Solving Inequalities and Representing Solutions on Number Lines | 解不等式及在数轴上表示解集

📚 Solving Inequalities and Representing Solutions on Number Lines | 解不等式及在数轴上表示解集

Inequalities are mathematical statements that compare two expressions using symbols such as <, >, ≤, or ≥. Solving inequalities is a core skill in IGCSE Mathematics, and representing their solutions on number lines is essential for clear communication and exam success.

不等式是使用 <、>、≤ 或 ≥ 等符号比较两个表达式的数学陈述。解不等式是 IGCSE 数学的核心技能，而在数轴上表示其解集对于清晰表达和考试成功至关重要。

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

Before solving inequalities, you must be confident with the meaning of each symbol. An open circle on a number line means the endpoint is not included, while a closed circle means the endpoint is included.

在解不等式之前，你必须对每个符号的含义充满信心。数轴上的空心圆表示不包含端点，而实心圆表示包含端点。

< (less than) | 小于 – the value on the left is strictly smaller; endpoint excluded.
> (greater than) | 大于 – the value on the left is strictly larger; endpoint excluded.
≤ (less than or equal to) | 小于或等于 – endpoint included.
≥ (greater than or equal to) | 大于或等于 – endpoint included.

For example, x > 3 means all numbers strictly greater than 3, so 3 is not a solution. On a number line, use an open circle at 3 and shade to the right.

例如，x > 3 表示所有严格大于 3 的数，因此 3 不是解。在数轴上，在 3 处画空心圆并向右侧阴影。

2. Solving Linear Inequalities | 解线性不等式

Solving a linear inequality is similar to solving a linear equation: add, subtract, multiply, or divide both sides by the same value. However, there is one crucial difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.

解线性不等式类似于解线性方程：两边同时加、减、乘或除以同一个数。然而，有一个关键区别：如果你两边同时乘以或除以一个负数，你必须反转不等号方向。

Operation \| 运算	Example \| 示例	Result \| 结果
Add/Subtract \| 加减	x − 5 > 2 → add 5 both sides	x > 7
Multiply/Divide by positive \| 乘除正数	3x ≤ 12 → divide by 3	x ≤ 4
Multiply/Divide by negative \| 乘除负数	−2x > 6 → divide by −2	x < −3 (sign reversed)

Example: Solve 2x − 7 ≥ 3.
Add 7 to both sides: 2x ≥ 10. Then divide by 2: x ≥ 5. The solution set is all numbers greater than or equal to 5.

示例：解 2x − 7 ≥ 3。
两边加 7：2x ≥ 10。然后除以 2：x ≥ 5。解集为所有大于或等于 5 的数。

3. The Golden Rule: Reversing the Sign | 黄金法则：反号

The most common mistake in solving inequalities is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always check your final answer by substituting a sample value.

解不等式时最常见的错误是在乘以或除以负数时忘记反转不等号。务必通过代入一个样本值来检查最终答案。

Consider −3x < 12. Dividing both sides by −3 gives x > −4. If you forgot to reverse, you would get x < −4, which is incorrect. Test x = 0: −3(0) = 0 < 12 is true, and 0 > −4 is indeed true.

考虑 −3x < 12。两边除以 −3 得 x > −4。如果忘记反转，你会得到 x < −4，这是错误的。检验 x = 0：−3(0) = 0 < 12 成立，且 0 > −4 也确实成立。

If a < b, then −a > −b, and 1/a > 1/b (if both positive).

4. Double Inequalities | 双不等式

A double inequality combines two inequalities, such as −2 ≤ x < 8. This means x ≥ −2 and x < 8 simultaneously. You can solve double inequalities by performing the same operation on all three parts.

双不等式结合两个不等式，例如 −2 ≤ x < 8。这意味着 x ≥ −2 且 x < 8 同时成立。你可以通过对三部分同时进行相同运算来解双不等式。

Example: Solve −3 ≤ 2x + 1 < 7.
Subtract 1 from all three parts: −4 ≤ 2x < 6.
Divide all three parts by 2: −2 ≤ x < 3.
On a number line, use a closed circle at −2, an open circle at 3, and shade between.

示例：解 −3 ≤ 2x + 1 < 7。
三部分同时减 1：−4 ≤ 2x < 6。
三部分同时除以 2：−2 ≤ x < 3。
在数轴上，在 −2 处画实心圆，在 3 处画空心圆，并阴影中间区域。

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

For a single inequality like x > 2, draw a number line, place an open circle above 2, and draw an arrow to the right. For x ≤ 5, use a closed circle at 5 and an arrow to the left.

对于像 x > 2 这样的单一不等式，画一条数轴，在 2 上方画一个空心圆，并向右侧画箭头。对于 x ≤ 5，在 5 处画实心圆并向左侧画箭头。

Open circle (空心圆) at endpoint – the endpoint is not included (< or >).
Closed circle (实心圆) at endpoint – the endpoint is included (≤ or ≥).
Shading 阴影 – extends in the direction of values that satisfy the inequality.

For a double inequality, the shading is a segment between two endpoints, with appropriate circles at each end.

对于双不等式，阴影位于两个端点之间的线段，并在每个端点使用对应的圆。

6. Solving Inequalities with Fractions and Decimals | 解含分数与小数的不等式

When an inequality contains fractions, multiply every term by the least common multiple (LCM) of the denominators to eliminate fractions. This process is called clearing the denominator.

当不等式含有分数时，将每一项乘以分母的最小公倍数（LCM）以消除分数。这个过程称为清除分母。

Example: Solve (x/3) − 1 < 2.
Multiply all terms by 3: x − 3 < 6.
Add 3 to both sides: x < 9.

示例：解 (x/3) − 1 < 2。
所有项乘以 3：x − 3 < 6。
两边加 3：x < 9。

For decimals, you may multiply both sides by powers of 10 to convert decimals to integers, making the algebra easier.

对于小数，你可以将两边乘以 10 的幂，将小数转换为整数，从而使代数运算更简便。

7. Inequalities with Variables on Both Sides | 变量在两侧的不等式

If the variable appears on both sides of the inequality, collect the unknown terms on one side and the constant terms on the other side, then solve as usual.

如果变量出现在不等式两侧，应将未知项集中在一侧，常数项集中在另一侧，然后照常求解。

Example: Solve 5x − 2 < 3x + 6.
Subtract 3x from both sides: 2x − 2 < 6.
Add 2 to both sides: 2x < 8.
Divide by 2: x < 4.

示例：解 5x − 2 < 3x + 6。
两边减 3x：2x − 2 < 6。
两边加 2：2x < 8。
除以 2：x < 4。

5x − 2 < 3x + 6 → 2x < 8 → x < 4

8. Word Problems Involving Inequalities | 涉及不等式的应用题

Many IGCSE problems present inequalities in words. Translating word phrases into inequality symbols is a key step.

许多 IGCSE 问题以文字形式呈现不等式。将文字短语转换为不等式符号是关键步骤。

“At least” 至少 → ≥
“No more than” 不超过 → ≤
“More than” 多于 → >
“Less than” 少于 → <
“Between” 介于 → double inequality

Example: A student needs at least 70 points to pass. If she already has 45 points, let x be the points needed on the final test. Then 45 + x ≥ 70, so x ≥ 25.

示例：一名学生至少需要 70 分才能通过。她已经有了 45 分，设 x 为期末考试所需的分数。则 45 + x ≥ 70，所以 x ≥ 25。

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

Here are frequent errors students make when solving inequalities, along with strategies to avoid them.

以下是学生在解不等式时常犯的错误，以及避免这些错误的策略。

Forgetting to reverse the sign when multiplying/dividing by a negative number (忘记反号) – always check with a test value.
Using the wrong circle on a number line (用错圆圈) – closed for ≤/≥, open for .
Misreading double inequalities (误读双不等式) – read them as two separate conditions.
Not simplifying before solving (未先化简) – expand brackets and combine like terms first.

10. Exam-Style Worked Example | 考试风格例题

Let us work through a complete exam-style question step by step.

让我们逐步完成一道完整的考试风格例题。

Question 问题: Solve the inequality 3(2x − 1) ≥ 4x + 5, and represent the solution on a number line.

Step 1: Expand brackets: 6x − 3 ≥ 4x + 5.

步骤 1：展开括号：6x − 3 ≥ 4x + 5。

Step 2: Subtract 4x from both sides: 2x − 3 ≥ 5.

步骤 2：两边减 4x：2x − 3 ≥ 5。

Step 3: Add 3 to both sides: 2x ≥ 8.

步骤 3：两边加 3：2x ≥ 8。

Step 4: Divide by 2: x ≥ 4.

步骤 4：除以 2：x ≥ 4。

On the number line, place a closed circle at 4 and shade the arrow pointing to the right, indicating all values greater than or equal to 4.

在数轴上，在 4 处画实心圆，并向右侧画出箭头，表示所有大于或等于 4 的值。

x ≥ 4

11. Practice Questions | 练习题目

Try the following questions to test your understanding. Solve each inequality and sketch its solution on a number line.

尝试以下题目以测试你的理解。解每个不等式并在数轴上画出其解集。

3x − 7 > 8
2 − 5x ≤ 12
−1 ≤ 3x + 2 ≤ 11
4(x − 2) ≥ 2x + 6
(2x + 1)/3 < 5

Answers: 1. x > 5. 2. x ≥ −2. 3. −1 ≤ x ≤ 3. 4. x ≥ 7. 5. x < 7.

答案：1. x > 5。2. x ≥ −2。3. −1 ≤ x ≤ 3。4. x ≥ 7。5. x < 7。

12. Summary | 总结

Solving inequalities is a systematic process: simplify, apply inverse operations, reverse the sign when multiplying or dividing by a negative, and then represent the solution on a number line using open and closed circles correctly.

解不等式是一个系统的过程：化简，应用逆运算，当乘以或除以负数时反转不等号，然后用空心圆和实心圆正确地在数轴上表示解集。

Mastering this topic will also help you with more advanced topics such as quadratic inequalities and linear programming. Practice with a variety of questions to build confidence and speed.

掌握这一主题也将帮助你进一步学习二次不等式和线性规划等更高级的主题。通过多种题目进行练习，以建立信心和速度。

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Solving Inequalities and Representing Solutions on Number Lines | 解不等式及在数轴上表示解集