IGCSE Mathematics: Mastering Inequalities | IGCSE数学：不等式精讲

📚 IGCSE Mathematics: Mastering Inequalities | IGCSE数学：不等式精讲

Inequalities are a fundamental topic in IGCSE Mathematics, appearing in algebra, coordinate geometry, and problem-solving. This guide covers linear inequalities, number lines, and quadratic inequalities, with exam-style examples to build your confidence.

不等式是IGCSE数学中的基础主题，出现在代数、坐标几何和问题求解中。本指南涵盖线性不等式、数轴表示和二次不等式，并通过考试风格的例题帮助你建立信心。

1. Inequality Symbols and Notation | 不等号与记法

In mathematics, we compare quantities using inequality symbols. The symbol ‘>’ means ‘greater than’, ‘<‘ means ‘less than’, ‘≥’ means ‘greater than or equal to’, and ‘≤’ means ‘less than or equal to’. An open circle on a number line means the value is not included, while a closed circle means it is included.

在数学中，我们使用不等号来比较数量。符号’>’表示“大于”，'<‘表示“小于”，’≥’表示“大于或等于”，’≤’表示“小于或等于”。在数轴上，空心圆圈表示该值不包含在内，实心圆圈表示包含在内。

The key difference between an equation and an inequality is that an equation has exactly one solution (or a finite set), while an inequality typically has infinitely many solutions. For example, x = 3 is a single point, but x > 3 includes all numbers greater than 3.

方程与不等式之间的关键区别在于：方程恰好有一个解（或有限个解），而不等式通常有无限多个解。例如，x = 3 是一个单独的点，但 x > 3 包含所有大于3的数。

‘>’ – strictly greater than / 严格大于
‘<‘ – strictly less than / 严格小于
‘≥’ – greater than or equal to / 大于或等于
‘≤’ – less than or equal to / 小于或等于

2. Solving Linear Inequalities | 解线性不等式

To solve a linear inequality, we use the same steps as solving an equation: expand brackets, collect like terms, and isolate the unknown variable. However, there is one critical difference: if we multiply or divide both sides by a negative number, we must reverse the inequality sign.

解线性不等式时，我们使用与解方程相同的步骤：去括号、合并同类项、分离未知数。然而，有一个关键区别：如果我们将两边乘以或除以一个负数，必须反转不等号方向。

For example, solve -2x > 6. Dividing both sides by -2 gives x < -3. The sign flips because dividing by a negative reverses order. Many students forget this rule, so always check your final answer by substituting a test value.

例如，解 -2x > 6。两边除以-2得到 x < -3。因为除以负数会颠倒顺序，所以不等号反转。许多学生忘记这一规则，因此务必通过代入测试值来检查最终答案。

-2x > 6 → x < -3

Another common operation is adding or subtracting the same value from both sides, which does not affect the inequality direction. For example, x + 5 ≤ 9 → x ≤ 4.

另一个常见操作是在两边加上或减去同一个值，这不会影响不等号方向。例如，x + 5 ≤ 9 → x ≤ 4。

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

An inequality can be shown visually on a number line. Use an open circle (○) for strict inequalities (> or <) and a closed circle (●) for inclusive inequalities (≥ or ≤). Draw an arrow in the direction of the solution set.

不等式可以在数轴上直观地表示。严格不等式（>或<）使用空心圆圈（○），包含性不等式（≥或≤）使用实心圆圈（●）。沿解集方向画箭头。

For x ≥ 2, place a closed circle at 2 and draw an arrow to the right. For x < 5, place an open circle at 5 and draw an arrow to the left. Double inequalities like 2 < x ≤ 5 have both an open circle at 2 and a closed circle at 5, with a line joining them.

对于 x ≥ 2，在2处放置实心圆点并向右画箭头。对于 x < 5，在5处放置空心圆点并向左画箭头。对于像 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，并在两者之间画一条连线。

Strict inequality: open circle / 严格不等式：空心圆
Inclusive inequality: closed circle / 包含性不等式：实心圆
Compound inequality: segment between two endpoints / 复合不等式：两端点之间的线段

4. Compound Inequalities | 复合不等式

A compound inequality combines two or more inequalities. The word ‘and’ means both conditions must hold simultaneously, while ‘or’ means at least one condition must hold. In IGCSE, you often encounter double-ended inequalities like 3 ≤ 2x + 1 < 7.

复合不等式将两个或多个不等式组合在一起。“且”表示两个条件必须同时满足，而“或”表示至少满足一个条件。在IGCSE中，你常会遇到如 3 ≤ 2x + 1 < 7 这样的双端不等式。

To solve a double-ended inequality, perform the same operation on all three parts. For 3 ≤ 2x + 1 < 7, subtract 1 from all parts: 2 ≤ 2x < 6. Then divide by 2: 1 ≤ x < 3. The solution is all real numbers from 1 (included) to 3 (not included).

解双端不等式时，对所有三个部分进行相同的运算。对于 3 ≤ 2x + 1 < 7，所有部分都减1：2 ≤ 2x < 6。然后除以2：1 ≤ x < 3。解是从1（包含）到3（不包含）的所有实数。

3 ≤ 2x + 1 < 7 → 1 ≤ x < 3

5. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities involve terms like x². The standard method is to rearrange the inequality to zero on one side, factorise the quadratic, find critical values, and test intervals. Alternatively, you can sketch a graph to determine where the expression is positive or negative.

二次不等式涉及 x² 这样的项。标准方法是将不等式整理为一侧为零，分解二次式，找到临界值，并测试区间。或者，你可以画草图来确定表达式在何处为正或负。

Consider x² – x – 6 > 0. Factorise: (x – 3)(x + 2) > 0. The critical values are x = 3 and x = -2. Test the three intervals: x < -2, -2 < x < 3, and x > 3.

考虑 x² – x – 6 > 0。分解因式：(x – 3)(x + 2) > 0。临界值为 x = 3 和 x = -2。测试三个区间：x < -2、-2 < x < 3 和 x > 3。

For x = -3 (in the first interval), (-3-3)(-3+2) = (-6)(-1) = 6 > 0, so this interval works. For x = 0 (middle interval), (0-3)(0+2) = (-3)(2) = -6, which is not > 0. For x = 4 (last interval), (4-3)(4+2) = (1)(6) = 6 > 0. Therefore, the solution is x < -2 or x > 3.

对于 x = -3（第一区间），(-3-3)(-3+2) = (-6)(-1) = 6 > 0，所以该区间成立。对于 x = 0（中间区间），(0-3)(0+2) = (-3)(2) = -6，不满足 > 0。对于 x = 4（最后区间），(4-3)(4+2) = (1)(6) = 6 > 0。因此，解为 x < -2 或 x > 3。

6. Graphical Approach to Quadratic Inequalities | 二次不等式的图形方法

Sketcing the graph of y = ax² + bx + c provides a quick visual check. The curve is a parabola. Determine its roots and, based on the coefficient ‘a’, identify whether it opens upward (a > 0) or downward (a < 0).

画出 y = ax² + bx + c 的草图可以快速进行直观检查。曲线是抛物线。确定其根，并根据系数’a’判断开口方向：a > 0 开口向上，a < 0 开口向下。

For a > 0 and inequality > 0, the solution is the x-values where the curve is above the x-axis: the two outer intervals. For inequality < 0, the solution is the interval between the roots. For example, x² – 4 < 0 has roots at x = -2 and x = 2, so -2 < x < 2.

对于 a > 0 且不等式 > 0，解是曲线在x轴上方的区域，即两个外侧区间。对于不等式 < 0，解是两根本之间的区间。例如，x² – 4 < 0 的根为 x = -2 和 x = 2，所以 -2 < x < 2。

When the quadratic cannot be factorised easily, use the quadratic formula or complete the square to find exact critical values. Always check whether the inequality is strict or inclusive, as this affects whether the endpoints are included.

当二次式不容易分解时，使用求根公式或配方法来找到精确的临界值。始终检查不等式是严格还是包含性的，因为这会影响到端点是否包含在解中。

7. Inequalities with Fractions | 含分数的不等式

Rational inequalities involve fractions with unknowns in the denominator. The safest method is to multiply both sides by the denominator squared, which is always positive, thus avoiding sign reversal issues. However, this is beyond the core IGCSE syllabus; a simpler case involves multiplying by a positive variable.

有理不等式涉及分母中含未知数的分数。最安全的方法是两边乘以分母的平方，该值始终为正，从而避免符号反转问题。然而，这超出了IGCSE核心大纲范围；更简单的情况是乘以正的变量。

For example, consider 2/x > 4. Since x could be negative, we cannot simply multiply by x. Instead, multiply both sides by x² (which is positive): 2x > 4x², then 4x² – 2x < 0, so 2x(2x – 1) < 0. The critical values are x = 0 and x = 1/2. Testing intervals gives 0 < x < 1/2.

例如，考虑 2/x > 4。由于x可能为负，我们不能直接乘以x。相反，两边乘以x²（正数）：2x > 4x²，然后 4x² – 2x < 0，即 2x(2x – 1) < 0。临界值为 x = 0 和 x = 1/2。测试区间得到 0 < x < 1/2。

2/x > 4 → 0 < x < 1/2

8. Word Problems Involving Inequalities | 不等式应用题

In exam questions, inequalities often model real-world constraints such as budget limits, minimum scores, or temperature ranges. The first step is to assign a variable, then translate the condition into an inequality, solve it, and interpret the result in context.

在考试问题中，不等式通常用来模拟现实约束，如预算限额、最低分数或温度范围。第一步是设变量，然后将条件转化为不等式，求解，并在实际情境中解释结果。

Suppose a student scores 78 and 85 on two tests. What score on the third test is needed to average at least 82? Let x be the third score. The average is (78 + 85 + x)/3 ≥ 82. Multiply by 3: 163 + x ≥ 246, so x ≥ 83. The student needs at least 83.

假设一个学生两次测验分别得了78分和85分。第三次测验需要多少分才能使平均分至少达到82？设第三次分数为x。平均分为 (78 + 85 + x)/3 ≥ 82。两边乘以3：163 + x ≥ 246，所以 x ≥ 83。该学生需要至少得83分。

Always consider the domain of the variable in real-world contexts. Scores, ages, and counts are non-negative integers, which may further restrict the solution set. In this case, the smallest integer score is 83.

在现实情境中，始终考虑变量的定义域。分数、年龄和数量是非负整数，这可能会进一步限制解集。在这个例子中，最小的整数分数是83。

9. Solving Inequalities with Absolute Values | 解绝对值不等式

Absolute value inequalities are a common extension. The expression |x| < a means -a < x < a, while |x| > a means x < -a or x > a. This concept is often introduced in IGCSE Extended level.

绝对值不等式是常见的扩展内容。表达式 |x| < a 表示 -a < x < a，而 |x| > a 表示 x < -a 或 x > a。这一概念常在IGCSE扩展级别中引入。

For example, solve |x – 3| ≤ 5. This means -5 ≤ x – 3 ≤ 5. Adding 3 to all parts gives -2 ≤ x ≤ 8. Similarly, |2x + 1| > 7 gives two cases: 2x + 1 > 7 → x > 3, or 2x + 1 < -7 → x < -4. Thus, x < -4 or x > 3.

例如，解 |x – 3| ≤ 5。这意味着 -5 ≤ x – 3 ≤ 5。所有部分加3得到 -2 ≤ x ≤ 8。类似地，|2x + 1| > 7 给出两种情况：2x + 1 > 7 → x > 3，或 2x + 1 < -7 → x < -4。因此，x < -4 或 x > 3。

|x – 3| ≤ 5 → -2 ≤ x ≤ 8

10. Exam Tips and Common Mistakes | 考试提示与常见错误

Many students lose marks on inequalities due to simple errors. Always reverse the inequality sign when multiplying or dividing by a negative number. When writing the solution set, use the correct brackets or intervals as required by the question. Finally, always check your answer by substituting a value from the solution set and a value outside it.

许多学生由于简单错误在不等式题目上失分。当乘以或除以负数时，务必反转不等号。在写解集时，根据题目要求使用正确的区间括号。最后，始终通过从解集内和一个解集外的值进行代入来检查答案。

Another common mistake is forgetting that open circles are used for strict inequalities on number lines. In quadratic inequalities, students sometimes confuse which intervals satisfy the inequality. Drawing a quick sign diagram helps avoid this.

另一个常见错误是忘记在数轴上严格不等式使用空心圆圈。在二次不等式中，学生有时会混淆哪些区间满足不等式。画一个符号图表有助于避免这个问题。

Always flip the sign when × or ÷ by a negative / 乘以或除以负数时始终翻转不等号
Open circle for > and <; closed circle for ≥ and ≤ / >和<用空心圆；≥和≤用实心圆
Test intervals for quadratic inequalities / 二次不等式要测试区间

11. Practice Questions | 练习题

The best way to master inequalities is through consistent practice. Try the following exam-style questions, then check your answers with the methods above.

掌握不等式的最佳方法是通过持续练习。尝试以下考试风格的题目，然后用上述方法检查你的答案。

1. Solve 3x – 7 > 2x + 1 and represent the solution on a number line.

1. 解 3x – 7 > 2x + 1，并在数轴上表示解。

2. Solve the compound inequality -1 ≤ 2 – 3x ≤ 8.

2. 解复合不等式 -1 ≤ 2 – 3x ≤ 8。

3. Solve x² + 2x – 8 ≥ 0.

3. 解 x² + 2x – 8 ≥ 0。

4. A rectangle has length (x + 3) cm and width x cm. Its perimeter is at least 30 cm. Find the smallest integer value of x.

4. 一个矩形的长为(x + 3)厘米，宽为x厘米。其周长至少为30厘米。求x的最小整数值。

Answers: 1) x > 8; 2) -2 ≤ x ≤ 1; 3) x ≤ -4 or x ≥ 2; 4) x ≥ 6, so smallest integer is 6.

答案：1) x > 8；2) -2 ≤ x ≤ 1；3) x ≤ -4 或 x ≥ 2；4) x ≥ 6，所以最小整数为6。

12. Summary | 总结

Inequalities are a versatile tool in IGCSE Mathematics. Master solving linear inequalities, representing solutions on number lines, handling compound inequalities, and solving quadratic inequalities by factorisation and sign testing. Remember the golden rule: flip the inequality sign when multiplying or dividing by a negative number.

不等式是IGCSE数学中一个多用途的工具。掌握解线性不等式、在数轴上表示解、处理复合不等式，以及通过因式分解和符号测试解二次不等式。记住黄金法则：乘以或除以负数时翻转不等号。

With regular practice and careful attention to detail, you can avoid common pitfalls and secure full marks in inequality questions. Revise the worked examples in this guide, attempt the practice questions, and use your textbook’s past-paper questions for further drill.

通过定期练习和对细节的仔细关注，你可以避免常见陷阱并在不等式题目中获得满分。复习本指南中的例题，尝试练习题，并利用教材中的历年真题进行进一步训练。

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IGCSE Mathematics: Mastering Inequalities | IGCSE数学：不等式精讲