T-K-1053: Mastering Equations & Inequalities in IGCSE Mathematics | T-K-1053：IGCSE数学中方程与不等式精通

📚 T-K-1053: Mastering Equations & Inequalities in IGCSE Mathematics | T-K-1053：IGCSE数学中方程与不等式精通

Equations and inequalities are essential tools in IGCSE Mathematics. They allow us to model real-world situations, uncover unknown values, and describe ranges of possible answers. This article presents a structured revision guide covering linear and quadratic equations, simultaneous equations, inequalities, and common problem-solving strategies.

方程与不等式是IGCSE数学中不可或缺的工具。它们帮助我们建立现实情境的模型、求解未知数值，并描述可能答案的范围。本文将提供一个结构化的复习指南，涵盖线性方程、二次方程、联立方程、不等式以及常见的解题策略。

1. Linear Equations | 线性方程

A linear equation is an equation where the unknown variable has exponent 1. The goal is to isolate the variable using inverse operations such as addition, subtraction, multiplication, and division.

线性方程是指未知数指数为1的方程。解题目标是通过加法、减法、乘法和除法等逆运算将变量单独分离出来。

Example: Solve 3x − 7 = 2x + 5.

示例：解方程 3x − 7 = 2x + 5。
First subtract 2x from both sides: 3x − 2x − 7 = 5, giving x − 7 = 5.

首先两边同时减去 2x：3x − 2x − 7 = 5，得到 x − 7 = 5。
Then add 7 to both sides: x = 12.

然后两边同时加 7：x = 12。

General method: ax + b = cx + d → x = (d − b) / (a − c), provided a ≠ c.

一般方法：ax + b = cx + d → x = (d − b) / (a − c)，前提是 a ≠ c。

When equations contain fractions, multiply every term by the common denominator to simplify before solving.

当方程中含有分数时，先将每一项乘以公分母，以简化后再求解。

2. Quadratic Equations | 二次方程

A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. There are three main solution methods: factorisation, the quadratic formula, and completing the square.

二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。主要的三种解法是：因式分解法、求根公式法和配方法。

x = (−b ± √(b² − 4ac)) / (2a)

The expression b² − 4ac is called the discriminant. If it is positive, there are two distinct real roots; if zero, one repeated root; if negative, no real roots.

表达式 b² − 4ac 称为判别式。若其值为正，则有两个不同的实数根；若为零，则有一个重根；若为负，则没有实数根。

Method \| 方法	Example \| 示例
Factorisation 因式分解	x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3
Quadratic Formula 求根公式	2x² + 5x − 3 = 0 → x = 0.5 or x = −3
Completing Square 配方法	x² + 6x + 1 = 0 → (x + 3)² − 8 = 0 → x = −3 ± 2√2

Always check solutions by substituting back into the original equation, especially when squaring or using the formula.

务必通过将解代回原方程进行验算，特别是在使用平方或求根公式的情况下。

3. Simultaneous Equations | 联立方程

Simultaneous equations involve two or more equations with multiple unknowns. The two most common techniques are elimination and substitution.

联立方程组包含两个或多个含有多个未知数的方程。最常用的两种方法是消元法和代入法。

Elimination is efficient when coefficients of one variable are opposites or can be made opposites by multiplication.

当某个变量的系数互为相反数，或可以通过乘法使其互为相反数时，消元法最为高效。

2x + y = 7
3x − y = 8

Add the two equations to eliminate y: 5x = 15 → x = 3. Substitute x = 3 into 2x + y = 7 to get 6 + y = 7 → y = 1.

将两个方程相加以消去 y：5x = 15 → x = 3。将 x = 3 代入 2x + y = 7，得 6 + y = 7 → y = 1。

Substitution is useful when one equation is already solved for a variable, or when dealing with non-linear equations.

当某个方程已经解出其中一个变量，或方程组中包含非线性方程时，代入法更为方便。

For simultaneous equations involving a quadratic, substitute the linear expression into the quadratic, solve the resulting quadratic, then find the corresponding y-values.

对于含有二次方程的联立方程组，将线性表达式代入二次方程中，求解所得的二次方程，再求出对应的 y 值。

4. Linear Inequalities | 线性不等式

Linear inequalities are similar to linear equations but use the symbols <, >, ≤, or ≥. The solution is often a range of values rather than a single number.

线性不等式与线性方程类似，但使用 <、>、≤ 或 ≥ 符号。其解通常是一个取值范围，而非单个数值。

Rule: Adding or subtracting the same number from both sides does not change the inequality direction.

规则：两边同时加上或减去同一个数，不等号方向不变。
Rule: Multiplying or dividing both sides by a positive number does not change the direction.

规则：两边同时乘以或除以同一个正数，不等号方向不变。
Critical rule: Multiplying or dividing both sides by a negative number reverses the inequality symbol.

关键规则：两边同时乘以或除以同一个负数，不等号方向必须反转。

Example: Solve 5 − 2x ≥ 9. Subtract 5: −2x ≥ 4. Divide by −2 and reverse the sign: x ≤ −2.

示例：解 5 − 2x ≥ 9。两边减 5：−2x ≥ 4。两边除以 −2 并反转不等号：x ≤ −2。

5. Quadratic Inequalities | 二次不等式

Quadratic inequalities involve an unknown squared term and require careful sign analysis. A standard approach is to factorise, find critical values, and test intervals.

二次不等式包含未知数的平方项，需要仔细进行符号分析。标准做法是先因式分解，找出临界值，再检验各区间。

x² − x − 6 < 0

Factorise: (x − 3)(x + 2) < 0. The critical values are x = 3 and x = −2. These values split the number line into three intervals.

因式分解：(x − 3)(x + 2) < 0。临界值为 x = 3 和 x = −2。这两个值将数轴分为三个区间。

Interval 区间	x < −2	−2 < x < 3	x > 3
Sign of product 乘积符号	Positive 正	Negative 负	Positive 正

Since the inequality is < 0, the solution is −2 < x < 3. For > 0, the solution would be x < −2 or x > 3.

因为不等式是 < 0，所以解为 −2 < x < 3。若为 > 0，则解为 x < −2 或 x > 3。

6. Number Line Representation | 数轴表示

Inequalities are often represented on a number line. An open circle indicates that the endpoint is not included; a closed circle indicates that it is included.

不等式常用数轴表示。空心圆表示端点不包含在解集内；实心圆表示端点包含在解集内。

Open circle 空心圆: for < or >, shown as a circle not filled in.

空心圆：用于 < 或 >，以未填充的圆圈表示。
Closed circle 实心圆: for ≤ or ≥, shown as a filled-in dot.

实心圆：用于 ≤ 或 ≥，以填充的黑点表示。
Shading: draw an arrow to the right for greater than, and to the left for less than.

阴影：大于时向右画箭头，小于时向左画箭头。

For a compound inequality such as −1 ≤ x < 4, use a closed circle at −1 and an open circle at 4, shading between them.

对于复合不等式如 −1 ≤ x < 4，在 −1 处用实心圆，在 4 处用空心圆，并将两点之间涂黑。

7. Word Problems with Equations and Inequalities | 方程与不等式应用题

Word problems require translating English statements into mathematical expressions. Identify unknown quantities, define variables, and build equations or inequalities based on the relationships described.

应用题需要将文字描述转化为数学表达式。先确定未知量、定义变量，再根据描述的关系建立方程或不等式。

Example: A rectangle has length 3 cm more than twice its width. Its perimeter is 34 cm. Find the width.

示例：一个矩形的长比宽的2倍多3厘米，其周长为34厘米。求宽。

Let width = w. Length = 2w + 3. Perimeter: 2(w + 2w + 3) = 34 → 6w + 6 = 34 → w = 28/6 = 14/3 cm.

设宽为 w。长 = 2w + 3。周长：2(w + 2w + 3) = 34 → 6w + 6 = 34 → w = 28/6 = 14/3 厘米。

For inequalities, look for phrases such as ‘at least’ (≥), ‘at most’ (≤), ‘more than’ (>), and ‘less than’ (<).

对于不等式，注意短语如“至少”对应 ≥，“至多”对应 ≤，“大于”对应 >，“小于”对应 <。

8. Graphical Methods | 图像法

Graphical methods allow us to solve equations and inequalities visually. The solutions correspond to intersection points or regions on a graph.

图像法使我们能够直观地求解方程和不等式。解对应于图像上的交点或区域。

To solve a pair of simultaneous linear equations graphically, plot both lines on the same axes. The coordinates of their intersection point give the solution.

要在图像上求解一对线性联立方程，可将两条直线绘制在同一坐标系中，它们的交点坐标即为方程组的解。

For a quadratic equation, plot y = ax² + bx + c and find where it crosses the x-axis. These x-values are the roots.

对于二次方程，绘制 y = ax² + bx + c，找出其与 x 轴的交点，这些 x 值即为方程的根。

To solve a quadratic inequality graphically, identify the portions of the curve that lie above or below the x-axis, depending on the inequality sign.

要用图像求解二次不等式，需判断曲线位于 x 轴上方或下方的部分，具体取决于不等号方向。

Always label axes and key points clearly when presenting graphical solutions in an exam.

在考试中展示图像解法时，务必清楚标注坐标轴和关键点。

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students lose marks through avoidable sign errors and procedural mistakes. Understanding the most common pitfalls helps improve accuracy.

许多学生由于可避免的符号错误和步骤错误而失分。了解最常见的陷阱有助于提高准确性。

Mistake 错误	Correction 纠正
Forgetting to reverse inequality when multiplying/dividing by a negative number 乘以或除以负数时忘记反转不等号	Always reverse the symbol when multiplying or dividing by a negative value 乘以或除以负数时务必反转符号
Incorrect expansion of (x − 3)² as x² − 9 错误展开 (x − 3)² 为 x² − 9	Correct expansion: (x − 3)² = x² − 6x + 9 正确展开应为 x² − 6x + 9
Dropping a negative sign when moving terms across the equals sign 移项时漏掉负号	Change the sign of every term that moves to the other side 每一项移项后都要变号
Not checking solutions in word problems 解应用题后未检验	Substitute back into the original conditions and discard impossible values 代回原条件验证，并舍弃不合理的值

In exams, show all working steps clearly. Even if the final answer is incorrect, method marks can still be awarded.

考试中应清晰展示所有解题步骤。即使最终答案不正确，仍可能获得步骤分。

10. Practice Strategies and Summary | 练习策略与总结

Consistent practice is the key to mastering equations and inequalities. Work through past paper questions, categorise mistakes, and revisit weak topics regularly.

持续练习是掌握方程与不等式的关键。通过练习历年真题、对错误进行分类，并定期复习薄弱知识点。

Start with linear equations to build confidence, then move to quadratics and simultaneous equations.

从线性方程开始建立信心，再过渡到二次方程和联立方程组。
Practice sketching graphs of linear and quadratic functions to support graphical solutions.

练习绘制一次和二次函数图像，以辅助图像解法。
For inequalities, always test a value from each interval to confirm the correct range.

对于不等式，始终在每个区间中取一个值进行验证，以确认正确的范围。
Review your errors without looking at the solution first; self-correction deepens understanding.

先不看答案，自行检查错误；自我纠错能加深理解。

Remember the core ideas: isolate the unknown for equations, preserve or reverse the inequality sign correctly, and always interpret the result in the context of the problem.

记住核心要点：解方程时分离未知数；处理不等式时正确保持或反转符号；并始终结合题目背景解释结果。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

T-K-1053: Mastering Equations & Inequalities in IGCSE Mathematics | T-K-1053：IGCSE数学中方程与不等式精通