📚 Solving Linear Equations and Inequalities | 线性方程与不等式的求解
Linear equations and inequalities are fundamental algebraic skills in IGCSE Mathematics. They appear in many forms, from simple one-step equations to word problems and number-line representations. This guide explains how to solve them systematically, avoid common errors, and present answers clearly in the exam.
线性方程与不等式是 IGCSE 数学中基础且重要的代数技能。它们以多种形式出现，从简单的一步方程到文字题和数轴表示。本指南介绍如何系统地求解、避免常见错误，并在考试中清晰地展示答案。
1. Linear Equations: The Basics | 线性方程基础
ile x² - 5 = 0 is not linear. The solution is the value of x that makes the equation true.
线性方程是指变量的指数为 1 的方程。例如，2x + 3 = 7 是线性的，而 x² - 5 = 0 不是线性的。方程的解就是使等式成立的 x 的值。
such as 5x - 4 = 2x + 8. The goal is to isolate the variable on one side of the equation.
在 IGCSE 考试中，你可能需要解形如 5x - 4 = 2x + 8 的方程。目标是将变量单独放在等式的一侧。
2. Balancing Method and Inverse Operations | 平衡法与逆运算
The balancing method treats an equation like a balance scale. Whatever you do to one side, you must also do to the other side. Use inverse operations in the correct order: undo addition or subtraction first, then undo multiplication or division.
平衡法把等式看作一架天平：对一侧进行任何操作，也必须对另一侧进行同样的操作。按正确顺序使用逆运算：先消去加法或减法，再消去乘法或除法。
0, subtract 5 from both sides to get 3x = 15, then divide both sides by 3 to obtain x = 5.
例如，解 3x + 5 = 20 时，两边先减去 5，得到 3x = 15，然后两边除以 3，得到 x = 5。
3x + 5 = 20 → 3x = 15 → x = 5
3. Equations with Brackets | 含括号的方程
mple, 2(3x - 1) = 4x + 6 expands to 6x - 2 = 4x + 6. Then collect like terms to isolate x.
当方程含有括号时，先用分配律展开括号。例如，2(3x - 1) = 4x + 6 展开为 6x - 2 = 4x + 6。然后合并同类项以分离 x。
Be careful with negative signs: -3(x + 4) means -3x - 12, not -3x + 12.
注意负号：-3(x + 4) 表示 -3x - 12，而不是 -3x + 12。
4. Equations with Fractions | 含分数的方程
To solve an equation with fractions, multiply every term by the lowest common denominator (LCD). This removes the denominators and creates a simpler linear equation.
要解含分数的方程，将每一项乘以最小公分母 (LCD)。这样可以消去分母，得到更简单的线性方程。
5. The LCD of 2 and 3 is 6. Multiplying through gives 3x + 2x = 30, so 5x = 30 and x = 6.
例如：解 x/2 + x/3 = 5。2 和 3 的最小公分母是 6。两边乘以 6 得 3x + 2x = 30，所以 5x = 30，x = 6。
x/2 + x/3 = 5 ×6 → 3x + 2x = 30 → x = 6
5. Forming Equations from Word Problems | 由文字题列方程
Many IGCSE questions ask you to form and then solve an equation from a real-life context. Start by letting the unknown quantity be x. Then translate the words into algebraic expressions and set up an equation.
许多 IGCSE 题目要求根据实际情境列出方程并求解。首先设未知量为 x，然后将文字信息转化为代数表达式并建立方程。
For example, "Three times a number increased by 4 equals 19." Let the number be x: 3x + 4 = 19. Solving gives x = 5.
例如，“一个数的 3 倍加 4 等于 19。”设这个数为 x：3x + 4 = 19。解得 x = 5。
- Read the question carefully and define the variable.
- Write each condition as an algebraic expression.
- Solve and check the answer in context.
- 仔细阅读题目并定义变量。
- 将每个条件写成代数表达式。
- 求解并在情境中检验答案。
6. Introduction to Linear Inequalities | 线性不等式入门
imilar to solving an equation, but the solution is usually a range of values rather than one value.
线性不等式用 <、>、≤ 和 ≥ 等符号比较两个表达式。解不等式与解方程类似，但解通常是一个取值范围，而不是一个值。
real numbers greater than 3, while x ≤ -1 means all real numbers less than or equal to -1.
例如，x > 3 表示所有大于 3 的实数，而 x ≤ -1 表示所有小于或等于 -1 的实数。
7. Solving Linear Inequalities | 解线性不等式
Use the same balancing method as equations, but with one extra rule: when you multiply or divide both sides by a negative number, you must reverse the inequality sign. For example, from -2x < 6, dividing by -2 gives x > -3.
使用与解方程相同的平衡法，但有一条额外规则：当两边乘以或除以一个负数时，必须反转不等号。例如，由 -2x < 6，两边除以 -2 得 x > -3。
- 3 ≤ 2x + 9. Subtract 2x from both sides: 3x - 3 ≤ 9. Add 3: 3x ≤ 12. Divide by 3: x ≤ 4.
例如：解 5x - 3 ≤ 2x + 9。两边减 2x：3x - 3 ≤ 9。加 3：3x ≤ 12。除以 3：x ≤ 4。
-2x < 6 ÷(-2) → x > -3
8. Inequality Graphs on a Number Line | 数轴上的不等式表示
> 2 or x < 2. Use a closed circle for inclusive inequalities such as x ≥ 2 or x ≤ 2.
不等式解常用数轴表示。严格不等式如 x > 2 或 x < 2 使用空心圆；包含等号的不等式如 x ≥ 2 或 x ≤ 2 使用实心圆。
t -1 and shade to the right. For x < 3, draw an open circle at 3 and shade to the left.
对于 x ≥ -1，在 -1 处画实心圆并向右画阴影；对于 x < 3，在 3 处画空心圆并向左画阴影。
9. Compound Inequalities | 复合不等式
x is greater than 2 and less than or equal to 5. It can be written as x > 2 and x ≤ 5.
复合不等式由两个不等式组合而成。例如，2 < x ≤ 5 表示 x 大于 2 且小于或等于 5。它可以写成 x > 2 且 x ≤ 5。
; 7, operate on all three parts at once. Add 3: 4 ≤ 2x < 10. Divide by 2: 2 ≤ x < 5.
解形如 1 ≤ 2x - 3 < 7 的复合不等式时，要同时对三部分进行运算。加 3：4 ≤ 2x < 10。除以 2：2 ≤ x < 5。
1 ≤ 2x - 3 < 7 → 4 ≤ 2x < 10 → 2 ≤ x < 5
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common mistakes include forgetting to reverse the inequality sign when dividing by a negative, losing a denominator when clearing fractions, and expanding brackets with negative signs incorrectly.
常见错误包括：除以负数时忘记反转不等号、去分母时漏乘、展开带负号的括号时出错。
In the exam, always show each step clearly. Check your solution by substituting it back into the original equation or inequality. If an answer seems unreasonable, go back and look for sign errors.
考试时要清楚地写出每一步。将解代回原方程或不等式进行检验。如果答案不合理，返回检查符号错误。
- Reverse the sign when multiplying or dividing by a negative.
- Multiply every term by the LCD for equations with fractions.
- Use open/closed circles correctly on number lines.
- Check final answers in the original expression.
- 乘以或除以负数时反转不等号。
- 解含分数方程时每一项都乘以最小公分母。
- 在数轴上正确使用空心圆和实心圆。
- 将最终答案代回原式检验。
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