📚 IGCSE Mathematics: Solving Linear Equations and Inequalities | IGCSE 数学：解一元一次方程与不等式
Linear equations and inequalities are among the most important algebra skills in IGCSE Mathematics. They appear in almost every exam paper, both as standalone questions and as steps in larger problems involving graphs, sequences, and word problems.
一元一次方程和不等式是 IGCSE 数学中最重要的代数技能之一。它们几乎出现在每份试卷中，既可以作为独立题目，也会出现在涉及图像、数列和应用题的较大问题中。
1. What Are Linear Equations? | 什么是一元一次方程？
can be written in the form ax + b = c, where a, b, and c are constants and a ≠ 0. The word 'linear' means that the variable x appears only to the first power.
一元一次方程是指可以写成 ax + b = c 形式的方程，其中 a、b、c 为常数且 a ≠ 0。"一元"表示方程中只有一个未知数，"一次"表示未知数 x 的指数为 1。
mple, 2x + 3 = 11, 5x − 4 = 6, and 3(x + 2) = 18 are all linear equations in one variable.
例如，2x + 3 = 11、5x − 4 = 6 和 3(x + 2) = 18 都是一元一次方程。
2. Solving Basic Linear Equations | 解基本的一元一次方程
To solve a linear equation, use inverse operations to isolate the unknown on one side of the equation. Whatever you do to one side, you must do to the other side to keep the equation balanced.
解一元一次方程时，需要使用逆运算将未知数单独留在方程的一边。对等式一边进行的任何操作，都必须同时对另一边进行，以保持等式平衡。
2x + 3 = 11 ⇒ 2x = 8 ⇒ x = 4
Subtract 3 from both sides first, then divide both sides by 2 to find x = 4.
先在两边同时减去 3，然后两边同时除以 2，得到 x = 4。
substituting it back into the original equation: 2(4) + 3 = 8 + 3 = 11, which is correct.
一定要把答案代入原方程进行检验：2(4) + 3 = 8 + 3 = 11，结果正确。
3. Equations with Unknowns on Both Sides | 含两边未知数的方程
When an equation has unknowns on both sides, collect the variable terms on one side and the constant terms on the other side. Add or subtract the same term from both sides to begin.
当方程两边都有未知数时，应把含未知数的项移到一边，把常数项移到另一边。首先在两边同时加上或减去同一项。
. Subtract 2x from both sides to get 2x + 5 = 13, then subtract 5 to get 2x = 8, so x = 4.
例如：4x + 5 = 2x + 13。两边同时减去 2x，得到 2x + 5 = 13，再同时减去 5，得到 2x = 8，因此 x = 4。
4. Equations with Brackets | 含括号的方程
distributive property a(b + c) = ab + ac. Be careful with negative signs outside brackets.
如果方程中含有括号，应先用分配律 a(b + c) = ab + ac 展开括号。当括号外有负号时，要特别小心。
Example: 3(x + 2) = 18. Expand: 3x + 6 = 18, subtract 6: 3x = 12, divide by 3: x = 4.
例如：3(x + 2) = 18。展开得 3x + 6 = 18，两边减 6 得 3x = 12，再除以 3 得 x = 4。
For 5 − 2(x − 1) = 3, expand carefully: 5 − 2x + 2 = 3, so 7 − 2x = 3, giving x = 2.
对于 5 − 2(x − 1) = 3，要小心展开：5 − 2x + 2 = 3，所以 7 − 2x = 3，解得 x = 2。
5. Equations with Fractions | 含分数的方程
To solve equations with fractions, multiply every term by the lowest common denominator (LCD) to clear the fractions. This produces a simpler linear equation.
解含分数的方程时，将每一项都乘以最小公分母（LCD），以消去分母。这样可以得到一个更简单的一元一次方程。
Example: x/2 + 3 = 7. Multiply all terms by 2: x + 6 = 14, so x = 8.
例如：x/2 + 3 = 7。所有项乘以 2，得 x + 6 = 14，因此 x = 8。
)/2, cross-multiply or multiply by 6: 2(x + 1) = 3(x − 2), then 2x + 2 = 3x − 6, so x = 8.
如果方程是 (x + 1)/3 = (x − 2)/2，可以交叉相乘或两边乘以 6：2(x + 1) = 3(x − 2)，展开得 2x + 2 = 3x − 6，所以 x = 8。
6. Introduction to Inequalities | 不等式入门
≥. Unlike an equation, an inequality usually has many possible solutions forming a range.
不等式使用 <、>、≤、≥ 等符号来比较两个表达式。与方程不同，不等式通常有多个解，形成一个范围。
| Symbol | Meaning | Example |
| < | Less than | x < 3 |
| > | Greater than | x > -2 |
| ≤ | Less than or equal to | x ≤ 5 |
| ≥ | Greater than or equal to | x ≥ 0 |
, x > 3 means all numbers greater than 3. x ≤ 5 means all numbers less than or equal to 5.
例如，x > 3 表示所有大于 3 的数。x ≤ 5 表示所有小于或等于 5 的数。
7. Solving Linear Inequalities | 解一元一次不等式
Solving a linear inequality is very similar to solving a linear equation. Use inverse operations to isolate the variable, but remember one key rule: if you multiply or divide both sides by a negative number, reverse the inequality symbol.
解一元一次不等式与解一元一次方程非常相似。使用逆运算将未知数单独留在一边，但有一条关键规则：如果两边同时乘以或除以一个负数，必须改变不等号的方向。
Example: 2x − 3 < 7. Add 3: 2x < 10, divide by 2: x < 5.
例如：2x − 3 < 7。两边加 3：2x < 10，再除以 2：x < 5。
Example with negative coefficient: −3x > 9. Divide by −3 and reverse the sign: x < −3.
含有负系数的例子：−3x > 9。两边除以 −3 并改变不等号方向：x < −3。
8. Representing Inequalities on a Number Line | 在数轴上表示不等式
inclusive inequalities (≤ or ≥). Then shade the line in the direction of the solution set.
在数轴上表示不等式时，严格不等式（< 或 >）使用空心圆圈，含等号的不等式（≤ 或 ≥）使用实心圆圈。然后向解集的方向延伸阴影线。
t 2 and shade to the right. For x ≤ −1, place a closed circle at −1 and shade to the left.
对于 x > 2，在 2 处画空心圆圈，并向右画阴影线。对于 x ≤ −1，在 −1 处画实心圆圈，并向左画阴影线。
2 < x ≤ 3 are shaded between −2 and 3, with an open circle at −2 and a closed circle at 3.
形如 −2 < x ≤ 3 的复合不等式应在 −2 和 3 之间画阴影线，在 −2 处用空心圆圈，在 3 处用实心圆圈。
9. Word Problems and Forming Equations | 应用题与列方程
Word problems require translating a written statement into an equation or inequality. Read the question carefully and define the unknown with a letter such as x.
应用题需要将文字描述转化为方程或不等式。仔细阅读题目，并用 x 等字母表示未知数。
Example: "Three more than twice a number is 15." Let the number be x, then 2x + 3 = 15, giving x = 6.
例如："一个数的两倍再加 3 等于 15。"设这个数为 x，则 2x + 3 = 15，解得 x = 6。
For inequalities, look for phrases such as 'at least' (≥), 'at most' (≤), 'more than' (>), and 'less than' (<).
对于不等式，要注意"至少"（≥）、"至多"（≤）、"多于"（>）和"少于"（<）等关键词。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Students often make mistakes when handling negative signs, reversing inequality symbols, or checking answers. Always write each line clearly and perform the same operation on both sides.
学生在处理负号、改变不等号方向或检验答案时经常出错。一定要清晰地写出每一步，并在两边同时进行相同的运算。
Common errors include forgetting to reverse the inequality when dividing by a negative, incorrectly expanding brackets, and not multiplying every term by the LCD when clearing fractions.
常见错误包括：除以负数时忘记改变不等号方向、括号展开错误、消去分母时没有将每一项都乘以最小公分母。
In the exam, show all working because method marks are awarded even if the final answer is wrong. Always re-read the question to check whether the answer should be an equation solution or an inequality range.
考试中要写出完整过程，因为即使最终答案错误，步骤分也会得到。最后一定重新审题，确认答案应是方程的解还是不等式的范围。
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