📚 GCSE Maths: Inequalities Revision Guide | GCSE 数学：不等式 考点精讲
Inequalities are a crucial part of GCSE Mathematics. They allow you to compare values and describe ranges of numbers that satisfy certain conditions. This guide walks you through every key topic — from basic inequality symbols to solving quadratic inequalities — with clear explanations and paired bilingual notes.
不等式是 GCSE 数学的重要考点。它用于比较数值、描述满足特定条件的数的范围。本文全面梳理不等式的基础符号、数轴表示、解法到二次不等式，配以中英双语精讲，助你轻松掌握。
1. Inequality Symbols | 不等式符号
ater than), ≤ (less than or equal to), ≥ (greater than or equal to), and ≠ (not equal to).
你必须熟悉 GCSE 考试中五种主要的不等式符号：<（小于）、>（大于）、≤（小于等于）、≥（大于等于）以及 ≠（不等于）。
| Symbol | Meaning | Example |
|---|---|---|
| < | less than 小于 | x < 5 |
| > | greater than 大于 | y > −2 |
| ≤ | less than or equal to 小于等于 | n ≤ 10 |
| ≥ | greater than or equal to 大于等于 | t ≥ 0 |
| ≠ | not equal to 不等于 | x ≠ 3 |
on a number line corresponds to < or >, while a closed circle corresponds to ≤ or ≥.
务必细心读取符号：在数轴上，空心圆对应 < 或 >，实心圆对应 ≤ 或 ≥。
2. Number Line Representation | 数轴表示
When you show an inequality on a number line, use an open circle (○) for strict inequality (< or >) and a closed circle (●) for weak inequality (≤ or ≥). Then shade the region that satisfies the inequality.
在数轴上表示不等式时，严格不等（< 或 >）用空心圆，弱不等（≤ 或 ≥）用实心圆，并将满足条件的区域涂上阴影。
−3 ≤ x < 2 means a closed circle at −3, an open circle at 2, and shading between them.
例如，x > −1 表示在 −1 处画空心圆，向右涂阴影；−3 ≤ x < 2 则表示 −3 处实心圆、2 处空心圆，并在两者之间涂阴影。
f the arrow. If the inequality is x ≤ 4, shade to the left; for x ≥ 4, shade to the right.
要确保阴影方向与不等号一致。若不等式为 x ≤ 4，向左涂阴影；若为 x ≥ 4，则向右涂阴影。
3. Solving Linear Inequalities | 解线性不等式
Solving a linear inequality is very similar to solving a linear equation. You can add, subtract, multiply or divide both sides, but remember one vital extra rule: if you multiply or divide by a negative number, you must reverse the inequality sign.
解线性不等式与解线性方程基本相似，可对两边进行加、减、乘、除运算，但有一条至关重要的额外规则：若乘以或除以负数，必须反转不等号方向。
Example: 2x + 3 > 7 ⇒ 2x > 4 ⇒ x > 2
Work step by step, isolating the variable on one side just as you would in an equation. The inequality sign stays the same unless you multiply or divide by a negative.
按步骤将变量项单独移到一边，就像解方程一样。不等号保持不变，除非你乘以或除以了负数。
Example with a negative: −3x ≤ 12 ⇒ x ≥ −4
becomes ≥. Forgetting to flip the sign is the most common mistake in inequality questions.
此例中我们两边同除以 −3，因此 ≤ 变为 ≥。忘记翻转不等号是不等式题目中最常见的错误。
4. Multiplying or Dividing by a Negative Number | 乘以或除以负数
The golden rule of inequalities: whenever you multiply or divide both sides by a negative number, reverse the direction of the inequality. This also applies when you multiply by −1 to change signs of all terms.
不等式的黄金法则：无论何时对两边同乘或同除一个负数，必须反转不等号方向。将整个式子乘以 −1 改变符号时同样适用。
Consider −x > 5. Multiply both sides by −1 to get x < −5. If you do not flip, you will obtain the wrong solution set and lose marks.
考虑 −x > 5，两边乘以 −1 得到 x < −5。若不翻转不等号，就会得到错误的解集，丢掉分数。
This rule does not apply when you add or subtract a negative number; the sign remains unchanged. Being disciplined about checking will save you from careless errors.
该规则只对乘除负数有效，加减负数无需改变不等号。养成检查的习惯，能避免许多无谓的粗心失分。
5. Compound (Double) Inequalities | 复合（双联）不等式
lve it, perform the same operation on all three parts simultaneously, keeping the variable in the middle.
复合不等式将两个不等式合并为一个，通常形如 a < bx + c ≤ d。解题时需对三个部分同时进行相同运算，始终把变量项留在中间。
Solve: −3 < 2x + 1 ≤ 5
solution can be shown on a number line with an open circle at −2 and a closed circle at 2.
三部分同时减去 1：−4 < 2x ≤ 4，然后同除以 2：−2 < x ≤ 2。解集可在数轴上用 −2 处的空心圆和 2 处的实心圆表示。
earrange or solve as two separate inequalities. The key is to maintain balance across all parts.
若变量出现在边缘，如 −5 ≤ 3 − 2x < 1，可重新整理或拆成两个不等式分别求解。关键是始终保持三部分平衡。
6. Integer Solutions from Inequalities | 不等式的整数解
Many GCSE questions ask: "List the integer values that satisfy the inequality." After solving, identify the range and write down every whole number that fits.
很多 GCSE 题目会问：“列出满足该不等式的所有整数值。”解题后，先定出范围，再写出该范围内每一个符合条件的整数。
ons are −2, −1, 0, 1, 2. Note that x = 3 is not included because of the strict inequality.
例如 −2 ≤ x < 3 的整数解是 −2, −1, 0, 1, 2。注意由于不等式不包含等号，x = 3 不属于解集。
e boundary integer is included; if it is < or >, it is not. Use a quick mental number line to avoid omissions.
一定要复查边界值。带等号则边界整数包含在内，仅为 < 或 > 则不包含。脑中快速画出迷你数轴可有效避免遗漏。
7. Quadratic Inequalities (Higher Tier) | 二次不等式（高阶）
For higher tier students, quadratic inequalities such as x² − 4x + 3 < 0 often appear. The method involves factorising the quadratic, finding the roots, then determining where the quadratic is negative or positive.
对于高阶考生，形如 x² − 4x + 3 < 0 的二次不等式很常见。解法通常为因式分解、求出根，然后判断二次式在哪些区间为负或为正。
First, solve the corresponding equation: x² − 4x + 3 = 0 ⇒ (x − 1)(x − 3) = 0 ⇒ x = 1 or x = 3. These roots split the number line into three regions.
首先解对应方程：x² − 4x + 3 = 0 ⇒ (x − 1)(x − 3) = 0 ⇒ x = 1 或 x = 3。这两个根把数轴分成三个区间。
Test a value from each region in the original inequality. The solution to x² − 4x + 3 < 0 is 1 < x < 3, where the quadratic is below the x‑axis.
在每个区间取一个测试值代入原不等式，可知 x² − 4x + 3 < 0 的解为 1 < x < 3，即二次函数图像位于 x 轴下方的部分。
8. Solving Quadratic Inequalities by Sketching Graphs | 通过草图解二次不等式
solve any quadratic inequality is to sketch the graph of y = ax² + bx + c. Identify the x‑epts (roots) and note whether the parabola opens upward (a > 0) or downward (a < 0).
解二次不等式万无一失的方法是画出 y = ax² + bx + c 的草图。标出与 x 轴的交点（根），并观察抛物线开口向上（a > 0）还是向下（a < 0）。
≥ 0, the roots are x = 2 and x = 3. The parabola opens upward, so the graph is above the x‑axis when x ≤ 2 or x ≥ 3. Thus the solution is x ≤ 2 or x ≥ 3.
以 x² − 5x + 6 ≥ 0 为例，根为 x = 2 和 x = 3。开口向上，故当 x ≤ 2 或 x ≥ 3 时图像在 x 轴上方，解集为 x ≤ 2 或 x ≥ 3。
If the quadratic has no real roots (discriminant < 0), then it is either always positive or always negative. The inequality ax² + bx + c > 0 will either be true for all real x or have no solution, depending on the sign of a.
如果二次式无实数根（判别式 < 0），则它要么恒正、要么恒负。如 ax² + bx + c > 0，视 a 的正负，解集可能是全体实数，也可能无解。
9. Real‑World Inequality Problems | 不等式应用题
Word problems often require you to form an inequality from a description, then solve and interpret the answer. Typical contexts include temperature ranges, weights, costs, and geometric constraints.
应用题通常要求你根据文字描述建立不等式，然后求解并解释答案。常见的背景有温度范围、重量限制、费用以及几何约束等。
Example: " h length (2x + 1) cm and width (x − 3) cm is at most 50 cm. Find the possible values of x." Set up: 2[(2x + 1) + (x − 3)] ≤ 50, then solve.
例如："一个矩形的长为 (2x + 1) 厘米，宽为 (x − 3) 厘米，其周长至多为 50 厘米。求 x 的可能取值范围。" 建立不等式 2[(2x + 1) + (x − 3)] ≤ 50，然后求解。
Always state your final answer in the context of the problem. For geometry, side lengths must be positive, so x > 3 is an extra constraint that must be combined with the inequality solution.
最后一定要根据题意给出答案。几何题中边长必须为正，因此 x > 3 作为额外约束，必须与不等式解集相结合。
10. Common Pitfalls and How to Check | 常见错误与检查方法
Mistake 1: Forgetting to reverse the inequality when multiplying or dividing by a negative. Always pause before the final line and ask, "Did I multiply or divide by a negative?"
常见错误一：乘除负数时忘记反转不等号。每次写出最后一行解答前，请暂停自问：“我是不是乘或除了一个负数？”
s (< or >) need open circles; weak inequalities (≤ or ≥) need closed circles. Double‑check your diagram.
常见错误二：数轴表示中混淆实心圆与空心圆。严格不等用空心，弱不等用实心，完成后务必再次检查图示。
Mistake 3: In quadratic inequalities, assuming the solution is simply between the roots for all cases. Always test a value or sketch the graph to confirm the correct region.
常见错误三：解二次不等式时，直接假设答案必为“两根之间”。请一定要取点验证或画草图，以确定正确的区间。
Quick checking tip: substitute a number from your final solution back into the original inequality. If it works, you are likely correct. Also check a value just outside your solution to ensure it does not satisfy the inequality.
快速检验小技巧：将解集中的某一个数代回原不等式，若成立说明你很可能正确；再取一个恰在解集外的数代入，应不满足原不等式。
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