📚 IGCSE OCR Maths: Inequalities Key Points Explained | IGCSE OCR 数学：不等式 考点精讲
Inequalities are used to compare two expressions when they are not equal. In the OCR IGCSE Mathematics syllabus, you must be able to solve linear and quadratic inequalities, represent solutions on a number line, interpret compound inequalities, and apply inequalities to graphical regions and word problems. Mastering these skills requires careful attention to sign rules and logical reasoning.
不等式用于比较两个不相等的表达式。在 OCR IGCSE 数学大纲中，你必须能够解一元一次和二次不等式，在数轴上表示解，理解复合不等式，并将不等式应用到图形区域和文字题中。掌握这些技能需要仔细关注符号规则和逻辑推理。
1. What Is an Inequality? | 什么是不等式？
An inequality states that one quantity is greater than, less than, greater than or equal to, or less than or equal to another. The notation uses the symbols > (greater than), < (less than), ≥ (greater than or equal to) and ≤ (less than or equal to). For example, x > 4 means “x is greater than 4’
不等式表示一个量大于、小于、大于等于或小于等于另一个量。符号使用 >（大于）、<（小于）、≥（大于等于）和 ≤（小于等于）。例如，x > 4 表示“x 大于 4”。
2. Representing Inequalities on a Number Line | 在数轴上表示不等式
On a number line, an open circle indicates that the endpoint is not included (> or <), while a closed circle means the endpoint is included (≥ or ≤). The line is shaded in the direction of all possible values. An inequality such as 1 < x ≤ 5 is drawn with an open circle at 1 and a closed circle at 5, with the line shaded between them.
在数轴上，空心圆表示端点不包含在内（> 或 <），实心圆表示包含端点（≥ 或 ≤）。数轴沿所有可能值的方向涂上阴影。不等式如 1 < x ≤ 5 在 1 处画空心圆，在 5 处画实心圆，中间部分涂阴影。
3. Solving Linear Inequalities | 解一元一次不等式
Linear inequalities are solved in almost the same way as linear equations: you can add, subtract, multiply or divide both sides by the same positive number without changing the direction of the inequality sign. For example, to solve 3x – 5 < 10, add 5 to both sides to get 3x < 15, then divide by 3 to obtain x < 5.
一元一次不等式的解法与一元一次方程几乎相同：你可以两边加上、减去、乘以或除以同一个正数，而不改变不等号的方向。例如，解 3x – 5 < 10，两边加 5 得 3x < 15，然后除以 3 得 x < 5。
- Always keep the inequality sign pointing the same way when using positive operations. | 使用正数运算时，始终保持不等号方向不变。
- Check your solution by substituting a value from the solution set. | 从解集中取一个值代入验证解是否正确。
4. Reversing the Inequality Sign | 不等式符号反转
When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is the crucial difference from solving equations. For instance, -2x ≤ 8 becomes x ≥ -4 after dividing by -2 and flipping the sign.
当不等式两边乘以或除以一个负数时，必须反转不等号的方向。这是与解方程的关键区别。例如，-2x ≤ 8 两边除以 -2 并反转符号后变为 x ≥ -4。
If a < b, then -a > -b (multiplying by -1 reverses the sign).
如果 a < b，那么 -a > -b（乘以 -1 反转符号）。
5. Compound Inequalities | 复合不等式
A compound inequality combines two inequalities into one statement, such as -3 ≤ 2x + 1 < 5. To solve it, apply the same operation to all three parts simultaneously. Subtract 1 from all parts: -4 ≤ 2x < 4, then divide by 2: -2 ≤ x < 2. Always keep the variable in the middle.
复合不等式将两个不等式组合成一个陈述，例如 -3 ≤ 2x + 1 < 5。求解时，对三部分同时进行相同运算。所有部分减 1：-4 ≤ 2x < 4，然后除以 2：-2 ≤ x < 2。始终将变量保持在中间。
Another type uses ‘or’: x < 1 or x > 4. Solutions are separate intervals. | 另一种使用“或”：x < 1 或 x > 4。解是分开的区间。
6. Solving Quadratic Inequalities | 解二次不等式
To solve a quadratic inequality like x² – 4x + 3 > 0, first find the roots of the corresponding equation x² – 4x + 3 = 0, which are x = 1 and x = 3. Then sketch the parabola y = x² – 4x + 3 (a positive U-shape). The inequality is > 0 where the graph is above the x-axis, so x < 1 or x > 3.
要解二次不等式如 x² – 4x + 3 > 0，首先求对应方程 x² – 4x + 3 = 0 的根，得 x = 1 和 x = 3。然后画出抛物线 y = x² – 4x + 3（开口向上的 U 形）。不等式 > 0 在图像位于 x 轴上方时成立，因此 x < 1 或 x > 3。
- For ax² + bx + c < 0 with a > 0, the solution lies between the roots. | 若 a > 0，解 ax² + bx + c < 0，则解在两根之间。
- Always test a value between the roots to confirm the sign. | 始终在两根之间取一个值测试符号以确认。
7. Graphical Interpretation of Quadratic Inequalities | 二次不等式的图形解释
Drawing a rough graph is the safest method. Mark the roots on the x-axis, note the shape (U-shaped for a > 0, n-shaped for a < 0), and then shade the region satisfying the inequality. If the inequality includes equality (≥ or ≤), the roots are included in the solution set.
画一个粗略图形是最安全的方法。在 x 轴上标出根，注意形状（a > 0 时 U 形，a < 0 时 ∩ 形），然后涂上满足不等式的区域。若不等式含等号（≥ 或 ≤），则根包含在解集中。
Example: -x² + 2x + 3 ≤ 0 → x ≤ -1 or x ≥ 3 (n-shaped, below axis).
例：-x² + 2x + 3 ≤ 0 → x ≤ -1 或 x ≥ 3（∩ 形，轴下方）。
8. Inequalities and Regions on Graphs | 不等式与图形区域
Inequalities can describe regions in the coordinate plane. For a straight line, y > 2x + 1 means the region above the line; y ≤ -x + 3 means the region on or below the line. Use a dashed line for strict inequalities (> or <) and a solid line for inclusive inequalities (≥ or ≤).
不等式可以描述坐标平面上的区域。对于一条直线，y > 2x + 1 表示直线上方的区域；y ≤ -x + 3 表示直线及其下方的区域。严格不等式（> 或 <）用虚线，包含等号的不等式（≥ 或 ≤）用实线。
Shade the unwanted region or the wanted region as instructed in the exam. Often, multiple inequalities form a bounded area.
根据题目要求涂上不需要的区域或需要的区域。通常，多个不等式围成一个有界区域。
9. Systems of Linear Inequalities | 线性不等式组
To solve a system like x + y ≤ 6, x ≥ 1, y > 2, graph each inequality on the same axes. Identify the region where all inequalities overlap. This is the feasible region, and its vertices can be found by solving the boundary equations simultaneously.
要解不等式组如 x + y ≤ 6, x ≥ 1, y > 2，在同一坐标系上画出每个不等式。找出所有不等式重叠的区域。这就是可行区域，其顶点可通过联立边界方程求得。
| Inequality | Line style and shading |
|---|---|
| x ≥ 1 | Solid vertical line, shade right |
| y > 2 | Dashed horizontal line, shade above |
| x + y ≤ 6 | Solid line, shade below |
中文：不等式 x ≥ 1：实线垂直线，右侧涂阴影；y > 2：虚线水平线，上方涂阴影；x + y ≤ 6：实线，下方涂阴影。
10. Word Problems Involving Inequalities | 涉及不等式的文字题
Inequality word problems require translating a written condition into mathematical symbols. Phrases like ‘at least’ become ≥, ‘at most’ become ≤, ‘more than’ becomes >, ‘fewer than’ becomes <. Define a variable, set up the inequality, solve it, and interpret the result in context.
不等式文字题需要将文字条件转化为数学符号。“至少”变为 ≥，“至多”变为 ≤，“多于”变为 >，“少于”变为 <。定义变量，建立不等式，求解，并结合语境解释结果。
Example: A student needs at least 75 marks to pass. If she has already scored 48, and the remaining test is out of 40, write an inequality and find the minimum marks needed. (48 + x ≥ 75, x ≥ 27, so she needs at least 27 out of 40.)
例如：一个学生需要至少 75 分才能通过。如果她已经得 48 分，剩下的测试满分 40 分，写出不等式并求所需最低分。（48 + x ≥ 75，x ≥ 27，所以她至少需要 27 分（满分 40 分）。）
11. Common Mistakes and Tips | 常见错误与技巧
Many students forget to reverse the inequality sign when multiplying or dividing by a negative. Always double-check this step. In compound inequalities, apply operations to all three parts, not just the middle one. For quadratic inequalities, do not just solve the equation and assume the answer is between the roots; the shape of the graph dictates the solution intervals.
许多学生在乘以或除以负数时忘记反转不等号。务必反复检查这一步。在复合不等式中，要对三部分同时运算，而非只对中间运算。对于二次不等式，不要仅仅解方程并假设答案在两根之间；图形的形状决定了解区间。
- Always test a value from your final solution in the original inequality. | 始终从最终解中取一个值代入原不等式检验。
- When shading regions, read exam instructions carefully: some ask to shade the region that satisfies the inequalities, others ask to shade the unwanted region. | 涂区域时，仔细阅读考试说明：有的要求涂上满足不等式的区域，有的要求涂上不需要的区域。
12. Summary of Key Inequality Skills | 不等式关键技能总结
To excel in OCR IGCSE inequalities, you must be fluent in manipulating inequality symbols, solving linear and quadratic forms, interpreting compound statements, and applying inequalities to coordinate geometry. Practice with both pure algebra problems and real-life contexts to build confidence and accuracy.
要在 OCR IGCSE 不等式部分取得优异成绩，你必须熟练操控不等号，求解一次和二次不等式，理解复合语句，并将不等式应用于坐标几何。通过练习纯代数问题和实际应用情境，建立信心与准确性。
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