📚 Unit 5 Higher Homework: Equations, Inequalities and Sequences | GCSE 数学高阶第5单元：方程、不等式与数列
This unit covers the core algebra skills from pages 100-123 of the Higher textbook, focusing on equations, inequalities and sequences. You will learn how to solve linear and quadratic equations, manipulate inequalities, and find the nth term of different sequences. These topics appear regularly on both calculator and non-calculator papers.
本单元涵盖教材第100-123页的高阶核心代数技能，重点包括方程、不等式与数列。你将学习如何解一元一次方程与二次方程、处理不等式，以及求不同类型数列的第n项公式。这些知识点在计算器与非计算器试卷中都频繁出现。
1. Solving Linear Equations | 解一元一次方程
A linear equation has the variable raised only to the power 1, for example 3x + 5 = 17. To solve it, use inverse operations in the correct order: undo addition or subtraction first, then multiplication or division.
一元一次方程中变量的指数仅为1，例如 3x + 5 = 17。解此类方程时，需按正确顺序使用逆运算：先移项处理加减法，再处理乘除法。
For the equation 3x + 5 = 17, subtract 5 from both sides to get 3x = 12, then divide both sides by 3 to obtain x = 4. Always check your answer by substituting it back into the original equation.
对于方程 3x + 5 = 17，两边同时减去5得到 3x = 12，然后两边同时除以3得到 x = 4。务必把答案代回原方程进行检验。
3x + 5 = 17 → 3x = 12 → x = 4
2. Equations with Brackets and Fractions | 含括号与分数的方程
When an equation contains brackets, expand them as a first step. For example, 2(x + 3) = 14 becomes 2x + 6 = 14 after expansion, then solve to get x = 4.
当方程含有括号时，第一步应先去括号展开。例如 2(x + 3) = 14 展开后为 2x + 6 = 14，然后求解得到 x = 4。
If fractions are present, multiply every term by the lowest common denominator (LCD). For (x/3) + 1 = 2, multiply each term by 3 to obtain x + 3 = 6, so x = 3.
若方程含有分数，应将每一项乘以最小公分母（LCD）。例如 (x/3) + 1 = 2，每一项乘以3得到 x + 3 = 6，因此 x = 3。
2(x + 3) = 14 → 2x + 6 = 14 → x = 4
(x/3) + 1 = 2 → x + 3 = 6 → x = 3
3. Forming Equations from Word Problems | 列方程解应用题
Word problems require you to translate English statements into algebraic equations. Identify the unknown quantity and assign it a letter, usually x. Then build an equation using the relationships given in the problem.
应用题要求你将文字语句转化为代数方程。首先确定未知量并用字母表示，通常设为 x。然后根据题目给出的关系建立方程。
Example: ‘Three times a number added to 7 gives 22.’ Let the number be x, then the equation is 3x + 7 = 22. Solving gives x = 5.
例如：“一个数的3倍加上7等于22。”设这个数为 x，则方程为 3x + 7 = 22。解方程得到 x = 5。
Always define the variable clearly and write the equation before solving. This shows method marks even if the final answer is wrong.
务必清晰定义变量并先列出方程再求解。即使最终答案错误，正确的列式过程也能获得方法分。
4. Solving Quadratic Equations | 解二次方程
A quadratic equation has the general form ax² + bx + c = 0, where a ≠ 0. The three main methods are factorising, completing the square, and using the quadratic formula.
二次方程的一般形式为 ax² + bx + c = 0，其中 a ≠ 0。三种主要解法为因式分解法、配方法与求根公式法。
If the quadratic can be factorised, set each factor equal to zero. For x² + 5x + 6 = 0, factorise to (x + 2)(x + 3) = 0, giving x = -2 or x = -3.
若二次式可以因式分解，则令每个因式等于零。例如 x² + 5x + 6 = 0，因式分解为 (x + 2)(x + 3) = 0，得到 x = -2 或 x = -3。
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2, x = -3
The quadratic formula works for any quadratic equation: x = (-b ± √(b² – 4ac)) / (2a). Use it when factorising is difficult or impossible.
求根公式适用于任何二次方程：x = (-b ± √(b² – 4ac)) / (2a)。当因式分解困难或无法进行时使用该公式。
x = (-b ± √(b² – 4ac)) / (2a)
5. Inequalities on a Number Line | 数轴上的不等式
Inequalities compare two expressions using the symbols < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). They can be represented on a number line using open or closed circles.
不等式使用符号 <（小于）、>（大于）、≤（小于等于）和 ≥（大于等于）来比较两个表达式。它们可以在数轴上用空心圈或实心圈表示。
An open circle is used for strict inequalities < or >, while a closed circle is used for ≤ or ≥. The arrow shows the direction of all possible values.
严格不等式 < 或 > 使用空心圈，而 ≤ 或 ≥ 使用实心圈。箭头方向表示所有可能取值的范围。
For example, x > 2 means all numbers greater than 2, with an open circle at 2 and an arrow pointing to the right. For x ≤ -1, use a closed circle at -1 and an arrow pointing left.
例如，x > 2 表示所有大于2的数，在2处画空心圈并向右箭头。对于 x ≤ -1，在 -1 处画实心圈并向左箭头。
6. Solving Linear Inequalities | 解线性不等式
Solving a linear inequality is similar to solving an equation, but with one important difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
解一元一次不等式与解方程类似，但有一个重要区别：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。
For 3x – 4 < 11, add 4 to both sides to get 3x < 15, then divide by 3 to obtain x < 5. No reversal is needed because division is by a positive number.
对于 3x – 4 < 11，两边同时加4得到 3x < 15，然后除以3得到 x < 5。因为除的是正数，所以不需要改变不等号方向。
For -2x ≤ 6, divide both sides by -2 and reverse the sign to get x ≥ -3. Always remember this reversal rule when the coefficient of x is negative.
对于 -2x ≤ 6，两边同时除以 -2 并改变符号方向得到 x ≥ -3。当 x 的系数为负数时，务必记住这一变号规则。
-2x ≤ 6 → x ≥ -3
7. Quadratic Inequalities | 二次不等式
A quadratic inequality involves an expression of the form ax² + bx + c and uses inequality symbols. The solution is often a range of x values, found by sketching a parabola or using sign tests.
二次不等式包含 ax² + bx + c 形式的不等式符号。解通常是一个 x 的取值范围，可通过绘制抛物线草图或使用符号测试法求得。
To solve x² – 4x – 5 < 0, first solve the related equation x² – 4x – 5 = 0, which factorises to (x – 5)(x + 1) = 0, giving x = 5 or x = -1. The curve is a positive parabola, so the expression is negative between the roots.
解 x² – 4x – 5 < 0 时，先解相关方程 x² – 4x – 5 = 0，因式分解为 (x – 5)(x + 1) = 0，得到 x = 5 或 x = -1。该曲线为开口向上的抛物线，因此表达式在两个根之间为负。
The solution is -1 < x < 5. Use a quick sketch to confirm the region where the curve lies below the x-axis.
解为 -1 < x < 5。快速绘制草图以确认曲线位于 x 轴下方的区域。
x² – 4x – 5 < 0 → -1 < x < 5
8. Sequences and the nth Term | 数列与第n项公式
A sequence is an ordered list of numbers that follows a rule. The nth term is a formula that allows you to find any term in the sequence without listing all previous terms.
数列是按照某种规则排列的一列数。第n项公式可以让你直接求出数列中任意一项，而无需列出前面所有项。
For an arithmetic sequence, the nth term is given by a + (n – 1)d, where a is the first term and d is the common difference. For example, in the sequence 4, 7, 10, 13, … a = 4 and d = 3, so the nth term is 3n + 1.
对于等差数列，第n项公式为 a + (n – 1)d，其中 a 为首项，d 为公差。例如数列 4, 7, 10, 13, … 中 a = 4，d = 3，因此第n项为 3n + 1。
nth term = a + (n – 1)d
For a quadratic sequence, the second difference is constant. The nth term has the form an² + bn + c, and you can find a, b, c by using the first few terms.
对于二次数列，二阶差分为常数。第n项公式形式为 an² + bn + c，可利用前几项求出 a、b、c 的值。
9. Special Sequences | 特殊数列
Certain sequences appear frequently in Higher exams: square numbers 1, 4, 9, 16, 25, … with nth term n², and cube numbers 1, 8, 27, 64, … with nth term n³.
高阶考试中经常出现某些特殊数列：平方数 1, 4, 9, 16, 25, … 的第n项为 n²，立方数 1, 8, 27, 64, … 的第n项为 n³。
Triangular numbers 1, 3, 6, 10, 15, … have nth term n(n + 1)/2. Fibonacci-type sequences are also tested, where each term is the sum of the previous two.
三角形数 1, 3, 6, 10, 15, … 的第n项为 n(n + 1)/2。斐波那契型数列也常被考查，其中每一项为前两项之和。
Triangular numbers: Tₙ = n(n + 1)/2
10. Exam Tips for Unit 5 | 单元5考试技巧
In the exam, always show clear method steps when solving equations or inequalities. Even if you make an arithmetic error, you can still earn marks for a correct method.
考试中解方程或不等式时，务必展示清晰的解题步骤。即使出现计算错误，正确的方法仍可获得步骤分。
Check your answers when time allows, especially for quadratic equations where two solutions are expected. Substitute each solution back into the original equation.
时间允许时务必检查答案，尤其是二次方程通常有两个解。将每个解代回原方程进行验证。
For inequality questions, pay close attention to whether the circle should be open or closed on the number line. Reversing the sign when dividing by a negative is a common mistake – highlight this in your working.
对于不等式题，要特别注意数轴上应使用空心圈还是实心圈。除以负数时改变不等号方向是常见错误，解题时请重点标注。
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