Solving Linear Equations and Inequalities | 一次方程与不等式的解法

📚 Solving Linear Equations and Inequalities | 一次方程与不等式的解法

Linear equations and inequalities form the backbone of Year 9 algebra. They appear in nearly every school assessment, and the skills you build here will be used again in IGCSE and A-Level mathematics. This revision guide explains the key methods step by step, with examples, common mistakes, and exam-ready strategies.

一次方程与不等式是九年级代数的核心内容。它们几乎出现在每一次校内测验中，你在这一单元建立的技能也将在 IGCSE 和 A-Level 数学中反复使用。本篇复习指南将逐步讲解关键方法，并配有例题、常见错误和应考策略。

1. What Is a Linear Equation? | 什么是一次方程

A linear equation is an equation in which the unknown, usually written as x, is only raised to the power 1. The graph of a linear equation is a straight line, which is why it is called linear. The main goal when solving any equation is to isolate the unknown on one side of the equals sign.

一次方程是指未知数（通常写作 x）的指数仅为 1 的方程。一次方程的图像是一条直线，因此被称为线性方程。解任何方程的主要目标都是把未知数单独移到等号的一边。

For example, 3x + 2 = 14 is a linear equation. The expression 3x + 2 is equal to 14 for exactly one value of x, and our job is to find it. Below are the key differences between an expression and an equation.

例如，3x + 2 = 14 是一个一次方程。当 x 取某一个特定值时，3x + 2 才等于 14，我们的任务就是找出这个值。下面列出了表达式与方程的主要区别。

Equation: 2x + 3 = 11 (has an equals sign and can be solved)
Expression: 2x + 3 (no equals sign; it can only be simplified or evaluated)
方程：2x + 3 = 11（含有等号，可以求解）
表达式：2x + 3（没有等号，只能化简或代入求值）

2. One-step and Two-step Equations | 一步方程与两步方程

A one-step equation needs only one inverse operation. If the equation says x + 7 = 15, subtract 7 from both sides to get x = 8. If it says 5x = 20, divide both sides by 5 to get x = 4.

一步方程只需要一种逆运算。例如 x + 7 = 15，两边同时减去 7，得到 x = 8。若是 5x = 20，两边同时除以 5，得到 x = 4。

A two-step equation such as 2x + 3 = 11 is solved by reversing the order of operations. First subtract 3, then divide by 2. The solution is x = 4.

两步方程如 2x + 3 = 11 需要按照运算顺序的相反方向来解。先减去 3，再除以 2，得到 x = 4。

2x + 3 = 11 → 2x = 8 → x = 4

3. Equations with Brackets | 括号方程

When an equation contains brackets, expand them first using the distributive law. For 3(x + 2) = 18, multiply both x and 2 by 3, giving 3x + 6 = 18. Then subtract 6 and divide by 3 to get x = 4.

当方程含有括号时，先用乘法分配律展开。例如 3(x + 2) = 18，把 x 和 2 都乘以 3，得到 3x + 6 = 18。然后减去 6，再除以 3，得到 x = 4。

If the bracket has a negative coefficient, keep the sign with the number. For -2(x – 5) = 8, expand to -2x + 10 = 8, then solve to get x = 1.

如果括号前是负系数，展开时要保留符号。例如 -2(x – 5) = 8，展开为 -2x + 10 = 8，然后解得 x = 1。

-2(x – 5) = 8 → -2x + 10 = 8 → -2x = -2 → x = 1

4. Unknown on Both Sides | 两边含未知数的方程

When the unknown appears on both sides, collect like terms onto the side where the coefficient is larger. For 7x – 3 = 3x + 13, subtract 3x from both sides to get 4x – 3 = 13, then add 3 and divide by 4 to get x = 4.

当未知数出现在等号两边时，把未知数项合并到系数较大的一边。例如 7x – 3 = 3x + 13，两边同时减去 3x，得到 4x – 3 = 13，再加上 3，最后除以 4，得到 x = 4。

Always check your answer by substituting it back into the original equation. Here the left side becomes 7 × 4 – 3 = 25 and the right side becomes 3 × 4 + 13 = 25.

一定要把答案代回原方程检验。这里左边为 7 × 4 – 3 = 25，右边为 3 × 4 + 13 = 25。

7x – 3 = 3x + 13 → 4x = 16 → x = 4

5. Equations with Fractional Coefficients | 分数系数方程

Fractional coefficients often appear in harder questions. For x/3 + 2 = 6, first subtract 2 to get x/3 = 4, then multiply both sides by 3 to get x = 12.

分数系数的方程常出现在较难的题目中。例如 x/3 + 2 = 6，先减去 2，得到 x/3 = 4，然后两边同时乘以 3，得到 x = 12。

If an equation has two fractions, multiply every term by the lowest common denominator. For x/2 + x/3 = 5, multiply by 6 to get 3x + 2x = 30, so 5x = 30 and x = 6.

如果方程含有两个分数，可以把每一项都乘以最小公分母。例如 x/2 + x/3 = 5，两边乘以 6，得到 3x + 2x = 30，所以 5x = 30，x = 6。

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

6. Introduction to Inequalities | 不等式入门

An inequality compares two expressions using the symbols < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Inequalities do not have one single solution; they have a range of solutions.

不等式用符号 <（小于）、>（大于）、≤（小于等于）和 ≥（大于等于）来比较两个表达式。不等式通常不止一个解，而是一个解的范围。

For example, x > 3 means any value greater than 3 is allowed, such as 4, 5.2, or 100. The value 3 itself is not included.

例如，x > 3 表示任何大于 3 的数都满足条件，如 4、5.2 或 100，但 3 本身不包含在内。

Symbol 符号	Meaning 含义	Example 示例
<	less than 小于	x < 5
>	greater than 大于	x > -2
≤	less than or equal to 小于等于	x ≤ 10
≥	greater than or equal to 大于等于	x ≥ 0

7. Solving One-step Inequalities | 解一步不等式

One-step inequalities are solved like one-step equations. If x + 4 < 10, subtract 4 from both sides to get x < 6. If 5x ≥ 20, divide both sides by 5 to get x ≥ 4.

一步不等式的解法与一步方程相似。例如 x + 4 < 10，两边同时减去 4，得到 x < 6。若 5x ≥ 20，两边除以 5，得到 x ≥ 4。

The only difference is that when you multiply or divide both sides by a negative number, the inequality sign must be reversed. For -2x < 8, divide by -2 and flip the sign to get x > -4.

唯一的区别是：当两边同时乘以或除以一个负数时，不等号的方向必须改变。例如 -2x < 8，两边除以 -2 并翻转符号，得到 x > -4。

-2x < 8 → x > -4

8. Solving Two-step Inequalities | 解两步不等式

Two-step inequalities follow the same reverse-order method. For 3x – 5 ≤ 10, add 5 to both sides to get 3x ≤ 15, then divide by 3 to get x ≤ 5.

两步不等式也使用逆运算顺序。例如 3x – 5 ≤ 10，两边加 5，得到 3x ≤ 15，再除以 3，得到 x ≤ 5。

Be careful when the coefficient of x is negative. For 7 – 2x > 9, first subtract 7 to get -2x > 2, then divide by -2 and flip the sign to get x < -1.

当 x 的系数为负数时要特别小心。例如 7 – 2x > 9，

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Solving Linear Equations and Inequalities | 一次方程与不等式的解法