它。例如：’一个矩形的周长小于 50 厘米。长是 (2x+3) 厘米，宽是 (x-1) 厘米。求 x 的可能范围。’

Set up the inequality: 2(2x+3 + x-1) < 50, which simplifies to 2(3x+2) < 50, then 6x+4 < 50, giving 6x < 46 and x < 7⅔. Also, since lengths must be positive, x-1 > 0, so x > 1. The solution is 1 < x < 7⅔. Always check for hidden constraints like positivity in geometry problems.

建立不等式：2(2x+3 + x-1) < 50，化简为 2(3x+2) < 50，然后 6x+4 < 50，得到 6x < 46 且 x < 7⅔。此外，由于长度必须为正，x-1 > 0，因此 x > 1。解是 1 < x < 7⅔。务必检查几何问题中如正数等隐藏的限制条件。

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