📚 IGCSE Maths: Linear Equations and Inequalities | IGCSE 数学：线性方程与不等式
Linear equations and inequalities are the building blocks of IGCSE algebra. They appear in almost every exam paper, both as standalone questions and as steps within larger problems. This guide covers the essential methods, common pitfalls, and exam techniques you need to solve linear equations and inequalities with confidence.
线性方程和不等式是 IGCSE 代数的基础模块。它们几乎出现在每一份试卷中，无论是作为独立题目还是作为较大问题的中间步骤。本指南涵盖你需要掌握的基本方法、常见错误和考试技巧，帮助你自信地求解线性方程和不等式。
1. Solving Basic Linear Equations | 基本线性方程的求解
A basic linear equation has the form ax + b = c, where a, b and c are constants and x is the unknown. To solve it, use inverse operations in the reverse of BIDMAS order: first add or subtract, then multiply or divide.
基本线性方程的形式为 ax + b = c，其中 a、b、c 是常数，x 是未知数。求解时，按照 BIDMAS 的逆序进行逆运算：先加减，后乘除。
Solve: 3x + 5 = 20
Subtract 5 from both sides: 3x = 15. Then divide both sides by 3: x = 5.
两边同时减去 5：3x = 15。然后两边同时除以 3：x = 5。
he original equation to check it. Here, 3(5) + 5 = 20 is true, so the solution is correct.
始终将答案代回原方程进行检验。此处 3(5) + 5 = 20 成立，因此解是正确的。
it is written as ax = b, divide both sides by a. Keep the equation balanced at every step.
如果方程写成 x/a = b，两边同时乘以 a；如果写成 ax = b，两边同时除以 a。每一步都要保持方程平衡。
2. Equations with Variables on Both Sides | 两边含未知数的方程
When an equation has variables on both sides, first collect all variable terms on one side and all constant terms on the other. Choose the side with the larger coefficient of x to keep x positive.
当方程两边都有未知数时，首先将所有含未知数的项移到一边，将所有常数项移到另一边。选择 x 系数较大的一边，使 x 保持为正。
Solve: 5x − 3 = 2x + 9
Subtract 2x from both sides: 3x − 3 = 9. Add 3 to both sides: 3x = 12. Divide by 3: x = 4.
两边同时减去 2x：3x − 3 = 9。两边同时加 3：3x = 12。除以 3：x = 4。
Remember that whatever operation you do to one side, you must do to the other side. This keeps the equation balanced.
记住，对一边进行任何运算，都必须对另一边进行相同的运算。这样才能保持方程平衡。
x − 5 gives 12 = 6x, so x = 2. Check: 7 − 4 = 3 and 8 − 5 = 3, so the solution is correct.
例如，解 7 − 2x = 4x − 5 得到 12 = 6x，所以 x = 2。检验：7 − 4 = 3，8 − 5 = 3，因此解正确。
3. Equations with Brackets | 含括号的方程
When brackets appear, expand them first unless both sides share a common factor that can be divided out. Be careful with negative signs in front of brackets.
当方程中出现括号时，先展开括号，除非两边有公因数可以直接约去。特别注意括号前的负号。
Solve: 2(3x − 1) = 4x + 8
e left side: 6x − 2 = 4x + 8. Subtract 4x: 2x − 2 = 8. Add 2: 2x = 10. Divide by 2: x = 5.
展开左边：6x − 2 = 4x + 8。减去 4x：2x − 2 = 8。加 2：2x = 10。除以 2：x = 5。
e number, every term inside changes sign. For example, −3(x − 4) = −3x + 12, not −3x − 12.
如果括号乘以负数，括号内每一项都要变号。例如，−3(x − 4) = −3x + 12，而不是 −3x − 12。
nother example: 5 − 2(x − 3) = 11. Expand: 5 − 2x + 6 = 11, so 11 − 2x = 11, giving x = 0.
另一个例子：5 − 2(x − 3) = 11。展开：5 − 2x + 6 = 11，即 11 − 2x = 11，得到 x = 0。
4. Equations with Fractions | 含分数的方程
For equations with fractions, multiply every term by the lowest common multiple of the denominators. This clears the fractions and turns the equation into a simpler linear equation.
对于含分数的方程，将每一项乘以分母的最小公倍数。这样可以消去分数，把方程变成更简单的线性方程。
Solve: x/2 + x/3 = 5
The LCM of 2 and 3 is 6. Multiply every term by 6: 3x + 2x = 30, so 5x = 30 and x = 6.
2 和 3 的最小公倍数是 6。每一项乘以 6：3x + 2x = 30，所以 5x = 30，x = 6。
, cross-multiply: 4(x + 1) = 3(2x − 1). This gives 4x + 4 = 6x − 3, so 7 = 2x and x = 3.5.
对于 (x + 1)/3 = (2x − 1)/4 这样的方程，交叉相乘：4(x + 1) = 3(2x − 1)。得到 4x + 4 = 6x − 3，所以 7 = 2x，x = 3.5。
If a denominator contains x, always check that your final answer does not make any denominator zero. Such values must be excluded from the solution set.
如果分母中含有 x，一定要检验最终答案是否会使任何分母为零。这样的值必须从解集中排除。
5. Rearranging Formulas | 公式变形
Rearranging a formula means making a different variable the subject. Use the same inverse operations as when solving equations, treating all other letters as constants.
公式变形是指把另一个变量变为subject（主元）。使用与解方程相同的逆运算，把所有其他字母当作常数处理。
Make x the subject: 2x + 3y = 12
Subtract 3y from both sides: 2x = 12 − 3y. Divide by 2: x = (12 − 3y)/2.
两边同时减去 3y：2x = 12 − 3y。除以 2：x = (12 − 3y)/2。
, make t the subject: subtract u to get v − u = at, then divide by a to get t = (v − u)/a.
对于 v = u + at，令 t 为主元：减去 u 得 v − u = at，再除以 a 得 t = (v − u)/a。
ne term, factorise first. For example, ax + bx = c becomes x(a + b) = c, so x = c/(a + b).
如果未知数出现在多个项中，先因式分解。例如，ax + bx = c 变为 x(a + b) = c，所以 x = c/(a + b)。
6. Introduction to Linear Inequalities | 线性不等式简介
s: <, ≤, >, ≥ or ≠. A linear inequality contains only terms in x to the first power.
不等式用以下符号之一比较两个表达式：<、≤、>、≥ 或 ≠。线性不等式只含有 x 的一次项。
x < 5 means x is less than 5. x ≥ −2 means x is greater than or equal to −2.
The solution of an inequality is a range of values, not a single value. You can represent this range on a number line or using interval notation.
不等式的解是一个取值范围，而不是单个值。你可以用数轴或区间表示法来表示这个范围。
facing outward means the endpoint is not included; a square bracket means it is included.
常见区间表示法：(a, b) 表示 a < x < b，而 [a, b] 表示 a ≤ x ≤ b。向外开口的括号表示端点不包含；方括号表示端点包含。
7. Solving Linear Inequalities | 解线性不等式
Linear inequalities are solved in almost the same way as linear equations. Add, subtract, multiply or divide both sides to isolate x, but keep the inequality sign in place.
线性不等式的解法与线性方程几乎相同。通过加减乘除两边来孤立 x，但保持不等号不变。
Solve: 3x + 5 < 20
Subtract 5 from both sides: 3x < 15. Divide by 3: x < 5.
两边同时减去 5：3x < 15。除以 3：x < 5。
For 5x − 3 ≥ 2x + 9, subtract 2x: 3x − 3 ≥ 9. Add 3: 3x ≥ 12. Divide by 3: x ≥ 4.
对于 5x − 3 ≥ 2x + 9，减去 2x：3x − 3 ≥ 9。加 3：3x ≥ 12。除以 3：x ≥ 4。
The only extra rule you must remember is the reversal rule when multiplying or dividing by a negative number. This is covered in the next section.
唯一需要额外记住的规则是乘以或除以负数时的变号规则。下一节将详细讲解。
8. The Reversal Rule for Negative Coefficients | 负系数变号规则
When you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed. This is the most common source of lost marks in IGCSE inequalities.
当不等式两边同时乘以或除以负数时，不等号必须反向。这是 IGCSE 不等式题目中最常见的失分点。
Solve: −2x > 8
Divide both sides by −2 and reverse the sign: x < −4.
两边同时除以 −2，并反向不等号：x < −4。
For 5 − 3x ≤ 11, subtract 5: −3x ≤ 6. Divide by −3 and reverse: x ≥ −2.
对于 5 − 3x ≤ 11，减去 5：−3x ≤ 6。除以 −3 并反向：x ≥ −2。
the solution set. For x ≥ −2, try x = 0: 5 − 0 = 5 ≤ 11 is true, so the answer is correct.
检验时，在解集中取一个值测试。对于 x ≥ −2，取 x = 0：5 − 0 = 5 ≤ 11 成立，因此答案正确。
9. Representing Inequalities on a Number Line | 在数轴上表示不等式
; and a filled circle for ≤ or ≥. Then draw an arrow in the direction of the solution set.
在数轴上，用空心圆表示 < 或 >，用实心圆表示 ≤ 或 ≥。然后朝解集方向画箭头。
; −2: open circle at −2, arrow to the right. x ≤ 3: filled circle at 3, arrow to the left.
x ≤ 3, draw an open circle at −2, a filled circle at 3, and join them with a line segment.
对于 −2 < x ≤ 3 这样的组合不等式，在 −2 处画空心圆，在 3 处画实心圆，并用线段连接。
ritten as (−2, 3]. The round bracket means −2 is not included, and the square bracket means 3 is included.
用区间表示法，−2 < x ≤ 3 写作 (−2, 3]。圆括号表示 −2 不包含，方括号表示 3 包含。
Always label the number line clearly and mark the key values. Examiners award method marks for correct circles and arrows.
务必清晰标注数轴并标出关键值。正确的圆圈和箭头可以获得方法分。
10. Integer Solutions and Word Problems | 整数解与应用题
Some exam questions ask for integer solutions that satisfy an inequality. Solve the inequality first, then list all integers in the solution set.
有些考试题目要求求出满足不等式的整数解。先解不等式，然后列出解集中的所有整数。
Find the integer solutions of 3 < 2x + 1 ≤ 11.
arts: 2 < 2x ≤ 10. Divide by 2: 1 < x ≤ 5. The integer solutions are x = 2, 3, 4, 5.
三边同时减去 1：2 < 2x ≤ 10。除以 2：1 < x ≤ 5。整数解为 x = 2, 3, 4, 5。
For word problems, define the unknown clearly, form an equation or inequality, solve it, and check that the answer makes sense in the context. For example, age problems often use expressions such as x + 5 or 3x.
对于应用题，先明确定义未知数，列出方程或不等式，求解，并检查答案在情境中是否合理。例如，年龄问题常用 x + 5 或 3x 这样的表达式。
th is 3 cm more than the width. Let the width be w, then 2(w + w + 3) = 36, so w = 7.5 cm.
一个常见应用题：矩形周长为 36 cm，长比宽多 3 cm。设宽为 w，则 2(w + w + 3) = 36，所以 w = 7.5 cm。
11. Common Exam Mistakes | 常见考试错误
One of the most frequent mistakes is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. Always pause and check the sign before writing the final answer.
最常见的错误之一是在乘以或除以负数时忘记反向不等号。写出最终答案前，一定要停下来检查符号。
example, in x/2 + x/3 = 5, you must multiply all three terms by 6, not just the fractions.
另一个错误是在分数方程中没有将每一项都乘以公分母。例如，在 x/2 + x/3 = 5 中，必须将三项都乘以 6，而不仅仅是分数项。
n. Remember that subtracting a bracket changes every sign inside: a − (b + c) = a − b − c.
展开括号时的符号错误也非常常见。记住，减去一个括号会改变括号内每一项的符号：a − (b + c) = a − b − c。
Finally, do not forget to check whether a solution makes any denominator zero in an equation with fractions involving x. Such values are not valid solutions.
最后，在含有 x 的分数方程中，不要忘记检查解是否使某个分母为零。这样的值不是有效解。
12. Exam Technique and Checklist | 考试技巧与检查清单
Show every step of your working. Even if your final answer is wrong, correct method steps can earn valuable marks. Write equations in a clear, vertical layout so the examiner can follow your reasoning.
写出每一步过程。即使最终答案错误，正确的方法步骤也能获得宝贵的分数。将方程竖直、清晰地排列，方便考官理解你的推理。
Always check your answer by substituting it back into the original equation or inequality. For inequalities, test a value inside the solution set and a value outside it if you have time.
始终将答案代回原方程或不等式进行检验。对于不等式，如果有时间，可以测试解集内和解集外各一个值。
Use a number line to visualise inequalities, especially combined inequalities. This helps you avoid mistakes with included and excluded endpoints.
用数轴将不等式可视化，尤其是组合不等式。这有助于避免端点的包含与排除错误。
- Checklist: Expand brackets correctly. Clear fractions by multiplying every term. Collect variables on one side. Reverse the sign when multiplying or dividing by a negative. Check your solution.
- 检查清单：正确展开括号。每一项乘以公分母消去分数。将未知数项移到一边。乘以或除以负数时反向不等号。检验你的解。
With regular practice, linear equations and inequalities become quick and reliable marks in the IGCSE exam. Focus on accuracy, show clear steps, and always verify your final answer.
通过规律练习，线性方程和不等式会成为 IGCSE 考试中快速而可靠的得分点。注重准确性，写出清晰步骤，并始终验证最终答案。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图