📚 Inequalities: Key Concepts and Exam Tips for IB & CIE | 不等式：IB CIE 考点精讲
Inequalities constitute a fundamental part of the IB and CIE Mathematics syllabi, appearing in both pure and applied contexts. Students must not only solve linear and quadratic inequalities but also handle rational, absolute value, and parametric forms with confidence. This guide summarises the essential principles, common pitfalls, and exam-oriented strategies to help you tackle inequality questions efficiently.
不等式是 IB 和 CIE 数学课程的基础内容，既出现在纯数，也出现在应用情境中。学生不仅要会解一元一次、一元二次不等式，还要能熟练处理有理不等式、绝对值不等式和含参不等式。本文梳理了核心原理、常见错误和应试策略，助你高效攻克不等式考题。
1. Fundamental Properties & Notation | 基本性质与符号
Inequalities maintain their direction when adding or subtracting the same quantity, or multiplying/dividing by a positive number. For example, if a > b, then a + c > b + c, and if c > 0, ac > bc. However, multiplying or dividing by a negative number reverses the inequality sign: if c < 0, then ac < bc. The transitive property holds: if a > b and b > c, then a > c.
不等式两边同时加、减同一个数，不等号方向不变；同乘、除以一个正数，不等号方向也不变。例如，若 a > b，则 a + c > b + c；若 c > 0，则 ac > bc。但乘、除以负数时，必须反转不等号：若 c < 0，则 ac < bc。传递性也成立：若 a > b 且 b > c，则 a > c。
nd ≠ (not equal to). On a number line, an open circle indicates strict inequality, while a closed dot indicates inclusion.
常用符号：< (小于), > (大于), ≤ (小于等于), ≥ (大于等于), ≠ (不等于)。在数轴上，空心圈表示严格不等，实心点表示包含端点。
2. Solving Linear Inequalities | 解线性不等式
To solve a linear inequality, isolate the variable on one side using inverse operations, just as in equations. The only extra caution: if you multiply or divide by a negative number in the final step, reverse the inequality sign. Express the solution using an inequality or interval notation.
解线性不等式与解方程类似，通过逆运算将变量移到一边孤立。唯一需要特别注意的是：若最后一步乘或除以负数，要反转不等号。解集可用不等式表示，也可用区间表示。
Example: Solve 3 – 2x ≥ 7. Subtract 3: –2x ≥ 4. Divide by –2 (and flip sign): x ≤ –2. The solution set is (–∞, –2].
示例：解 3 – 2x ≥ 7。移项：–2x ≥ 4。除以 –2 并反转：x ≤ –2。解集为 (–∞, –2]。
Always check by substituting a value from the solution set, e.g., x = –3 gives 3 – 2(–3) = 9 ≥ 7, valid.
务必选取解集内的值检验，例如取 x = –3，得 3 – (–6) = 9 ≥ 7，成立。
3. Compound Inequalities & Interval Notation | 复合不等式与区间表示
A compound inequality like – parate inequalities or solved as a whole. When performing an operation on all three parts, remember to reverse both inequality signs if multiplying/dividing by a negative.
诸如 –2 < 3x + 1 ≤ 10 的复合不等式，可拆成两个独立不等式求解，也可整体操作。对三部分同时进行乘除运算时，若乘除以负数，两个不等号都需反转。
Example: Solve –2 < 3x + 1 ≤ 10. Subtract 1: –3 < 3x ≤ 9. Divide by 3: –1 < x ≤ 3. So x ∈ (–1, 3].
示例：解得 –1 < x ≤ 3，即 x ∈ (–1, 3]。
Interval notation: use parentheses ( ) for strict inequalities and brackets [ ] for inclusive endpoints. Infinity always takes a parenthesis.
区间表示法：严格不等用圆括号 ( )，包含端点用方括号 [ ]；无穷大总使用圆括号。
4. Quadratic Inequalities | 二次不等式
+ c, noting whether it opens upwards (a > 0) or downwards (a < 0). The solution corresponds to the x‑intervals where the graph lies above (>0) or below (<0) the x‑axis.
解二次不等式 ax² + bx + c > 0 时，先将所有项移到一边，因式分解（或用求根公式）求出表达式的零点（临界值）。画出抛物线 y = ax² + bx + c，注意开口方向（a > 0 开口向上，a < 0 开口向下）。不等式的解集即为图像在 x 轴上方 (>0) 或下方 (<0) 的 x 区间。
Example: Solve x² – 5x + 6 < 0. Factorise: (x – 2)(x – 3) < 0. The parabola y = (x – 2)(x – 3) opens upwards and cuts the x‑ and 3. The curve is below the axis between the roots, so the solution is 2 < x < 3.
示例：解 x² – 5x + 6 < 0。因式分解得 (x – 2)(x – 3) < 0。抛物线 y = (x – 2)(x – 3) 开口向上，与 x 轴交于 2 和 3。在两交点之间图像在 x 轴下方，故解为 2 < x < 3。
If the discriminant Δ = b² – olds for all real x, while ax² + bx + c < 0 has no solution (and vice versa when a <
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