KS3 Maths: Inequalities | KS3 数学：不等式考点精讲

📚 KS3 Maths: Inequalities | KS3 数学：不等式考点精讲

In KS3 Maths, inequalities are a fundamental building block that extends your understanding of equations. Instead of stating that two expressions are exactly equal, an inequality shows that one side is greater than, less than, or at least/at most the other side. Mastering inequalities will help you solve real-world problems, interpret graphs, and lay the groundwork for algebra topics at GCSE and beyond.

在 KS3 数学中，不等式是扩展方程理解的重要基础模块。不等式并不表示两个表达式严格相等，而是表明一边大于、小于、至少或至多等于另一边。掌握不等式将帮助你解决实际问题、解读图像，为 GCSE 及更高阶段的代数内容打下基础。

1. Understanding Inequality Symbols | 认识不等式符号

An inequality uses special symbols to compare two values. The four main symbols you need to know are: ‘>’ (greater than), ‘<‘ (less than), ‘≥’ (greater than or equal to), and ‘≤’ (less than or equal to).

不等式使用特殊符号来比较两个值。你需要知道的四个主要符号是：’>’（大于），'<‘（小于），’≥’（大于或等于）和 ‘≤’（小于或等于）。

For example, ‘7 > 3’ means 7 is strictly greater than 3. ‘x < 5’ means the variable x can be any number smaller than 5, such as 4.9, 0, or -2, but not 5 itself. When the symbol includes an equals bar, like ‘y ≥ 8’, y can be 8 or any number larger than 8.

例如，’7 > 3′ 表示 7 严格大于 3。’x < 5′ 表示变量 x 可以是任何小于 5 的数，比如 4.9、0 或 -2，但不能等于 5。当符号包含等号线时，如 ‘y ≥ 8’，y 可以是 8 或任何大于 8 的数。

The ‘strict’ inequalities (> and <) exclude the boundary value, while the ‘inclusive’ symbols (≥ and ≤) include it. This distinction is vital when representing solutions on a number line.

‘严格’不等式（> 和 <）不包含边界值，而’包含型’符号（≥ 和 ≤）则包含边界值。这一区别在数轴上表示解集时至关重要。

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

We often visualise the solutions of an inequality using a number line. For a strict inequality like x < 2, we draw an open circle at 2 (showing 2 is not included) and shade the line to the left, with an arrow continuing to negative infinity.

我们常使用数轴直观表示不等式的解集。对于像 x < 2 这样的严格不等式，我们在数字 2 处画一个空心圆圈（表示 2 不包含在内），并向左涂色线条，用箭头一直延伸到负无穷。

For x ≥ -1, we draw a closed (filled) circle at -1 and shade to the right, because all numbers greater than -1, including -1 itself, satisfy the inequality. The closed circle indicates that -1 is part of the solution set.

对于 x ≥ -1，我们在 -1 处画一个实心（填充）圆圈，并向右涂色，因为所有大于 -1 的数，包括 -1 本身，都满足不等式。实心圆圈表示 -1 是解集的一部分。

A double inequality like 3 < x ≤ 7 is shown by an open circle at 3, a closed circle at 7, and a line segment between them. This tells us x is strictly greater than 3 but less than or equal to 7.

像 3 < x ≤ 7 这样的双不等式，在数轴上显示为：3 处为空心圆圈，7 处为实心圆圈，两者之间画一条线段。这告诉我们 x 严格大于 3 但小于或等于 7。

3. Solving Simple Inequalities | 解简单的一元一次不等式

Solving basic inequalities works exactly like solving equations, with one key difference: we can add or subtract the same amount from both sides without changing the direction of the inequality sign.

解简单的不等式与解方程几乎完全相同，但有一个关键区别：我们可以在两边同时加上或减去同一个数，而不改变不等号的方向。

Consider x + 6 > 10. To isolate x, subtract 6 from both sides: x + 6 – 6 > 10 – 6, which simplifies to x > 4. The open circle at 4 and shading to the right represents all numbers greater than 4.

考虑 x + 6 > 10。为了将 x 单独移至左边，两边同时减去 6：x + 6 – 6 > 10 – 6，化简得 x > 4。在数轴上，4 处画空心圆圈并向右涂色，表示所有大于 4 的数。

Similarly, for x – 3 ≤ 2, add 3 to both sides to get x ≤ 5. The inequality sign stays the same because addition and subtraction do not affect the order.

类似地，对于 x – 3 ≤ 2，两边同时加上 3 得到 x ≤ 5。不等号保持不变，因为加法和减法不会影响次序。

4. The Golden Rule: Multiplying or Dividing by a Negative | 黄金法则：乘以或除以负数要反向

This is the most important rule in inequalities: when you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality sign. Without this step, your solution will be incorrect.

这是不等式最重要的法则：当你对不等式的两边乘以或除以一个负数时，你必须反转（翻转）不等号的方向。如果不进行这一步，你的解就是错误的。

Why does this happen? Think about simple numbers: 2 < 5 is true. If we multiply both sides by -1, we get -2 and -5. But -2 is greater than -5, so we must write -2 > -5. The relationship flipped because the number line order reverses for negatives.

为什么会这样？用简单的数字想一想：2 < 5 成立。如果我们两边乘以 -1，得到 -2 和 -5。但是 -2 大于 -5，因此必须写成 -2 > -5。因为负数在数轴上的顺序反转，大小关系就颠倒了。

Example: Solve -3y < 12. Divide both sides by -3 and flip the sign: y > -4. Always check your answer: if y = -3 (which is greater than -4), -3(-3)=9, and 9 < 12 is true.

示例：解 -3y < 12。两边除以 -3 并翻转符号：y > -4。务必检验你的答案：如果 y = -3（它大于 -4），-3(-3)=9，而 9 < 12 成立。

Remember: if you are multiplying or dividing by a positive number, the sign remains unchanged. Only negative multipliers/divisors trigger the flip.

请记住：如果你乘以或除以一个正数，不等号方向保持不变。只有乘以或除以负数时，才需要反转符号。

5. Solving Two-Step Inequalities | 解两步不等式

Two-step inequalities involve two operations, such as 2x + 3 ≤ 11. First, subtract 3 from both sides: 2x ≤ 8. Then divide both sides by 2 (positive, so sign stays): x ≤ 4.

两步不等式包含两次运算，例如 2x + 3 ≤ 11。首先，两边减去 3：2x ≤ 8。然后两边除以 2（正数，所以符号不变）：x ≤ 4。

When the variable term is negative, it is often safer to move the variable to the other side to avoid flipping early. For 10 – 3x > 1, you can add 3x to both sides: 10 > 1 + 3x. Then subtract 1: 9 > 3x. Finally divide by 3: 3 > x, which is the same as x < 3. Notice the sign flipped at the end when we rearranged, but that is simply a rewrite, not a multiply/divide by negative.

当变量项带有负号时，通常更安全的做法是将变量移到另一边，以避免过早翻转符号。对于 10 – 3x > 1，可以两边同时加上 3x：10 > 1 + 3x。然后减去 1：9 > 3x。最后除以 3：3 > x，这与 x < 3 相同。请注意，我们在重写时符号翻转了，但这只是改写，并非乘以或除以负数导致。

Alternatively, you can subtract 10 from both sides: -3x > -9, then divide by -3, which does require a flip: x < 3. Both methods lead to the same correct solution.

你也可以两边减去 10：-3x > -9，然后除以 -3，此时确实需要反转符号：x < 3。两种方法得到相同的正确解。

6. Compound Inequalities (Double Inequalities) | 复合不等式（双不等式）

A compound inequality combines two inequalities into one statement, such as 5 < x ≤ 10. This means x is greater than 5 and less than or equal to 10. The variable sits in the middle, and both conditions must be satisfied at the same time.

复合不等式将两个不等式合并为一个语句，例如 5 < x ≤ 10。这意味着 x 大于 5 并且小于或等于 10。变量位于中间，两个条件必须同时满足。

These inequalities are extremely useful for describing bounded ranges, like ages between 12 and 16 inclusive on one side. When graphing, an open circle at 5 and a closed circle at 10, with a line connecting them, correctly shows the solution set.

这类不等式在描述有边界范围时非常有用，比如年龄介于 12 到 16 岁之间，且包含某一侧边界。在数轴上绘制时，5 处为空心圆圈，10 处为实心圆圈，用线段连接，准确地表示了解集。

It is important to write compound inequalities in the correct order: the smaller number on the left, the larger on the right. Writing 3 > x > 7 is meaningless because it contradicts itself; always check that the symbols point the same way and the numbers increase from left to right.

用正确顺序书写复合不等式很重要：较小的数在左边，较大的数在右边。写成 3 > x > 7 是毫无意义的，因为它自相矛盾；务必检查符号方向是否一致，并且数字从左到右是递增的。

7. Solving Compound Inequalities | 解复合不等式

To solve a compound inequality like 4 ≤ 2x < 10, you perform the same operation on all three parts simultaneously. First, divide everything by 2 (positive, so signs stay): 2 ≤ x < 5. This gives the solution set where x is at least 2 and strictly less than 5.

要解像 4 ≤ 2x < 10 这样的复合不等式，你需要同时对三部分执行相同的运算。首先，整体除以 2（正数，所以符号不变）：2 ≤ x < 5。这样便得出解集：x 至少为 2 且严格小于 5。

If a negative coefficient appears, you must apply the flip rule to all inequality signs. For -6 < -2x ≤ 4, divide the entire compound inequality by -2. Remember to flip both signs: 3 > x ≥ -2. It is conventional to rewrite the solution with the smaller number on the left: -2 ≤ x < 3.

如果出现负系数，必须对所有不等号应用翻转规则。对于 -6 < -2x ≤ 4，整个复合不等式除以 -2。记住要翻转两个不等号：3 > x ≥ -2。习惯上我们会将较小的数写在左边来重写解集：-2 ≤ x < 3。

Sometimes you need to handle addition or subtraction in a compound inequality. For 1 < x/3 + 2 ≤ 5, first subtract 2 from all parts: -1 < x/3 ≤ 3. Then multiply by 3 (positive): -3 < x ≤ 9. The solution is now clear.

有时你需要在复合不等式中处理加减法。对于 1 < x/3 + 2 ≤ 5，首先所有部分减去 2：-1 < x/3 ≤ 3。然后乘以 3（正数）：-3 < x ≤ 9。解集现在就清晰了。

8. Real-World Applications | 实际应用问题

Inequalities model many real-life constraints, such as minimum height requirements, spending limits, or speed limits. For example, a rectangular garden has a width w metres. The length is 3 metres more than the width. If the perimeter must be at least 22 metres, we can form an inequality.

不等式可以模拟许多现实生活中的约束条件，例如最低身高要求、消费限制或速度限制。例如，一个矩形花园的宽为 w 米。长比宽多 3 米。如果周长至少为 22 米，我们就可以建立一个不等式。

Perimeter = 2(length + width) = 2((w+3) + w) = 2(2w+3) = 4w + 6. The condition ‘at least 22’ translates to 4w + 6 ≥ 22. Solve: subtract 6, then divide by 4: 4w ≥ 16, so w ≥ 4. The width must be 4 metres or greater.

周长 = 2(长 + 宽) = 2((w+3) + w) = 2(2w+3) = 4w + 6。’至少 22′ 的条件转换为 4w + 6 ≥ 22。求解：减去 6，然后除以 4：4w ≥ 16，因此 w ≥ 4。宽度必须为 4 米或更大。

Similarly, a mobile phone plan costs £10 per month plus 5p per minute of calls. If you want to spend no more than £15 in a month, the inequality is 10 + 0.05m ≤ 15, where m is the number of minutes. Solving gives m ≤ 100 minutes.

类似地，一个手机套餐每月收费 10 英镑，外加每分钟通话费5便士。如果你想每月花费不超过15英镑，不等式为 10 + 0.05m ≤ 15，其中 m 是通话分钟数。求解得 m ≤ 100 分钟。

9. Common Mistakes to Avoid | 常见错误及避免方法

One frequent error is forgetting to flip the inequality sign when multiplying or dividing by a negative number. Always pause and ask: ‘Am I using a negative number?’ If yes, flip the sign immediately.

一个常见错误是在乘以或除以负数时忘记翻转不等号。时刻停下来问自己：’我在用负数吗？’ 如果是，立即翻转不等号。

Another mistake is misreading the inequality symbol on a number line. Students sometimes use an open circle for ≥ or a closed circle for >. Remember: open = strict (>,<); closed = inclusive (≥,≤).

另一个错误是看错数轴上的不等号。学生有时对 ≥ 使用空心圆圈，或对 > 使用实心圆圈。请记住：空心 = 严格不等（>，<）；实心 = 包含等号（≥，≤）。

When writing compound inequalities, some students write 8 < x < 4, which is impossible. Always ensure the smaller number is on the left and the larger on the right, with the inequality signs pointing the same way.

在书写复合不等式时，有些学生会写成 8 < x < 4，这是不可能的。始终确保较小的数在左边，较大的数在右边，并且不等号方向一致。

Finally, always check your solution by substituting a value into the original inequality. If x > 3, test x = 4 and also a boundary value like 3 (which should not work for strict). This catches sign errors early.

最后，始终通过代入一个值到原不等式来检验你的解。如果解是 x > 3，测试 x = 4，也可以测试边界值如 3（对于严格不等，它应该不成立）。这能及早发现符号错误。

10. Summary and Key Takeaways | 总结与关键要点

Inequalities are a powerful tool for describing ranges of values. The core principles to remember are: treat them like equations when adding or subtracting; always flip the sign when multiplying/dividing by a negative; use open/closed circles correctly on a number line; and master compound inequalities by operating on all parts simultaneously.

不等式是描述数值范围的强大工具。需要记住的核心原则是：在加减运算时将它们视为方程处理；在乘以/除以负数时务必翻转符号；在数轴上正确使用空心/实心圆圈；通过同时对所有部分进行运算来掌握复合不等式。

Whether you are solving simple linear inequalities or tackling real-world word problems, consistent practice and careful checking will build your confidence. The skills you develop now will form the foundation for more advanced algebra, including quadratic inequalities and graphing linear inequalities in later years.

无论你是在解简单的一元一次不等式，还是在处理实际应用题，持续练习和仔细检查都将增强你的信心。你现在培养的这些技能，将为更高级的代数内容——包括二次不等式和线性不等式图像——打下基础。

Keep a special note on the ‘negative flip’ rule, as it is the most tested concept at KS3 level. With these revision points, you are well-prepared to tackle any inequality question with accuracy.

请特别记牢’负数翻转’这一规则，因为这是 KS3 阶段考查最多的概念。掌握了这些复习要点，你就能精准地解决任何不等式题目了。

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KS3 Maths: Inequalities | KS3 数学：不等式 考点精讲