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Year 8 Maths Series 200: Linear Equations and Inequalities | 八年级数学系列200：线性方程与不等式

2026-08-21 18:41 · 本文部分内容由AI辅助创作

📚 Year 8 Maths Series 200: Linear Equations and Inequalities | 八年级数学系列200：线性方程与不等式

Linear equations and inequalities are central to the Year 8 algebra syllabus and appear frequently in IGCSE Foundation and Higher papers. This revision guide covers solving methods, word problems, and common pitfalls step by step.

线性方程与不等式是八年级代数教学大纲的核心内容，也频繁出现在 IGCSE 基础卷和进阶卷中。本复习指南分步骤讲解求解方法、文字题以及常见易错点。

1. Algebraic Expressions Recap | 代数式复习

Before solving equations, you must be confident with algebraic expressions. Like terms have the same variable part, such as 3x and 5x, so they can be combined by adding or subtracting their coefficients. Unlike terms, such as 3x and 3y or 4x and 4x², cannot be combined into a single term.

在解方程之前，你必须熟练掌握代数式。同类项具有相同的字母部分，例如 3x 和 5x，可以通过加减系数进行合并。不同类项，例如 3x 和 3y 或 4x 和 4x²，不能合并为一项。

you multiply the term outside the bracket by every term inside the bracket. It works the same way when there is a minus sign before the bracket, but you must carry the negative sign carefully.

分配律可以展开括号：a(b + c) = ab + ac。也就是说，括号外的项要乘以括号内的每一项。括号前有负号时规则相同，但必须小心处理负号。

Common mistakes include confusing x² with 2x, or forgetting to multiply every term inside a bracket. Always check signs and powers carefully before moving on to equations.

常见错误包括混淆 x² 与 2x，或在展开括号时漏乘某项。在进入方程之前，务必仔细检查符号和幂次。

3(x + 4) − 2x = 3x + 12 − 2x = x + 12

2. Solving Basic Linear Equations | 解一元一次方程

s and x is the unknown. To solve it, you need to isolate x on one side of the equation by using inverse operations in the correct order.

一元一次方程的一般形式为 ax + b = c，其中 a、b 和 c 是常数，x 是未知数。解方程时，需要按照正确顺序使用逆运算，将 x 单独留在等号一边。

Whatever you do to one side of the equation, you must also do to the other side. This keeps the equation balanced. It is best to undo addition or subtraction first, then undo multiplication or division.

等式一边进行的操作，另一边必须同样进行，这样才能保持等式平衡。最好先处理加减法，再处理乘除法。

For example, subtract 3 from both sides, then divide by 2. Always present your steps vertically and clearly to avoid arithmetic errors.

例如，两边先减去 3，再除以 2。步骤要竖排书写并保持清晰，以减少计算错误。

2x + 3 = 11 → 2x = 8 → x = 4

Check your answer by substituting it back into the original equation. If both sides are equal, the solution is correct.

将答案代回原方程进行检验。如果两边相等，说明解是正确的。

3. Equations with Brackets | 含括号的方程

When an equation contains brackets, expand them first using the distributive law. Then collect like terms on each side and solve the simplified equation using inverse operations.

如果方程含有括号，先用分配律展开，再在各边合并同类项，然后利用逆运算求解化简后的方程。

Be careful with negative signs outside brackets. For example, −(x − 2) equals −x + 2 because the negative sign multiplies every term inside the bracket. A common error is writing −(x − 2) as −x − 2, which is wrong.

注意括号外有负号的情况。例如 −(x − 2) 等于 −x + 2，因为负号会乘以括号内的每一项。常见错误是把 −(x − 2) 写成 −x − 2，这是不正确的。

Expanding first removes hidden structure and reduces mistakes in later steps. Once the brackets are gone, the problem becomes a basic linear equation.

先展开可以消除隐藏的结构，减少后续步骤中的错误。括号去掉后，题目就变成基础的一元一次方程。

3(x − 2) = 12 → 3x − 6 = 12 → 3x = 18 → x = 6

4. Equations with Fractions | 含分数的方程

Fractions make equations look harder, but multiplying every term by the lowest common denominator clears them quickly. This gives a simpler linear equation without fractions.

分数让方程看起来更复杂，但将每一项都乘以最小公分母可以迅速去分母，得到一个不含分数的更简单的一元一次方程。

Multiply every term by 6 to get 3x + 2x = 30. Then combine like terms and solve as usual.

例如，方程 x/2 + x/3 = 5 中，2 和 3 的最小公分母是 6。将每一项乘以 6，得到 3x + 2x = 30。然后合并同类项并按常规方法求解。

Always multiply all terms, including any constant on the other side of the equals sign. Missing a term is a very common error when clearing fractions.

一定要乘以所有项，包括等号另一边的常数。去分母时漏乘某一项是非常常见的错误。

x/2 + x/3 = 5 → 3x + 2x = 30 → 5x = 30 → x = 6

If the equation has a single fraction, you can also multiply both sides by the denominator directly. Both methods give the same result.

如果方程只有一个分数，也可以直接两边乘以分母。两种方法得到的结果相同。

5. Forming Equations from Word Problems | 由文字题建立方程

Word problems require you to define an unknown, usually using x, and translate the words into a mathematical equation. Keywords such as 'sum', 'difference', 'product' and 'is' indicate the operations you need to use.

文字题需要设未知数，通常用 x 表示，并将文字翻译成数学方程。例如 '和'、'差'、'积' 和 '等于' 等关键词提示你需要使用的运算。

Always state what x represents before writing the equation. This helps you form the correct equation and makes your working clearer to an examiner.

在列出方程之前，始终说明 x 代表什么。这有助于你建立正确的方程，也让考官更容易理解你的步骤。

After solving the equation, you must answer the original question. If the problem asks for a length, age or amount of money, include the appropriate unit in your final answer.

求出方程的解后，必须回答原题。如果题目问的是长度、年龄或金额，最终答案中要包含适当的单位。

egers example, solving gives 3x + 3 = 42, so x = 13. The three integers are 13, 14 and 15.

对于连续整数的例子，求解可得 3x + 3 = 42，因此 x = 13。这三个整数分别是 13、14 和 15。

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6. Introduction to Inequalities | 不等式入门

quations, which usually have one solution, inequalities usually have a range of possible solutions.

不等式使用 <、>、≤、≥ 等符号比较两个表达式。与方程通常只有一个解不同，不等式的解通常是一个范围。

For example, x > 3 includes every number greater than 3, such as 3.1, 4, 10 and 100. It does not include 3 itself because the symbol is strict.

例如，x > 3 包含所有大于 3 的数，例如 3.1、4、10 和 100。它不包括 3 本身，因为使用的是严格不等号。

s < or >. The direction of shading or the arrow shows which values satisfy the inequality.

在数轴上，实心圆表示 ≤ 或 ≥，空心圆表示 < 或 >。阴影或箭头的方向表示哪些值满足不等式。

x < 5 means x is any value less than 5.

7. Solving Linear Inequalities | 解一元一次不等式

You solve inequalities in the same way as equations: you can add, subtract, multiply or divide both sides by the same positive number without changing the inequality sign.

解不等式与解方程类似：两边可以加减同一个数，或乘以、除以同一个正数，不等号方向不变。

However, multiplying or dividing both sides by a negative number reverses the inequality sign. This is the most important rule for inequalities and a very common exam trap.

但是，两边乘以或除以一个负数时，不等号方向要反转。这是不等式最重要的规则，也是考试中非常常见的陷阱。

For example, divide both sides of −2x < 6 by −2 and reverse the sign to get x > −3. Always show the sign change clearly in your working.

例如，将 −2x < 6 两边除以 −2 并改变不等号方向，得到 x > −3。在解题步骤中要清楚地展示不等号方向的变化。

−2x < 6 → x > −3

ng a value inside the range. For example, x = 0 satisfies −2(0) < 6 because 0 < 6 is true.

可以通过在解集内取一个值来检验。例如 x = 0 满足 −2(0) < 6，因为 0 < 6 成立。

8. Number Lines and Solution Sets | 数轴与解集

Representing inequalities on a number line is a common IGCSE skill. Use an open circle for strict inequalities such as x > 2 or x < −1, because the endpoint is not included in the solution set.

在数轴上表示不等式是 IGCSE 的常见技能。对于 x > 2 或 x < −1 这类严格不等式，使用空心圆，因为端点不包含在解集内。

e the endpoint is part of the solution set. Shading should cover the region that satisfies the inequality.

对于 x ≥ 2 或 x ≤ −1 这类包含不等号的情况，使用实心圆，因为端点是解集的一部分。阴影应覆盖满足不等式的区域。

If the variable is on the left side of the inequality, the arrow on the number line points in the same direction as the inequality sign. This makes drawing and reading number lines easier.

如果变量在不等号的左边，数轴上的箭头方向与不等号方向一致。这样画图和读图都会更简单。

InequalityNumber line representation
x > 2Open circle at 2, arrow to the right
x ≤ −1Closed circle at −1, arrow to the left

Always label the key number on the number line and make sure your arrow is clear. If you are asked to list integer solutions, read the integers covered by the shaded region.

务必在数轴上标出关键数字，并确保箭头清晰。如果题目要求列出整数解，只需读出阴影区域覆盖的整数。

9. Compound Inequalities | 复合不等式

1 ≤ 5. This means 2x + 1 is greater than −3 and less than or equal to 5 at the same time.

复合不等式将两个不等式连接在一起，例如 −3 < 2x + 1 ≤ 5。这意味着 2x + 1 同时大于 −3 且小于等于 5。

To solve, apply the same operation to all three parts at once. Subtract 1 from all parts, then divide every part by the same positive number. If you divide by a negative number, you must reverse both inequality signs.

求解时需同时对三部分进行相同运算。先将三部分都减去 1，再将每一部分除以同一个正数。如果除以负数，则两个不等号都要反转。

This type often appears in questions asking for integer solutions. After solving, list the integers that satisfy the range, making sure you include or exclude endpoints correctly.

这类题常要求写出整数解。求出范围后，列出满足该范围的整数，并确保正确处理端点。

−3 < 2x + 1 ≤ 5 → −4 < 2x ≤ 4 → −2 < x ≤ 2

Integer solutions are −1, 0, 1 and 2. Notice that −2 is not included because the inequality is strict at that end, while 2 is included.

整数解为 −1、0、1 和 2。注意 −2 不包含在内，因为该端为严格不等号，而 2 包含在内。

10. Exam-Style Tips for Algebra | 代数题考试技巧

Always show each step of working, especially when signs, brackets or fractions are involved. IGCSE examiners award method marks for correct processes even if the final answer is wrong.

始终展示每一步计算过程，尤其是涉及符号、括号或分数时。即使最终答案错误，IGCSE 考官也会对正确步骤给予方法分。

Check your solution by substituting it back into the original equation or inequality. For an equation, both sides should be equal. For an inequality, test a value inside and a value outside the solution set.

将解代回原方程或不等式进行检验。对于方程，两边应相等。对于不等式，可在解集内和解集外各取一个值进行验证。