Unit 6: Algebraic Expressions, Equations and Inequalities | 第6单元：代数式、方程与不等式

📚 Unit 6: Algebraic Expressions, Equations and Inequalities | 第6单元：代数式、方程与不等式

Welcome to this Unit 6 revision guide for Grade 8C mathematics. In this unit, you will strengthen your skills in manipulating algebraic expressions, expanding brackets, factorising, solving linear equations and inequalities, and applying algebra to real-world problems. Mastery of these topics is essential for success in later units on graphs, quadratics and coordinate geometry.

欢迎使用本篇针对 8C 年级数学第 6 单元的复习指南。本单元将帮助你巩固代数式的变形、去括号、因式分解、解一元一次方程与不等式，以及将代数方法应用于实际问题等技能。掌握这些内容对后续学习图像、二次函数和坐标几何至关重要。

1. Learning Objectives | 学习目标

By the end of this unit, you should be able to simplify algebraic expressions by collecting like terms, expand single and double brackets, factorise linear and quadratic expressions, solve linear equations and inequalities, substitute into formulae, rearrange formulae, and translate word problems into algebraic equations.

完成本单元后，你应该能够通过合并同类项来化简代数式，展开单项式和双项式括号，对一次和二次表达式进行因式分解，解一元一次方程和不等式，代入公式，变形公式，并将实际问题转化为代数方程。

Simplify algebraic expressions confidently. — 自信地化简代数式。
Expand and factorise expressions using the distributive law. — 使用分配律展开和因式分解表达式。
Solve equations and inequalities with negative coefficients. — 解含有负系数的一次方程和不等式。
Apply algebra to perimeter, area and number problems. — 将代数应用于周长、面积和数字问题。

2. Algebraic Terminology | 代数术语

In algebra, a term is a combination of numbers, variables and powers multiplied together. A coefficient is the number part of a term, a variable is the letter or symbol used to represent an unknown, and a constant is a term with no variable. For example, in the expression 3x² + 5x – 2, the coefficient of x² is 3, the coefficient of x is 5, and -2 is a constant term.

在代数中，项是由数字、变量和幂相乘而成的组合。系数是项中的数字部分，变量是用来表示未知数的字母或符号，常数项是没有变量的项。例如，在代数式 3x² + 5x – 2 中，x² 的系数是 3，x 的系数是 5，-2 是常数项。

Like terms have exactly the same variable parts, including the same exponents. For instance, 4x and -7x are like terms, but 4x and 4x² are not. Understanding this distinction is the first step in simplifying expressions correctly.

同类项具有完全相同的变量部分，包括相同的指数。例如，4x 和 -7x 是同类项，但 4x 和 4x² 不是。正确理解这一区别是化简表达式的第一步。

3. Simplifying Algebraic Expressions | 化简代数式

Only like terms can be combined by adding or subtracting their coefficients. The variable part remains unchanged. This process is called collecting like terms, and it is used to make expressions shorter and easier to work with.

只有同类项可以通过相加或相减系数来合并。变量部分保持不变。这个过程称为合并同类项，用来使表达式更简短、更容易处理。

For example, the expression 4a + 3b – 2a + 5b can be simplified by grouping the a terms and the b terms separately:

例如，表达式 4a + 3b – 2a + 5b 可以通过分别合并 a 项和 b 项来化简：

4a + 3b – 2a + 5b = (4a – 2a) + (3b + 5b) = 2a + 8b

Always check that you have combined every pair of like terms and that no sign has been lost. A common error is to change a negative sign when rearranging terms, so keep each term’s sign attached to it.

务必检查是否已合并每一对同类项，并且没有丢失符号。一个常见错误是在重排各项时改变了负号，因此要确保每个项的符号始终跟随该项。

4. Expanding Brackets | 去括号

Expanding brackets means removing them by using the distributive law. For a single bracket, multiply each term inside the bracket by the term outside: a(b + c) = ab + ac. This is the same rule whether the outside term is positive or negative.

去括号是指使用分配律去掉括号。对于单项式括号，将括号内的每一项都乘以括号外的项：a(b + c) = ab + ac。无论括号外的项是正还是负，这一规则都相同。

For example, expand -2(3x – 4):

例如，展开 -2(3x – 4)：

-2(3x – 4) = -2 × 3x + (-2) × (-4) = -6x + 8

For double brackets, use the FOIL method or multiply each term in the first bracket by each term in the second bracket. The expansion of (x + a)(x + b) is x² + (a + b)x + ab.

对于双项式括号，可以使用 FOIL 法，或将第一个括号中的每一项与第二个括号中的每一项相乘。(x + a)(x + b) 的展开结果是 x² + (a + b)x + ab。

For example, expand (x + 3)(x – 2):

例如，展开 (x + 3)(x – 2)：

(x + 3)(x – 2) = x² – 2x + 3x – 6 = x² + x – 6

After expanding, always simplify by collecting like terms. Watch especially for the product of two negative terms, which gives a positive result.

展开后，务必通过合并同类项进行化简。尤其要注意两个负项相乘得到正项的情况。

5. Factorising Expressions | 因式分解

Factorising is the reverse of expanding. It means writing an expression as a product of its factors. The first step is to look for the highest common factor of all terms. For example, in 6x + 9, both terms share a factor of 3, so 6x + 9 = 3(2x + 3).

因式分解是展开的逆过程。它意味着将一个表达式写成其因式的乘积。第一步是找出所有项的最高公因式。例如，在 6x + 9 中，两项都有公因数 3，因此 6x + 9 = 3(2x + 3)。

For quadratic expressions of the form x² + bx + c, look for two numbers that multiply to give c and add to give b. For example, to factorise x² + 5x + 6, find two numbers that multiply to 6 and add to 5: the numbers are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3).

对于形如 x² + bx + c 的二次表达式，寻找两个数，它们的乘积为 c，和为 b。例如，要对 x² + 5x + 6 进行因式分解，找到两个数，它们的乘积为 6，和为 5：这两个数是 2 和 3，因此 x² + 5x + 6 = (x + 2)(x + 3)。

x² + 5x + 6 = (x + 2)(x + 3)

Always check your factorising by expanding the brackets. If you get back to the original expression, your factors are correct. This is a quick and reliable verification method in exams.

务必通过展开括号来检验因式分解。如果能够得到原来的表达式，就说明因式分解正确。这是在考试中快速且可靠的验证方法。

6. Solving Linear Equations | 解一元一次方程

A linear equation has variables raised only to the power of 1. To solve such an equation, use inverse operations to isolate the variable. Keep the equation balanced by performing the same operation on both sides.

一元一次方程中的变量指数仅为 1。解这类方程时，使用逆运算来隔离变量。通过在等式两边执行相同的运算来保持方程平衡。

For example, solve 3(x – 2) = 9:

例如，解方程 3(x – 2) = 9：

3(x – 2) = 9 → 3x – 6 = 9 → 3x = 15 → x = 5

When an equation contains fractions, multiply every term by the lowest common denominator to clear the fractions. For example, (x + 1)/3 = 2 can be solved by multiplying both sides by 3 to get x + 1 = 6, so x = 5.

当方程中含有分数时，将每一项都乘以最小公分母以消去分数。例如，(x + 1)/3 = 2 可以通过两边同时乘以 3 来求解，得到 x + 1 = 6，因此 x = 5。

Always substitute your solution back into the original equation to check it works. This helps catch arithmetic slips and sign errors before you move on.

始终将解代回原方程检验是否正确。这有助于在继续下一题之前发现算术错误和符号错误。

7. Solving Inequalities | 解不等式

Inequalities are solved in the same way as equations, except that multiplying or dividing both sides by a negative number reverses the inequality sign. The symbols <, >, ≤ and ≥ indicate less than, greater than, less than or equal to, and greater than or equal to.

不等式的解法与方程类似，但乘以或除以负数时需要反转不等号。符号 <、>、≤ 和 ≥ 分别表示小于、大于、小于或等于、大于或等于。

For example, solve -2x ≤ 6:

例如，解不等式 -2x ≤ 6：

-2x ≤ 6 → x ≥ -3

Inequality solutions can be shown on a number line using an open circle for < or > and a closed circle for ≤ or ≥. In this example, the closed circle at -3 and an arrow pointing to the right represent all numbers greater than or equal to -3.

不等式的解可以在数轴上表示：< 或 > 用空心圆圈，≤ 或 ≥ 用实心圆圈。在这个例子中，-3 处用实心圆圈，箭头指向右侧，表示所有大于或等于 -3 的数。

When solving double inequalities such as -3 < 2x + 1 ≤ 5, apply the same operation to all three parts, then isolate x in the middle.

解双不等式如 -3 < 2x + 1 ≤ 5 时，要对三个部分同时进行相同的运算，然后将中间部分中的 x 分离出来。

8. Working with Formulae | 运用公式

A formula is an equation that shows the relationship between different quantities. To use a formula, substitute known values for the variables and then evaluate the result. For example, the formula F = ma gives force when mass and acceleration are known: if m = 2 kg and a = 3 m/s², then F = 2 × 3 = 6 N.

公式是表示不同量之间关系的方程。使用公式时，将已知值代入变量，然后求值。例如，公式 F = ma 用于在质量和加速度已知时计算力：如果 m = 2 kg 且 a = 3 m/s²，那么 F = 2 × 3 = 6 N。

Rearranging a formula means changing its subject. For example, to make x the subject of y = mx + c, subtract c from both sides and divide by m:

变形公式意味着改变其主项。例如，要让 x 成为 y = mx + c 的主项，应先两边减去 c，再除以 m：

y = mx + c → y – c = mx → x = (y – c)/m

Practise rearranging common formulae such as the area of a rectangle A = lw, the circumference of a circle C = 2πr, and the speed formula v = d/t. Each rearrangement follows the same inverse-operation logic as solving equations.

练习变形常见公式，如矩形面积 A = lw、圆周长 C = 2πr 以及速度公式 v = d/t。每一种变形都遵循与解方程相同的逆运算逻辑。

9. Algebraic Fractions | 代数分式

Algebraic fractions can be simplified by factorising the numerator and denominator and cancelling common factors. For example, (x² – 1)/(x – 1) can be simplified to x + 1, but only when x ≠ 1, because the original fraction is undefined at x = 1.

代数分式可以通过对分子和分母进行因式分解并约去公因式来化简。例如，(x² – 1)/(x – 1) 可以化简为 x + 1，但前提是 x ≠ 1，因为原分式在 x = 1 处无意义。

To add or subtract algebraic fractions, first find a common denominator. For example, 1/x + 1/y = (y + x)/(xy). To multiply, multiply the numerators and denominators directly; to divide, multiply by the reciprocal of the second fraction.

代数分式的加减运算首先需要找到公分母。例如，1/x + 1/y = (y + x)/(xy)。乘法运算是直接将分子相乘、分母相乘；除法运算是乘以第二个分式的倒数。

When solving equations with algebraic fractions, multiply through by the common denominator to produce a simpler equation without fractions. This method is especially useful for checking solutions in exam conditions.

解含有代数分式的方程时，将方程两边同时乘以公分母，转化为不含分式的简单方程。这种方法在考试中特别有助于检验解。

10. Word Problems | 应用题

To solve word problems, translate the problem into algebra by defining a variable and writing an equation that represents the given information. For example, “three more than twice a number is 11” can be written as 2n + 3 = 11, where n is the unknown number. Solving gives n = 4.

解应用题时，先定义一个变量，写出表示已知信息的方程，将文字问题转化为代数问题。例如，“一个数的两倍再加 3 等于 11”可以写成 2n + 3 = 11，其中 n 是未知数。解方程得到 n = 4。

Geometry problems often use perimeter and area formulae. For instance, if the length of a rectangle is 3 cm more than its width, and the perimeter is 22 cm, let the width be w. Then the length is w + 3, and 2(w + w + 3) = 22, so 2(2w + 3) = 22, giving w = 4 cm and length = 7 cm.

几何问题常常用到周长和面积公式。例如，如果矩形的长比宽多 3 cm，周长为 22 cm，设宽为 w。那么长为 w + 3，且 2(w + w + 3) = 22，即 2(2w + 3) = 22，解得 w = 4 cm，长为 7 cm。

Always define your variable clearly and answer the exact question asked, including units where appropriate. A correct equation with no final statement is often marked down in exams.

务必清晰地定义变量，并准确回答题目所问，必要时包括单位。一个方程正确但没有最终结论的答案在考试中通常会被扣分。

11. Common Mistakes and Exam Tips | 常见错误与考试提示

Many marks are lost through small, avoidable errors. Being aware of these common mistakes can help you avoid them under exam pressure.

许多失分都源于一些小的、可以避免的错误。了解这些常见错误有助于你在

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Unit 6: Algebraic Expressions, Equations and Inequalities | 第6单元：代数式、方程与不等式